Abstract
Many applications, including some bifurcation problems of functional differential equations, lead to the problem of finding all zeros of a mapping f: U⊆X→X, where X is some (real) Banach space. In this type of nonlinear problem, we are interested in the solutions of
In most cases, it turns out that it is too much to ask to determine the zeros analytically and explicitly. Hence one looks for a more qualitative study of the zeros, such as the number, location, and multiplicity.
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Keywords
- Global Hopf Bifurcation
- Local Hopf Bifurcation Theorem
- Nonconstant Periodic Solution
- Global Continuation
- State-dependent Delay
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
6.1 Introduction
Many applications, including some bifurcation problems of functional differential equations, lead to the problem of finding all zeros of a mapping f: U ⊆ X → X, where X is some (real) Banach space. In this type of nonlinear problem, we are interested in the solutions of
In most cases, it turns out that it is too much to ask to determine the zeros analytically and explicitly. Hence one looks for a more qualitative study of the zeros, such as the number, location, and multiplicity.
To illustrate this and to motivate the topological degree, we consider the case \(f \in \mathcal{H}(\mathbb{C})\), where \(\mathcal{H}(\mathbb{C})\) denotes the set of holomorphic functions on a domain \(U \subset \mathbb{C}\). Recall that the winding number of a path γ: \([0,1] \rightarrow \mathbb{C}\) around a point \(z_{0} \in \mathbb{C}\) is defined by
It gives the number of times that z 0 is encircled, taking orientation into account (that is, encirclings in opposite directions are counted with opposite signs).
In particular, if we pick \(f \in \mathcal{H}(\mathbb{C})\), we compute (assuming 0 ∉ f(γ))
where z k denotes zeros of f, and α k their respective multiplicity. Moreover, if γ is a Jordan curve encircling a simply connected domain \(U \subset \mathbb{C}\), then \(n(\gamma,z_{k}) = 0\) if \(z_{k}\notin U\) and n(γ, z k ) = 1 if z k ∈ U. Hence n(f(γ), 0) counts the number of zeros inside U.
Let us also recall how we compute complex integrals along complicated paths using homotopy invariance (see [23, 240, 241]). In this approach, we look for a simpler path along which the integral can be computed that is homotopic to the original one. In particular, if f: \(\gamma \rightarrow \mathbb{C} \setminus \{ 0\}\) and g: \(\gamma \rightarrow \mathbb{C} \setminus \{ 0\}\) are homotopic, we have \(n(f(\gamma ),0) = n(g(\gamma ),0)\) (which is known as Rouché’s theorem). More explicitly, we need to find a mapping g for which n(g(γ), 0) can be computed and a homotopy H: \([0,1]\times \gamma \rightarrow C \setminus \{ 0\}\) such that H(0, z) = f(z) and H(1, z) = g(z) for z ∈ γ. For example, to see how many zeros of \(f(z) = \frac{1} {2}{z}^{6} + z -\frac{1} {3}\) lie inside the unit circle, we consider g(z) = z. Then H(t, z) = (1 − t)f(z) + tg(z) is the required homotopy, since | f(z) − g(z) | < | g(z) | , | z | = 1, implying H(t, z) ≠ 0 on [0, 1] ×γ. Hence f(z) has one zero inside the unit circle.
To summarize, given a (sufficiently smooth) domain U with enclosing Jordan curve ∂U, we have defined a degree \(\mathrm{deg}(f,U,z_{0}) = n(f(\partial U),z_{0}) = n(f(\partial U) - z_{0},0) \in \mathbb{Z}\) that counts the number of solutions of f(z) = z 0 inside U. The invariance of this degree with respect to certain deformations of f allow us to explicitly compute deg(f, U, z 0) even in nontrivial cases. Degree theory has been developed for various classes of mappings, not all of which are mentioned in the chapter. For relevant results on topological degree, see, for example, [24, 25, 177–182, 191–195]. Moreover, similar ideas also appears in the definitions of Fuller index. See, for example, Chow and Mallet-Paret [69].
6.2 The Brouwer Degree
In 1912, Brouwer [47] introduced the so-called Brouwer degree in \({\mathbb{R}}^{n}\). See Brouwder [46], Alexander et al. [8–10], Chow et al. [71], Krasnosel’skii [191], Sieberg [265] for historical developments. In this section, we introduce Brouwer degree theory. Throughout this section, U will be a bounded open subset of \({\mathbb{R}}^{n}\). For \(f \in {C}^{1}(U, {\mathbb{R}}^{n})\), the Jacobi matrix of f at x ∈ U is \(f^{\prime}(x) = (\frac{\partial f_{j}(x)} {\partial x_{i}} )_{1\leq i,j\leq n}\), and the Jacobi determinant of f at x ∈ U is
The set of regular values is
Its complement \(\mathrm{CV}(f) = {\mathbb{R}}^{n} \setminus \mathrm{ RV}(f)\) is called the set of critical values. Set \({C}^{r}(\bar{U}, {\mathbb{R}}^{n}) =\{ f \in {C}^{r}(U, {\mathbb{R}}^{n}) :\, {d}^{j}f \in C(\bar{U}, {\mathbb{R}}^{n})\ {\rm for\ all}\ 0 \leq j \leq r\}\) and
for \(y \in {\mathbb{R}}^{n}\).
Lemma 6.1 (Sard’s lemma).
Let \(U \subset {\mathbb{R}}^{n}\) be open and \(f \in {C}^{1}(U, {\mathbb{R}}^{n})\) . Then \(\mu _{n}(f(\mathrm{CV}(f))) = 0\) , where μ n is the n-dimensional Lebesgue measure.
A function deg that assigns each \(f \in D_{y}^{0}(U, {\mathbb{R}}^{n})\), \(y \in {\mathbb{R}}^{n}\), a real number deg(f, U, y) will be called a degree if it satisfies the following conditions:
- (BD1) (translation invariance) :
-
deg(f, U, y) = deg(f − y, U, 0).
- (BD2) (normalization) :
-
\(\mathrm{deg}(\mathbb{I},U,y) = 1\) if y ∈ U, where \(\mathbb{I}\) denotes the identity operator when the space involved is clear.
- (BD3) (additivity) :
-
If U 1 and U 2 are open, disjoint subsets of U such that \(y\notin f(U \setminus (U_{1} \cup U_{2}))\), then \(\mathrm{deg}(f,U,y) =\mathrm{ deg}(f,U_{1},y) +\mathrm{ deg}(f,U_{2},y)\).
- (BD4) (homotopy invariance) :
-
If \(H : [0,1] \times \bar{ U} \rightarrow {\mathbb{R}}^{n}\) is continuous, so that y ∉ H(t, ∂U) for evert t ∈ [0, 1], and f = H(0, ⋅), g = H(1, ⋅), then deg(f, U, y) = deg(g, U, y).
To compute the degree of a nonsingular matrix, we need the following lemma.
Lemma 6.2.
Two nonsingular matrices \(M_{1},M_{2} \in \mathrm{ GL}(n)\) are homotopic in GL(n) if and only if \(\mathrm{sgn}\det M_{1} =\mathrm{ sgn}\det M_{2}\).
Using this lemma, we can prove the following theorem.
Theorem 6.1.
Suppose \(f \in D_{y}^{1}(\bar{U}, {\mathbb{R}}^{n})\) and y ∉ CV (f). Then a degree satisfying (BD1)–(BD4) satisfies
where the sum is finite.
In fact, the determinant formula (6.4) can be extended to all \(f \in D_{y}^{0}(\bar{U}, {\mathbb{R}}^{n})\), that is, deg(f, U, y) as defined in (6.4) is locally constant with respect to both y and f. In particular, we have the following result.
Theorem 6.2.
There is a unique degree deg satisfying (BD1)–(BD4) . Moreover, for each given \(f \in D_{y}^{0}(\bar{U}, {\mathbb{R}}^{n})\) , we have
where \(\tilde{f} \in D_{y}^{2}(\bar{U}, {\mathbb{R}}^{n})\) is sufficiently close to f (with respect to the sup-norm topology in \({C}^{r}(\bar{U}, {\mathbb{R}}^{n})\) ), and \(y \in \mathrm{ RV}(\tilde{f})\) , and the above calculation is independent of the choice of \(\tilde{f}\).
To extend the formula (6.4) to all \(f \in D_{y}^{0}(\bar{U}, {\mathbb{R}}^{n})\), we first note that \(\epsilon :=\min \{ \vert f(x) - y\vert :\, x \in \partial U\} > 0\), and then apply the Weierstrass theorem to obtain \(\tilde{g} \in {C}^{2}(\bar{U},{R}^{n})\), so that \(\max \{\vert f(x) -\tilde{ g}(x)\vert :\, x \in \bar{ U}\} <\epsilon /2\). We then use Sard’s theorem to find a regular value \(y_{0} \in {\mathbb{R}}^{n}\) of \(\tilde{g}\) such that | y − y 0 | < ε ∕ 2. We then define \(g :\bar{ U} \rightarrow {\mathbb{R}}^{n}\) as \(g(x) =\tilde{ g}(x) - y_{0}\). Then \(g \in {C}^{2}(\bar{U}; {\mathbb{R}}^{n}),\max \{\vert g(x) - f(x)\vert \} <\epsilon\), and 0 is a regular value of g. We can define
and we need to check that this definition is independent of the choice of such g.
6.3 The Leray–Schauder Degree
In 1934, Leray and Schauder [207] generalized Brouwer degree theory to an infinite Banach space and established the so-called Leray–Schauder degree. It turns out that the Leray–Schauder degree is a powerful tool in proving various existence results for nonlinear differential equations (see, for example, [89, 90]). The objective of this section is to extend the Brouwer degree to general Banach spaces. We first extend the Brouwer degree to general finite-dimensional spaces.
Let X be a (real) Banach space of dimension n, and let n be an isomorphism between X and \({\mathbb{R}}^{n}\). Then for \(f \in D_{y}(\bar{U},X)\), U ⊂ X open, y ∈ X, we can define
provided this definition is independent of the basis chosen. To see this, let ψ be a second isomorphism. Then \(A =\psi {\circ \phi }^{-1} \in \mathrm{ GL}(n)\). Abbreviate \({f}^{{\ast}} =\phi \circ f {\circ \phi }^{-1}\), y ∗ = ϕ(y), and pick \(\tilde{{f}}^{{\ast}}\in D_{y}^{1}(\phi (\bar{U}), {\mathbb{R}}^{n})\) in the same component of \(D_{y}(\phi (\bar{U}), {\mathbb{R}}^{n})\) as f ∗ such that \({y}^{{\ast}}\in \mathrm{ RV}({f}^{{\ast}})\). Then \(A \circ \tilde{ {f}}^{{\ast}}\circ {A}^{-1} \in D_{y}^{1}(\psi (U),{R}^{n})\) is the same component of \(D_{y}(\psi (\bar{U}), {\mathbb{R}}^{n})\) as \(A \circ {f}^{{\ast}}\circ {A}^{-1} =\psi \circ f\circ \psi\) (since A is also a homeomorphism) and
by the chain rule. Thus we have \(\mathrm{deg}(\psi \circ f {\circ \psi }^{-1},\psi (U),\psi (y)) =\mathrm{ deg}(\phi \circ f {\circ \phi }^{-1},\phi (U),\phi (y))\), and our definition is independent of the basis chosen. In addition, it inherits all properties from the mapping degree in \({\mathbb{R}}^{n}\). Note also that the reduction property holds if \({\mathbb{R}}^{m}\) is replaced by an arbitrary subspace X 1, since we can always choose ϕ: \(X \rightarrow {\mathbb{R}}^{n}\) such that \(\phi (X_{1}) = {\mathbb{R}}^{m}\).
Our next aim is to tackle the infinite-dimensional case. The general idea is to approximate F by finite-dimensional operators (in the same spirit as we approximated continuous f by smooth functions). To do this, we need to know which operators can be approximated by finite-dimensional operators. Hence we have to recall some basic facts first.
Let X and Y be Banach spaces and U ⊂ X. An operator F: U ⊂ X → Y is called finite-dimensional if its range is finite-dimensional. In addition, it is called compact (completely continuous) if it is continuous and maps bounded sets into relatively compact ones. The set of all compact operators is denoted by \(\mathcal{C}(U,Y )\), and the set of all compact finite-dimensional operators is denoted by \(\mathcal{F}(U,Y )\). Both sets are normed linear spaces, and we have \(\mathcal{F}(U,Y ) \subseteq \mathcal{C}(U,Y ) \subseteq C(U,Y )\). If U is compact, then \(\mathcal{C}(U,Y ) = C(U,Y )\) (since the continuous image of a compact set is compact), and if dim(Y ) < ∞, then \(\mathcal{F}(U,Y ) = \mathcal{C}(U,Y )\). In particular, if \(U \subset {\mathbb{R}}^{n}\) is bounded, then \(\mathcal{F}(U,Y ) = \mathcal{C}(U, {\mathbb{R}}^{n}) = C(U, {\mathbb{R}}^{n})\).
For U ⊂ X, we set \(\mathcal{D}_{y}(\bar{U},X) =\{ F \in \mathcal{C}(\bar{U},X)\): \(y\notin (\mathbb{I} + F)(\partial U)\}\) and \(\mathcal{F}_{y}(\bar{U},X) =\{ F \in \mathcal{F}(\bar{U},X)\): \(y\notin (\mathbb{I} + F)(\partial U)\}\). Note that for \(F \in \mathcal{D}_{y}(\bar{U},X)\), we have \(\rho =\mathrm{ dist}(y,(\mathbb{I} + F)(\partial U)) > 0\), since \(\mathbb{I} + F\) maps closed sets to closed sets.
Pick \(F_{1} \in \mathcal{F}(\bar{U},X)\) such that | F − F 1 | < ρ. Hence, \(F_{1} \in \mathcal{F}_{y}(\bar{U},X)\). Next, let X 1 be a finite-dimensional subspace of X such that \(F_{1}(U) \subset X_{1}\), y ∈ X 1, and set \(U_{1} = U \cap X_{1}\). Then we have \(F_{1} \in \mathcal{F}_{y}(\bar{U}_{1},X_{1})\), and we may define
It is easy to verify that this definition is independent of F 1 and X 1.
Theorem 6.3.
Let U be a bounded open subset of a (real) Banach space X and let \(F \in \mathcal{F}_{y}(\bar{U},X)\) , y ∈ X. Then the following hold.
-
(i)
\(\mathrm{deg}(\mathbb{I} + F,U,y) =\mathrm{ deg}(\mathbb{I} + F - y,U,0)\).
-
(ii)
\(\mathrm{deg}(\mathbb{I},U,y) = 1\) if y ∈ U.
-
(iii)
If U 1,2 are open, disjoint subsets of U such that \(y\notin f(\bar{U} \setminus (U_{1} \cup U_{2}))\) , then \(\mathrm{deg}(\mathbb{I} + F,U,y) =\mathrm{ deg}(\mathbb{I} + F,U_{1},y) +\mathrm{ deg}(\mathbb{I} + F,U_{2},y)\).
-
(iv)
If H: \([0,1] \times \bar{ U} \rightarrow X\) and y: [0,1] → X are both continuous such that \(H(t) \in D_{y(t)}(U, {\mathbb{R}}^{n})\) , t ∈ [0,1], then \(\mathrm{deg}(\mathbb{I} + H(0),U,y(0)) =\mathrm{ deg}(\mathbb{I} + H(1),U,y(1))\).
6.4 Global Bifurcation Theorem
As in Sect. 5.1, we study the nonlinear parameter-dependent problem
where F: \(E \times \mathbb{R} \rightarrow X\) is a C 1-map such that F(0, α) = 0 for all \(\alpha \in \mathbb{R}\), and E ⊆ X is an open neighborhood of 0 (possibly E = X). Note that (6.9) has the trivial solution for all values of α. We shall now consider the question of bifurcation from this trivial branch of solutions and demonstrate the existence of global branches of nontrivial solutions bifurcating from the trivial branch. If \(X = {\mathbb{R}}^{n}\), then we use the Brouwer degree; if X is an infinite-dimensional (real) Banach space, then we assume that F(x, α) = x + f(x, α) and that f: \(E \times \mathbb{R} \rightarrow X\) is completely continuous. Thus for F( ⋅, α), the Leray–Schauder degree is applicable. The application of degree theory to bifurcation theory goes back to Krasnosel’skii [191]. Global bifurcation theorem of the following type were first proved by Rabinowitz [251]. Several generalizations have been given by Ize et al. [177–182], Krawcewicz et al. [192–195], and Nussbaum et al. [232–236].
Theorem 6.4.
Let there exist a, \(b \in \mathbb{R}\) with a < b such that u = 0 is an isolated solution of (6.9) for α = a and α = b, where a and b are not bifurcation points. Furthermore, assume that
where B r (0) ={ u ∈ E: \(\|u\| < r\}\) is an isolating neighborhood of the trivial solution. Let
and let \(\mathcal{C}\) be the maximal connected subset of \(\mathcal{S}\) that contains {0} × [a,b]. Then either
-
(i)
\(\mathcal{C}\) is unbounded in \(E \times \mathbb{R}\),
or else
-
(ii)
\(\mathcal{C}\cap \{ 0\} \times (\mathbb{R} \setminus [a,b])\neq \varnothing \).
Proof.
Define a class \(\mathcal{U}\) of subsets of \(E \times \mathbb{R}\) as follows:
where \(\varOmega _{0} = B_{r}(0) \times [a,b]\), and Ω ∞ is a bounded open subset of \((E \setminus \{ 0\}) \times \mathbb{R}\). We shall first show that (6.9) has a nontrivial solution (u, α) ∈ ∂Ω for every such \(\varOmega \in \mathcal{U}\). To accomplish this, let us consider the following sets:
We observe that K may be regarded as a compact metric space, and A and B are compact subsets of K. We hence may apply Whyburn’s lemma to deduce that either there exists a continuum in K connecting A to B, or else there is a separation K A , K B of K, with A ⊂ K A , B ⊂ K B . If the latter holds, we may find open sets U, V in \(E \times \mathbb{R}\) such that K A ⊂ U, K B ⊂ V, with U ∩ V = ∅. We let \({\varOmega }^{{\ast}} = \varOmega \cap (U \cup V )\) and observe that \({\varOmega }^{{\ast}}\in \mathcal{U}\). It follows by construction that there are no nontrivial solutions of (6.9) that belong to ∂Ω ∗ ; this, however, is impossible, since it would imply, by the generalized homotopy and the excision principle of the Leray–Schauder degree, that \(\mathrm{deg}(F(\cdot,a),B_{r}(0),0) =\mathrm{ deg}(F(\cdot,b),B_{r}(0),0)\), contradicting (6.10). We hence have that, for each \(\varOmega \in \mathcal{U}\), there is a continuum C of solutions of (6.9) that intersects ∂Ω in a nontrivial solution.
We assume now that neither of the alternatives of the theorem holds, that is, we assume that \(\mathcal{C}\) is bounded and \(\mathcal{C}\cap \{ 0\} \times (\mathbb{R} \setminus [a,b]) = \varnothing \). In this case, we may, using the boundedness of \(\mathcal{C}\), construct a set \(\varOmega \in \mathcal{U}\) containing no nontrivial solutions in its boundary, thus arriving once more at a contradiction. □
6.5 \({\mathbb{S}}^{1}\)-Equivariant Degree
Let \(\mathbb{E}\) be a real isometric Banach representation of the group \(G = {\mathbb{S}}^{1}\). The isotypical direct sum decomposition is denoted by
where \(\mathbb{E}_{0} = {\mathbb{E}}^{G}\stackrel{del}{=} \{x \in \mathbb{E};gx = x\) for all g ∈ G} is the subspace of G-fixed points, and for k ≥ 1, \(x \in \mathbb{E}_{k}\setminus \{0\}\) implies that G x , the isotropy group of x, is \(\mathbb{Z}_{k}\stackrel{def}{=} \{g \in G;{g}^{k} = 1\}\). Throughout this section, we assume the following:
- (SD1):
-
For each integer k = 0, 1, …, the subspace \(\mathbb{E}_{k}\) is of finite dimension.
All subspaces \(\mathbb{E}_{k}\), k ≥ 1, admit a natural structure of complex vector spaces such that an \(\mathbb{R}\)-linear operator \(A: \mathbb{E}_{k} \rightarrow \mathbb{E}_{k}\) is G-equivariant if and only if it is \(\mathbb{C}\)-linear with respect to this complex structure. Therefore, by choosing a \(\mathbb{C}\) basis in \(\mathbb{E}_{k}\), k ≥ 1, we can define an isomorphism between the group of all G-equivariant automorphisms of \(\mathbb{E}_{k}\), denoted by \(GL_{G}(\mathbb{E}_{k})\), and the general linear group \(GL(m_{k}, \mathbb{C})\), where \(m_{k} =\dim _{\mathbb{C}}\mathbb{E}_{k}\).
Let \(\mathbb{F}\) be another Banach isometric representation of G, and \(L: \mathbb{E} \rightarrow \mathbb{F}\) a given equivariant linear bounded Fredholm operator of index zero. We say that an equivariant compact operator \(K : \mathbb{E} \rightarrow \mathbb{F}\) is an equivariant compact resolvent of L if \(L + K : \mathbb{E} \rightarrow \mathbb{F}\) is an isomorphism. We shall denote by CR G(L) the set of all equivariant compact resolvents of L, and assume that
- (SD2):
-
CR G(L) ≠ ∅.
In what follows, a point of the Banach space \(\mathbb{E} \times {\mathbb{R}}^{2}\) is denoted by (x, λ) with \(x \in \mathbb{E}\) and \(\lambda \in {\mathbb{R}}^{2}\), and the action of G on \(\mathbb{E} \times {\mathbb{R}}^{2}\) is defined by g(x, λ) = (gx, λ) for every g ∈ G.
We consider a G-equivariant continuous map \(f : \mathbb{E} \times {\mathbb{R}}^{2} \rightarrow \mathbb{F}\) such that
where \(Q: \mathbb{E} \times {\mathbb{R}}^{2} \rightarrow \mathbb{F}\) is a completely continuous map and the following assumption is satisfied:
- (SD3):
-
There exists a two-dimensional submanifold \(M \subset \mathbb{E}_{0} \times {\mathbb{R}}^{2}\) such that (i) M ⊂ f − 1(0); (ii) if \((u_{0},\lambda _{0}) \in M\), then there exist an open neighborhood \(U_{\lambda _{0}}\) of λ 0 in \({\mathbb{R}}^{2}\), an open neighborhood \(U_{u_{0}}\) of u 0 in \(\mathbb{E}_{0}\), and a C 1-map \(\eta : U_{\lambda _{0}} \rightarrow \mathbb{E}_{0}\) such that \(M \cap (U_{u_{0}} \times U_{\lambda _{0}}) =\{ (\eta (\lambda ),\lambda );\lambda \in U_{\lambda _{0}}\}\).
In relation to the bifurcation problem of (6.13), we consider the structure of the set of solutions to the following equation:
All points (u, λ) ∈ M are called trivial solutions of (6.13) or (6.14), and all other solutions in f − 1(0) ∖ M are called nontrivial solutions. A point \((u_{0},\lambda _{0}) \in M\) is called a bifurcation point if in every neighborhood of \((u_{0},\lambda _{0}) \in M\) there is a nontrivial solution for (6.14).
Equation (6.14) can be transformed into the equivariant fixed-point problem
Let \(\mathcal{F}(u,\,\lambda ) = u - {(L + K)}^{-1} \circ [Q(\cdot,\,\lambda ) + K](u),\,(u,\,\lambda ) \in \mathbb{E} \times {\mathbb{R}}^{2}\). Then (6.14) is equivalent to the equation
The idea of finding nontrivial solutions to (6.16) in an open G-invariant neighborhood \(\mathcal{U}\subseteq \mathbb{E} \times {\mathbb{R}}^{2}\) of \((u_{0},\lambda _{0}) \in M\) is based on an auxiliary functionψ to (6.16), which is introduced to distinguish nontrivial solutions from trivial solutions. Here \(\mathcal{U}\) is said to be G-invariant if \((gx,\,\lambda ) \in \mathcal{U}\) for all g ∈ G, \((x,\,\lambda ) \in \mathcal{U}\). An auxiliary function to (6.16) on the set \(\mathcal{U}\) is an equivariant function (i.e., ψ(gx) = gψ(x) for all g ∈ G and \(x \in \overline{\mathcal{U}}\), where \(\overline{\mathcal{U}}\) denotes the closure of \(\mathcal{U}\); here and in what follows G acts on \({\mathbb{R}}^{2}\) trivially) \(\psi : \overline{\mathcal{U}}\subset \mathbb{E} \times {\mathbb{R}}^{2} \rightarrow \mathbb{R}\) satisfying ψ(u, λ) < 0 for all \((u,\,\lambda ) \in \overline{\mathcal{U}}\cap M\). Then every solution to the system
is a nontrivial solution to (6.13). This leads to the equivariant map \(\mathcal{F}_{\psi } : \overline{\mathcal{U}}\rightarrow \mathbb{E} \times \mathbb{R}\) defined by
and the problem of finding a nontrivial solution to (6.13) in \(\mathcal{U}\) can be reduced to the problem of finding a solution to the equation \(\mathcal{F}_{\psi }(u,\,\lambda ) = 0\) in \(\mathcal{U}\), which may be solved by the so-called \({\mathbb{S}}^{1}\) -equivariant degree as a topological invariant associated with the problem (6.17).
To describe the definition and basic properties of \({\mathbb{S}}^{1}\)-degree, we assume that V is an isometric Hilbert representation of \(G = {\mathbb{S}}^{1}\). If U is an open bounded invariant subset of \(V \oplus \mathbb{R}\) (where \({\mathbb{S}}^{1}\) acts trivially on \(\mathbb{R}\)) and \(F : (\overline{U},\partial U) \rightarrow (V,V \setminus \{ 0\})\) is an equivariant compact vector field on \(\overline{U}\), then there is defined the \({\mathbb{S}}^{1}\)-equivariant degree of F with respect to U, which is a sequence of integers
such that deg k (F, U) ≠ 0 for only a finite number of indices k. The basic properties of \({\mathbb{S}}^{1}\mbox{ -}\deg\) are as follows (see [24, 25, 112, 180, 194] for details):
-
(i)
Existence: If \({\mathbb{S}}^{1}\mbox{ -}\deg (F,U): =\{\deg _{k}(F,U)\}_{k=1}^{\infty }\neq 0\), i.e., there exists \(k \in \mathbb{N}\) such that deg k (F, U) ≠ 0, then \({F}^{-1}(0) \cap {U}^{H}\neq \varnothing \), where \(H = \mathbb{Z}_{k}\) and
$$\displaystyle{{U}^{H}: =\{ v \in U :\, gv = v\,\ \mbox{ for any}\ g \in H\}.}$$ -
(ii)
Additivity: If U 1 and U 2 are two open invariant subsets of U such that U 1 ∩ U 2 = ∅ and \({F}^{-1}(0) \cap U \subseteq U_{1} \cup U_{2}\), then \({\mathbb{S}}^{1}\mbox{ -}\deg (F,U) = {\mathbb{S}}^{1}\mbox{ -}\deg (F,U_{1}) + {\mathbb{S}}^{1}\mbox{ -}\deg (F,U_{2})\).
-
(iii)
Homotopy invariance: If \(\mathcal{H}: (\overline{U},\partial U) \times [0,1] \rightarrow (V,V \setminus \{ 0\})\) is an \({\mathbb{S}}^{1}\)-equivariant homotopy of compact vector fields, then \({\mathbb{S}}^{1}\mbox{ -}\deg (\mathcal{H}_{0},U) = {\mathbb{S}}^{1}\mbox{ -}\deg (\mathcal{H}_{1},U)\), where \(\mathcal{H}_{t}(\theta ) = \mathcal{H}(t,\theta )\) for t ∈ [0, 1] and \(\theta \in \overline{U}\).
-
(iv)
Contraction: Suppose that W is another isometric Hilbert representation of \({\mathbb{S}}^{1}\) and let Ω be an open, bounded, invariant subset of W such that 0 ∈ Ω. Define \(\varPhi : \overline{\varOmega } \times \overline{U} \rightarrow W \oplus V\) by Φ(y, x, t) = (y, F(x, t)). Then \({\mathbb{S}}^{1}\mbox{ -}\deg (\varPhi,U) = {\mathbb{S}}^{1}\mbox{ -}\deg (F,U)\).
Now we return to the problem (6.17). If the mapping \(\mathcal{F}_{\psi } : \overline{\mathcal{U}}\rightarrow \mathbb{E} \times \mathbb{R}\) has no solution on \(\partial \mathcal{U}\) and \(\mathcal{F} : \mathcal{U}\rightarrow \mathbb{E}\) is a condensing field (i.e., \(\pi -\mathcal{F}\) is a condensing map, where \(\pi : \mathcal{U}\rightarrow \mathbb{E}\) is the natural projection on \(\mathbb{E}\)), then the \({\mathbb{S}}^{1}\)-equivariant degree \({\mathbb{S}}^{1}\)-\(\deg (\mathcal{F}_{\psi },\,\mathcal{U})\) is well defined, and its nontriviality implies the existence of solutions of \(\mathcal{F}_{\psi }(u,\,\lambda ) = 0\) in \(\mathcal{U}\). Global continuation of the branch of nontrivial solutions (solutions in f − 1(0) ∖ M) bifurcating from (u 0, λ 0) can be characterized by the above \({\mathbb{S}}^{1}\)-degree at all bifurcation points along the closure of the branch if such a branch is bounded in \(\mathbb{E} \times {\mathbb{R}}^{2}\) (the so-called Fuller space).
To describe precisely this \({\mathbb{S}}^{1}\)-degree-based bifurcation theory, we need some additional information about: (i) the construction of the open neighborhood \(\mathcal{U}\), (ii) the auxiliary function ψ, (iii) the computation of \({\mathbb{S}}^{1}\)-\(\deg (\mathcal{F}_{\psi },\mathcal{U})\).
If \(\mathcal{F}(u,\,\lambda )\) is differentiable with respect to u, we are able to define singular points of system (6.16) through its linearization at the trivial solutions of (6.14). This is unfortunately not so for state-dependent DDEs. Therefore, we shall distinguish two cases.
6.5.1 Differentiability Case
Throughout this subsection, we further assume that at all points (u 0, λ 0) ∈ M, the derivative \(D_{u}f(u_{0},\lambda _{0}): \mathbb{E} \rightarrow \mathbb{F}\) of f with respect to u exists and is continuous on M. We say that \((u_{0},\lambda _{0}) \in M\) is \(\mathbb{E}\)-singular if \(D_{u}f(u_{0},\lambda _{0}): \mathbb{E} \rightarrow \mathbb{F}\) is not an isomorphism. An \(\mathbb{E}\)-singular point (u 0, λ 0) is isolated if there are no other \(\mathbb{E}\)-singular points in some neighborhood of (u 0, λ 0). It follows from the implicit function theorem that if (u 0, λ 0) is a bifurcation point, then (u 0, λ 0) is an \(\mathbb{E}\)-singular point.
We start with the construction of the open neighborhood \(\mathcal{U}\). We consider the open neighborhood of (u 0, λ 0) ∈ M defined by
where r > 0 is chosen such that
-
(i)
\(\mathcal{F}(u,\,\lambda )\neq 0\) for all \((u,\,\lambda ) \in \overline{B_{M}(u_{0},\lambda _{0};r,\rho )}\) such that \(\vert \lambda -\lambda _{0}\vert =\rho,\|u -\eta (\lambda )\|\neq 0\);
-
(ii)
\((u_{0},\,\lambda _{0})\) is the only \(\mathbb{E}\)-singular point of \(\mathcal{F}\) in \(\overline{B_{M}(u_{0},\,\lambda _{0};r,\,\rho )}\).
We call \(B_{M}(u_{0},\lambda _{0};r,\rho )\) a special neighborhood of \(\mathcal{F}\) determined by r and ρ. The existence of a special neighborhood \(\mathcal{U}\stackrel{def}{=} B_{M}(u_{0},\lambda _{0};r,\rho )\) follows from the assumption that the \(\mathbb{E}\)-singular point (u 0, λ 0) of \(\mathcal{F}\) is isolated. Note that the equivariant version of Dugundji’s extension theorem (see [193, p. 197]) implies that there exists a continuous \({\mathbb{S}}^{1}\)-equivariant function \(\theta : \overline{\mathcal{U}}\rightarrow \mathbb{R}\) such that
-
(i)
θ(η(λ), λ) = − | λ − λ 0 | for all \((\eta (\lambda ),\,\lambda ) \in \overline{\mathcal{U}}\cap M\);
-
(ii)
θ(u, λ) = r if \(\|u -\eta (\lambda )\| = r\).
Such a function θ is called a completing function (or Ize’s function). Clearly, if θ is a completing function, then \(\psi _{\delta }(u,\,\lambda )\stackrel{def}{=} \theta (u,\,\lambda )-\delta\) is negative on the subset of trivial solutions \(\mathcal{U}\cap M\), provided that δ > 0. So ψ δ is an auxiliary function to (6.16). For δ > 0 small enough, we can define \(\mathcal{F}_{\psi _{\delta }} : \overline{\mathcal{U}}\rightarrow \mathbb{E} \times \mathbb{R}\) by
and define the \({\mathbb{S}}^{1}\)-equivariant degree \({\mathbb{S}}^{1}\mbox{ -}\deg (\mathcal{F}_{\psi _{\delta }},\mathcal{U})\). By the homotopy invariance of the \({\mathbb{S}}^{1}\)-degree, \({\mathbb{S}}^{1}\mbox{ -}\deg (\mathcal{F}_{\psi _{\delta }},\mathcal{U}) = {\mathbb{S}}^{1}\mbox{ -}\deg (\mathcal{F}_{\theta },\mathcal{U})\). Therefore, the nontriviality of \({\mathbb{S}}^{1}\)-\(\deg (\mathcal{F}_{\theta },\mathcal{U})\) implies the existence of a nontrivial solution of (6.14) in \(\mathcal{U}\).
We now turn to the computation of \({\mathbb{S}}^{1}\)-\(\deg (\mathcal{F}_{\theta },\mathcal{U})\). We identify \({\mathbb{R}}^{2}\) with \(\mathbb{C}\), and for sufficiently small ρ > 0, we define α: D → M, \(D\stackrel{def}{=} \{z \in \mathbb{C} :\, \vert z\vert \leq 1\}\), by
Since we have assumed that \((x_{0},\lambda _{0}) = (\eta (\lambda _{0}),\lambda _{0}) \in M\) is an isolated \(\mathbb{E}\)-singular point, it is clear that we can choose sufficiently small ρ > 0 such that α(D) contains only one \(\mathbb{E}\)-singular point, namely (x 0, λ 0). Consequently, the formula \(\varPsi (z)\stackrel{def}{=} D_{u}\mathcal{F}(\alpha (z)),z \in {\mathbb{S}}^{1} \subseteq D\), defines a continuous map \(\varPsi : {\mathbb{S}}^{1} \rightarrow GL_{G}(\mathbb{E})\), which has the decomposition (see [88] for details) \(\varPsi = \varPsi _{0} \oplus \varPsi _{1} \oplus \cdots \oplus \varPsi _{k} \oplus \cdots \), where \(\varPsi _{k} = \varPsi \vert _{\mathbb{E}_{k}} : {\mathbb{S}}^{1} \rightarrow GL_{G}(\mathbb{E}_{k})\) for k = 1, 2, ⋯ and \(\varPsi _{0} : {\mathbb{S}}^{1} \rightarrow GL(\mathbb{E}_{0})\), with \(GL(\mathbb{E}_{0})\) the set of linear automorphisms of \(\mathbb{E}_{0}\). We now define
It is clear that ε 0 does not depend on the choice of \(z \in {\mathbb{S}}^{1}\).
We need one more notion, the crossing number, to calculate \(\mathrm{deg}_{B}(\mbox{ $\det _{\mathbb{C}}$}[\varPsi _{k}])\):
Lemma 6.3 ( [88]).
Suppose \(\alpha _{0},\,\beta _{0},\,\delta,\,\varepsilon\) are given numbers with \(\alpha _{0},\,\delta,\,\varepsilon > 0\) . Let \(\,\varOmega \stackrel{def}{=} (0,\,\alpha _{0}) \times (\beta _{0}-\varepsilon,\,\beta _{0}+\varepsilon ) \subseteq {\mathbb{R}}^{2}\) . Assume that \(H : [\sigma _{0}-\delta,\,\sigma _{0}+\delta ] \times \bar{ \varOmega } \rightarrow {\mathbb{R}}^{2}\) is a continuous function satisfying:
-
(i)
H(σ, α, β) ≠ 0 for all \(\sigma \in [\sigma _{0}-\delta,\,\sigma _{0}+\delta ]\) and \((\alpha,\,\beta ) \in \partial \varOmega \setminus \{ (0,\,\beta );\beta \in (\beta _{0}-\varepsilon,\,\beta _{0}+\varepsilon )\};\)
-
(ii)
if (α, β) ∈ Ω and \(H_{\sigma _{0}\pm \delta }(\alpha,\,\beta ) = 0\) , then α ≠ 0.
Let \(\varOmega _{1}\stackrel{def}{=} (\sigma _{0}-\delta,\,\sigma _{0}+\delta ) \times (\beta _{0}-\varepsilon,\,\beta _{0}+\varepsilon )\) and define the function \(\varPsi _{H} :\bar{ \varOmega }_{1} \rightarrow {\mathbb{R}}^{2}\) by Ψ H (σ,β) = H(σ,0,β), for \(\sigma \in [\sigma _{0}-\delta,\,\sigma _{0}+\delta ]\) , and \(\beta \in [\beta _{0}-\varepsilon,\,\beta _{0}+\varepsilon ]\) . Then Ψ H (σ,β) ≠ 0 for (σ,β) ∈ ∂Ω 1 and \(\mathrm{deg}_{B}(\varPsi _{H},\varOmega _{1}) =\gamma\) , where γ is the crossing number given by
Lemma 6.4.
Let \(\,\mathcal{U} = B_{M}(u_{0},\,\lambda _{0};r^{\prime},\rho ) \subseteq \mathbb{E} \times {\mathbb{R}}^{2}\) be a special neighborhood of \(\mathcal{F}\) , and θ a completing function. Then the \({\mathbb{S}}^{1}\) degree \({\mathbb{S}}^{1}\mbox{ -}\deg (\mathcal{F}_{\theta },\,\mathcal{U})\) is well defined, and
That is,
where μ(u 0 , λ 0 ) is defined by (6.20).
By Lemma 6.4, we have the following local bifurcation theorem of Krasnosel’skii type [191].
Theorem 6.5.
Suppose that \(f : \mathbb{E} \oplus {\mathbb{R}}^{2} \rightarrow \mathbb{F}\) is a G-equivariant continuous map that is continuously differentiable with respect to x at points (x,λ) ∈ M and satisfies (SD1) – (SD3) . If \((u_{0},\lambda _{0}) \in M\) is an isolated \(\mathbb{E}\) -singular point such that \(\varepsilon _{0}\mu _{k}(u_{0},\lambda _{0})\not =0\) for some k ≥ 1, then (u 0 , λ 0 ) is a bifurcation point of (6.13) . More precisely, there exists a sequence (u n , λ n ) of nontrivial solutions to (6.13) such that the isotropy group of u n contains \(\mathbb{Z}_{k}\) and \((u_{n},\,\lambda _{n}) \rightarrow (u_{0},\,\lambda _{0})\) as n →∞.
We remark that the above results hold when \({\mathbb{R}}^{2}\) is replaced by an open subset of \({\mathbb{R}}^{2}\). Geba and Marzantowicz [111] established the following global bifurcation theorem of Rabinowitz type [251] by applying the \({\mathbb{S}}^{1}\)-degree theory due to Dylawerski, Geba, Jodel, and Marzantowicz [85].
Theorem 6.6.
Suppose that \(f : \mathbb{E} \oplus {\mathbb{R}}^{2} \rightarrow \mathbb{F}\) is as in Theorem 6.5 and suppose further that M is complete and every \(\mathbb{E}\) -singular point in M is isolated. Let \(\mathcal{S}(f)\) denote the closure of the set of all nontrivial solutions of (6.13) . Then for each bounded component C of \(\mathcal{S}(f)\) , the set C ∩ M is a finite set, i.e.,
and
where \(\mathcal{U}_{i}\) is a special neighborhood of (u i , λ i ), θ i is a completing function defined on \(\overline{\mathcal{U}}_{i}\) , and ε i and \(\mu (u_{i},\,\lambda _{i})\) are defined by (6.20).
Proof.
If C is a bounded component of \(\mathcal{S}(f)\), then every point of C ∩ M is a bifurcation point that is also a \(\mathbb{E}\)-singular point of f. Since every \(\mathbb{E}\)-singular point of f is isolated and M is complete, C ∩ M is a bounded and closed subset of \(\mathbb{E}_{0} \times {\mathbb{R}}^{2} \supset M\). By (SD1), \(\mathbb{E}_{0} \times {\mathbb{R}}^{2}\) is finite-dimensional, and hence C ∩ M is compact. Therefore, C ∩ M is a finite set.
Choose r, ρ > 0 sufficiently small that for each i = 1, 2, ⋯ , q, \(U_{i} = B_{M}(u_{i},\,\lambda _{i};r,\,\rho )\) is a special neighborhood of (u i , λ i ) for f and \(U_{i} \cap U_{j} = \varnothing \) if i ≠ j. Let \(U = U_{1} \cup U_{2} \cup \cdots \cup U_{q}\) and find a bounded open set \(\varOmega _{1} \subset \mathbb{E} \times {\mathbb{R}}^{2}\) such that C ∖ U ⊆ Ω 1 and Ω 1 ⋂M = ∅. Put Ω 2 = U ∪Ω 1. Then C ⊆ Ω 2. We can (e.g., see [193, p. 174]) find an open invariant subset \(\varOmega \subseteq \mathbb{E} \times {\mathbb{R}}^{2}\) such that C ⊆ Ω ⊆ Ω 2 and \(\partial \varOmega \cap \mathcal{S}(f) = \varnothing \).
Note that Ω is an open, bounded, invariant subset. We now choose r 0 ∈ (0, r) and ρ 0 ∈ (0, ρ) such that for every i = 1, 2, ⋯ , q, we have
-
(i)
\(B_{M}(u_{i},\,\lambda _{i};r_{0},\,\rho _{0}) \subseteq \varOmega \);
-
(ii)
\(\mathcal{U}_{i}\stackrel{def}{=} B_{M}(u_{i},\,\lambda _{i};r_{0},\,\rho _{0})\) is a special neighborhood of \((u_{i},\,\lambda _{i})\) for f.
Set \(\mathcal{U} = \mathcal{U}_{1} \cup \mathcal{U}_{2} \cup \cdots \cup \mathcal{U}_{q}\) and
We note that r 0 > 0, and define an invariant function by
Now, \(\mathcal{U}_{i}\) is a special neighborhood, and hence we have \((C \setminus \mathcal{U}) \cap \bar{\mathcal{U}}_{i} = C \cap \partial \mathcal{U}_{i} \subseteq \partial \mathcal{U}_{r_{0}}\), where we have θ(u, λ) = r 0. Then by (6.21), θ(u, λ) is continuous on \((C \setminus \mathcal{U}) \cap \bar{\mathcal{U}}_{i}\). Also, by the construction of \(\mathcal{U}_{i}\), we have \(\bar{\mathcal{U}}_{i} \cap \bar{\mathcal{U}}_{j} = \varnothing \) if i ≠ j. Therefore, \(\theta : C \cup \bar{\mathcal{U}}\rightarrow \mathbb{R}\) is continuous.
By the equivariant version of Dugundji’s extension theorem (see [193, p. 197]), we can extend \(\theta : C \cup \bar{\mathcal{U}}\rightarrow \mathbb{R}\) to a continuous invariant function \(\theta : \overline{\varOmega } \rightarrow \mathbb{R}\) such that
-
(iii)
θ(u, λ) = − | λ − λ i | if \((u,\,\lambda ) \in \overline{\mathcal{U}}_{i}\bigcap M\);
-
(iv)
θ(u, λ) = r 0 if \((u,\,\lambda ) \in (C \setminus \mathcal{U}) \cup \partial \mathcal{U}_{r_{0}}\).
Let \(\mathcal{F}_{\theta }(u,\,\lambda ) = (\mathcal{F}(u,\,\lambda ),\,\theta (u,\,\lambda ))\). Then \(\mathcal{F}_{\theta }^{-1}(0) = {\mathcal{F}}^{-1}{(0)\bigcap \theta }^{-1}(0)\). By (iii), we know that \(\mathcal{F}_{\theta }^{-1}(0) \subseteq C\). Since C ∩ ∂Ω = ∅, \(\mathcal{F}_{\theta }^{-1}(0) \cap \partial \varOmega = \varnothing \). Therefore, \({\mathbb{S}}^{1}\mbox{ -}\deg (\mathcal{F}_{\theta },\,\varOmega )\) is well defined.
We now construct a homotopy \(H : \overline{\varOmega } \times [0,\,1] \rightarrow \mathbb{E} \times \mathbb{R}\) as follows:
Note that trivial solutions \((u,\,\lambda ) \in \bar{ \varOmega }\) outside \(\mathcal{S}(f)\) are contained in \(\bar{\mathcal{U}}_{i} \cap M\) for some i = 1, 2, ⋯ , q, and by (iii), we have
Then by the fact that \(\partial \varOmega \cap \mathcal{S}(f) = \varnothing \), we have H(u, λ, α) ≠ 0 for all (u, λ, α) ∈ ∂Ω ×[0, 1]. Note that θ is invariant and \(\mathcal{F}\) is equivariant. So H is an \({\mathbb{S}}^{1}\)-homotopy. Since \(H(u,\,\lambda,\,0) = \mathcal{F}_{\theta }(u,\,\lambda )\) and \(H(u,\,\lambda,\,1) = (\mathcal{F}(u,\,\lambda ),\,-\rho _{0})\neq 0\) for all \((u,\,\lambda ) \in \overline{\varOmega } \times [0,\,1]\), by the existence and homotopy invariance of the \({\mathbb{S}}^{1}\)-degree, we have \({\mathbb{S}}^{1}\mbox{ -}\deg (\mathcal{F}_{\theta },\,\varOmega ) = 0\). But (i)–(iv) imply that \(\mathcal{F}_{\theta }^{-1}(0) \subseteq C \cap \mathcal{U}\). Then it follows from the excision property of the \({\mathbb{S}}^{1}\mbox{ -}\)degree that
On the other hand, by the additivity property of the \({\mathbb{S}}^{1}\)-degree, we have
Let \(\theta _{i}(u,\,\lambda ) =\theta (u,\,\lambda )\vert _{\overline{\mathcal{U}}_{ i}}\). Note that \(\mathcal{U}\subseteq \overline{\varOmega }\) implies that \(((C \setminus \mathcal{U}) \cup \partial \mathcal{U}_{r_{0}})\bigcap \overline{\mathcal{U}}_{i} = \partial \mathcal{U}_{i} \cap \partial \mathcal{U}_{r_{0}}\). Then θ i (u, λ) is a completing function on \(\overline{\mathcal{U}}_{i}\), and we have
Therefore, it follows from Lemma 6.4 that
The proof is complete. □
6.5.2 Nondifferentiability Case
If f(u, λ) is not differentiable with respect to u, then we need to justify that the formal linearization can be utilized to detect the local Hopf bifurcation and to describe the global continuation of periodic solutions for such a system with state-dependent delay. Our approach to this justification of formal linearization is through a simple homotopy argument. Namely, we will consider the equation
for an \({\mathbb{S}}^{1}\)-equivariant C 1-map \(\tilde{\mathcal{F}} : \overline{\mathcal{U}}\rightarrow \mathbb{E}\) that is \({\mathbb{S}}^{1}\)-homotopic to \(\mathcal{F}\) in a sense to be detailed below. For the functional-analytic setting of the Hopf bifurcation of state-dependent DDEs, such a C 1-map is attained by extending a linear operator obtained through the formal linearization from a C 1-space to a C-space, an idea previously used by Eichmann [86] and Mallet-Paret–Nussbaum–Paraskevopoulos [215]. To be more precise, we assume that such a C 1-map is given by
where \(\tilde{Q} : \overline{\mathcal{U}}\rightarrow \mathbb{E}\) is an \({\mathbb{S}}^{1}\)-equivariant C 1-map and
- (SD4):
-
\(M \subseteq \tilde{{\mathcal{F}}}^{-1}(0)\), and for every \(\lambda \in {\mathbb{R}}^{2}\), \({(L + K)}^{-1} \circ (\tilde{Q}(\cdot,\lambda ) + K) : \mathbb{E} \rightarrow \mathbb{E}\) is a condensing map.
By the implicit function theorem, if \((u_{0},\,\lambda _{0}) \in M\) is a bifurcation point of system (6.23), then the derivative \(D_{u}\tilde{\mathcal{F}}(u_{0},\,\lambda _{0})\), which is G-equivariant, is not an automorphism of \(\mathbb{E}\). Therefore, all bifurcation points of (6.23) are contained in the set
Let (u 0, λ 0) be an isolated \(\mathbb{E}\)-singular point of \(\tilde{\mathcal{F}}\). To tie the \({\mathbb{S}}^{1}\)-equivariant degree of \(\mathcal{F}\) to that of \(\tilde{\mathcal{F}}\), we assume that:
- (SD5):
-
We can choose the constants r > 0 and ρ > 0 such that \(B_{M}(u_{0},\lambda _{0};r,\rho )\) is a special neighborhood of \(\tilde{\mathcal{F}}\) and there exists 0 < r′ ≤ r such that \(\mathcal{F}(u,\,\lambda )\neq 0\) for all \((u,\,\lambda ) \in B_{M}(u_{0},\lambda _{0};r^{\prime},\rho )\) with | λ − λ 0 | = ρ and \(\|u -\eta (\lambda )\|\neq 0\).
If ψ is an auxiliary function to (6.16), then by the construction of the \({\mathbb{S}}^{1}\mbox{ -}\)degree and the assumptions (SD2), (SD4), and (SD5), there exists a special neighborhood \(\mathcal{U}\stackrel{def}{=} B_{M}(u_{0},\lambda _{0};r^{\prime},\rho )\) of \(\tilde{\mathcal{F}}\) such that the continuous G-equivariant maps \(\mathcal{F}_{\psi }\) and \(\tilde{\mathcal{F}}_{\psi }\) are nonzero on the boundary of \(\mathcal{U}\), and therefore both \({\mathbb{S}}^{1}\)-\(\deg (\tilde{\mathcal{F}}_{\psi },\,\mathcal{U})\) and \({\mathbb{S}}^{1}\)-\(\deg (\mathcal{F}_{\psi },\,\mathcal{U})\) are well defined.
For a completing function θ defined on \(\mathcal{U}\), if \(\tilde{\mathcal{F}}_{\theta } = (\tilde{\mathcal{F}},\,\theta )\) is homotopic to \(\mathcal{F}_{\theta }\) on \(\mathcal{U}\), then the homotopy invariance of the \({\mathbb{S}}^{1}\)-degree ensures that \({\mathbb{S}}^{1}\)-\(\deg (\mathcal{F}_{\theta },\mathcal{U}) = {\mathbb{S}}^{1}\)-\(\deg (\tilde{\mathcal{F}}_{\theta },\mathcal{U})\). On the other hand, we can follow the approach presented in the previous subsection to calculate \({\mathbb{S}}^{1}\)-\(\mathrm{deg}(\tilde{\mathcal{F}}_{\theta },\,\mathcal{U})\).
Finally, in order to exclude bifurcation of solutions of (6.22) in \(\mathbb{E}_{0} \times {\mathbb{R}}^{2}\), we assume that
- (SD6):
-
\(D_{u}\tilde{\mathcal{F}}(u_{0},\,\lambda _{0})\vert _{\mathbb{E}_{0}} : \mathbb{E}_{0} \rightarrow \mathbb{E}_{0}\) is an isomorphism.
Theorem 6.7 ( [170]).
Assume that (SD1)–(SD6) hold and let \(\mathcal{U}=B_{M}(u_{0},\,\lambda _{0};r^{\prime},\rho )\subseteq \) \(\mathbb{E} \times {\mathbb{R}}^{2}\) be a special neighborhood for \(\tilde{\mathcal{F}}\) and θ a completing function. If \(\tilde{\mathcal{F}}_{\theta }\) is homotopic to \(\mathcal{F}_{\theta }\) on \(\overline{\mathcal{U}}\) and there exists k ≥ 1 such that \({\mathbb{S}}^{1}\) - \(\mathrm{deg}_{k}(\tilde{\mathcal{F}}_{\theta },\,\mathcal{U})\neq 0\) , then \((u_{0},\,\lambda _{0})\) is a bifurcation point for (6.13) . That is, there exists a sequence of nontrivial solutions (u n , λ n ) of (6.13) such that the isotropy group of u n contains \(\mathbb{Z}_{k}\) and \((u_{n},\,\lambda _{n}) \rightarrow (u_{0},\,\lambda _{0})\) as n →∞.
For global bifurcation, we assume further that both \(\mathcal{F}\) and \(\tilde{\mathcal{F}}\) are defined on \(\mathbb{E} \times {\mathbb{R}}^{2}\), and that:
- (SD7):
-
Every bifurcation point of (6.13) is a \(\mathbb{E}\)-singular point of \(\tilde{\mathcal{F}}\).
- (SD8):
-
\(\tilde{\mathcal{F}}_{\theta }\) is homotopic to \(\mathcal{F}_{\theta }\) on some special neighborhood \(\mathcal{U}\) of each isolated \(\mathbb{E}\)-singular point of \(\tilde{\mathcal{F}}\), where θ is a completing function defined on \(\mathcal{U}\).
Now we can state the following global bifurcation theorem of Rabinowitz type.
Theorem 6.8 ( [170]).
Assume that (SD1)–(SD8) hold and (SD5)–(SD6) hold for every \(\mathbb{E}\) -singular point (u 0 , λ 0 ) of \(\tilde{\mathcal{F}}\) . Assume further that every \(\mathbb{E}\) -singular point of \(\tilde{\mathcal{F}}\) in M is isolated and M is complete. Let \(\mathcal{S}\) denote the closure of the set of all nontrivial solutions of (6.13) . Then for each bounded component C of \(\mathcal{S}\) , the set C ∩ M is a finite set, i.e., \(C \cap M =\{ (u_{1},\,\lambda _{1}),\,(u_{2},\,\lambda _{2}),\cdots \,,(u_{q},\,\lambda _{q})\}\) , and
where \(\mathcal{U}_{i}\) is a special neighborhood of \((u_{i},\,\lambda _{i})\) , and θ i is a completing function defined on \(\overline{\mathcal{U}}_{i}\).
6.6 Global Hopf Bifurcation Theory of DDEs
In this section, we employ the \({\mathbb{S}}^{1}\)-equivariant degree to establish global Hopf bifurcations for general functional differential equations of mixed type with two parameters. We state our theory in a very general setting to allow for mixed type to ensure that the general theory can be used to address the issue of global bifurcations of bifurcated periodic solutions with additional features, such as spatial–temporal symmetry, for systems of DDEs with special symmetries.
Let X denote the Banach space of bounded continuous mappings \(x: \mathbb{R} \rightarrow {\mathbb{R}}^{n}\) equipped with the supremum norm. For reasons mentioned above, we will consider functional differential equations with both delayed and advanced arguments. Therefore, for x ∈ X and \(t \in \mathbb{R}\), we will use x t to denote an element in X defined by x t(s) = x(t + s) for \(s \in \mathbb{R}\).
Consider the functional differential equation
parameterized by two real numbers \((\alpha,p) \in \mathbb{R} \times \mathbb{R}_{+}\), where \(\mathbb{R}_{+} = (0,\infty )\), and \(F : X \times \mathbb{R} \times \mathbb{R}_{+} \rightarrow {\mathbb{R}}^{n}\) is completely continuous. Identifying the subspace of X consisting of all constant mappings with \({\mathbb{R}}^{n}\), we obtain a mapping \(\hat{F} = F\vert _{{\mathbb{R}}^{n}\times \mathbb{R}\times \mathbb{R}_{+}}: {\mathbb{R}}^{n} \times \mathbb{R} \times \mathbb{R}_{+} \rightarrow {\mathbb{R}}^{n}\). We require the following assumption:
- (GHB1):
-
\(\hat{F}\) is twice continuously differentiable.
Denote by \(\hat{x}_{0} \in X\) the constant mapping with the value \(x_{0} \in {\mathbb{R}}^{n}\). We call \((\hat{x}_{0},\alpha _{0},p_{0})\) a stationary solution of (6.24) if \(\hat{F}(x_{0},\alpha _{0},p_{0}) = 0\). We assume that:
- (GHB2):
-
At each stationary solution \((\hat{x}_{0},\alpha _{0},p_{0})\), the derivative of \(\hat{F}(x,\alpha,p)\) with respect to the first variable \(x\), evaluated at \((\hat{x}_{0},\alpha _{0},p_{0})\), is an isomorphism of \({\mathbb{R}}^{n}\).
Under (GHB1)–(GHB2), for each stationary solution \((\hat{x}_{0},\alpha _{0},p_{0})\), there exist \(\varepsilon _{0} > 0\) and a continuously differentiable mapping \(y: B_{\varepsilon _{0}}(\alpha _{0},p_{0}) \rightarrow {\mathbb{R}}^{n}\) such that \(\hat{F}(y(\alpha,p),\alpha,p) = 0\) for \((\alpha,p) \in B_{\varepsilon _{0}}(\alpha _{0},p_{0}) = (\alpha _{0} -\varepsilon _{0},\alpha _{0} +\varepsilon _{0}) \times (p_{0}-\varepsilon _{0},p_{0}+\varepsilon _{0})\).
We need the following smoothness condition:
- (GHB3):
-
\(F(\varphi,\alpha,p)\) is differentiable with respect to \(\varphi\), and the n ×n complex matrix function \(\Delta _{(\hat{y}(\alpha,p),\alpha,p)}(\lambda )\) is continuous in \((\alpha,p,\lambda ) \in B_{\varepsilon _{0}}(\alpha _{0},p_{0}) \times \mathbb{C}\). Here, for each stationary solution \((\hat{x}_{0},\alpha _{0},p_{0})\), we have \(\Delta _{(\hat{x}_{0},\alpha _{0},p_{0})}(\lambda ) =\lambda \mathrm{ Id} - DF(\hat{x}_{0},\alpha _{0},p_{0})(\mathrm{{e}}^{\lambda \cdot }\mathrm{Id})\), where \(DF(\hat{x}_{0},\alpha _{0},p_{0})\) is the complexification of the derivative of \(F(\varphi,\alpha,p)\) with respect to \(\varphi\), evaluated at \((\hat{x}_{0},\alpha _{0},p_{0})\).
For easy reference, we will again call \(\Delta _{(\hat{x}_{0},\alpha _{0},p_{0})}(\lambda )\) the characteristic matrix and the zeros of \(\det \Delta _{(\hat{x}_{0},\alpha _{0},p_{0})}(\lambda ) = 0\) the characteristic values of the stationary solution \((\hat{x}_{0},\alpha _{0},p_{0})\). So (GHB2) is equivalent to assuming that 0 is not a characteristic value of any stationary solution of (6.24).
Definition 6.1.
A stationary solution \((\hat{x}_{0},\alpha _{0},p_{0})\) is called a center if it has purely imaginary characteristic values of the form \(\mathrm{i}m \frac{2\pi } {p_{0}}\) for some positive integer m. A center \((\hat{x}_{0},\alpha _{0},p_{0})\) is said to be isolated if (i) it is the only center in some neighborhood of \((\hat{x}_{0},\alpha _{0},p_{0})\); (ii) it has only finitely many purely imaginary characteristic values of the form \(\mathrm{i}m \frac{2\pi } {p_{0}}\), m an integer.
Assume now that \((\hat{x}_{0},\alpha _{0},p_{0})\) is an isolated center. Let \(J(\hat{x}_{0},\alpha _{0},p_{0})\) denote the set of all positive integers m such that \(\mathrm{i}m \frac{2\pi } {p_{0}}\) is a characteristic value of \((\hat{x}_{0},\alpha _{0},p_{0})\). We assume that there exists \(m \in J(\hat{x}_{0},\alpha _{0},p_{0})\) such that:
- (GHB4):
-
There exist \(\varepsilon \in (0,\varepsilon _{0})\) and \(\delta \in (0,\varepsilon _{0})\) such that on \([\alpha _{0}-\delta,\alpha _{0}+\delta ] \times \partial \varOmega _{\varepsilon,p_{0}}\), \(\det \Delta _{(\hat{y}(\alpha,p),\alpha,p)}(u +\mathrm{ i}m\frac{2\pi } {p}) = 0\) if and only if α = α 0, u = 0, p = p 0, where \(\varOmega _{\varepsilon,p_{0}} =\{ (u,p) :\, 0 < u <\varepsilon,p_{0}-\varepsilon < p < p_{0}+\varepsilon \}\).
Let
Then (GHB4) implies that \(H_{m}^{\pm }(\hat{x}_{0},\alpha _{0},p_{0})\not =0\) on \(\partial \varOmega _{\varepsilon,p_{0}}\). Consequently, the integer
is well defined.
Definition 6.2.
\(\gamma _{m}(\hat{x}_{0},\alpha _{0},p_{0})\) is called the mth crossing number of \((\hat{x}_{0},\alpha _{0},p_{0})\).
We will show that \(\gamma _{m}(\hat{x}_{0},\alpha _{0},p_{0})\not =0\) implies the existence of a local bifurcation of periodic solutions with periods near p 0 ∕ m. More precisely, we have the following:
Theorem 6.9.
Assume that (GHB1)–(GHB3) are satisfied, and that there exist an isolated center \((\hat{x}_{0},\alpha _{0},p_{0})\) and an integer \(m \in J(\hat{x}_{0},\alpha _{0},p_{0})\) such that (GHB4) holds and \(\gamma _{m}(\hat{x}_{0},\alpha _{0},p_{0})\not =0\) . Then there exists a sequence \((\alpha _{k},p_{k}) \in \mathbb{R} \times \mathbb{R}_{+}\) such that
-
(i)
\(\lim _{k\rightarrow \infty }(\alpha _{k},p_{k}) = (\alpha _{0},p_{0});\)
-
(ii)
at each \((\alpha,p) = (\alpha _{k},p_{k})\), (6.24) has a nonconstant periodic solution x k (t) with period p k ∕m;
-
(iii)
\(\lim _{k\rightarrow \infty }x_{k}(t) =\hat{ x}_{0}\) , uniformly for \(t \in \mathbb{R}\).
To describe the global continuation of the local bifurcation obtained in Theorem 6.9, we need to assume that:
- (GHB5):
-
All centers of (6.24) are isolated and (GHB4) holds for each center \((\hat{x}_{0},\alpha _{0},p_{0})\) and each \(m \in J(\hat{x}_{0},\alpha _{0},p_{0})\).
- (GHB6):
-
For each bounded set \(W \subseteq X\times \mathbb{R} \times \mathbb{R}_{+}\), there exists a constant l > 0 such that \(\vert F(\varphi,\alpha,p)-F(\psi,\alpha,p)\vert \leq l\sup _{s\in R}\vert \varphi (s)-\psi (s)\vert \) for \((\varphi,\alpha,p),(\psi,\alpha,p) \in \ W\).
Theorem 6.10.
Let
Assume that \((\hat{x}_{0},\alpha _{0},p_{0})\) is an isolated center satisfying the conditions in Theorem 6.9 . Denote by \(C(\hat{x}_{0},\alpha _{0},p_{0})\) the connected component of \((\hat{x}_{0},\alpha _{0},p_{0})\) in Σ(F). Then either
-
(i)
\(C(\hat{x}_{0},\alpha _{0},p_{0})\) is unbounded, or
-
(ii)
\(C(\hat{x}_{0},\alpha _{0},p_{0})\) is bounded, \(C(\hat{x}_{0},\alpha _{0},p_{0}) \cap N(F)\) is finite, and
$$\displaystyle{ \sum _{(\hat{x},\alpha,p)\in C(\hat{x}_{0},\alpha _{0},p_{0})\cap N(F)}\gamma _{m}(\hat{x},\alpha,p) = 0 }$$(6.25)for all \(m = 1,2, \ldots,\) where \(\gamma _{m}(\hat{x},\alpha,p)\) is the mth crossing number of \((\hat{x},\alpha,p)\) if \(m \in J(\hat{x},\alpha,p)\) , and zero otherwise.
Proof of Theorems 6.9 and 6.10: Put \({\mathbb{S}}^{1} = \mathbb{R}/2\pi \mathbb{Z}\), \(\mathbb{E} = {L}^{1}({\mathbb{S}}^{1}; {\mathbb{R}}^{n})\), \(\mathbb{F} = {L}^{2}({\mathbb{S}}^{1}; {\mathbb{R}}^{n})\). Define \(L: \mathbb{E} \rightarrow \mathbb{F}\) and \(Q: \mathbb{E} \times \mathbb{R} \times \mathbb{R}_{+} \rightarrow \mathbb{F}\) by
where
Clearly, x(t) is a p-periodic solution of (6.24) if and only if \(z(t) = x(\frac{p} {2\pi }t)\) is a solution in \(\mathbb{E}\) of the operator equation Lz = Q(z, α, p).
The representations \(\mathbb{E}\) and \(\mathbb{F}\) are isometric Hilbert representations of the group \({\mathbb{S}}^{1}\), where \({\mathbb{S}}^{1}\) acts by shifting the argument. With respect to these \({\mathbb{S}}^{1}\)-actions, L is an equivariant bounded linear Fredholm operator of index zero with an equivariant compact resolvent K, and Q is an \({\mathbb{S}}^{1}\)-equivariant compact mapping. Moreover, at \((\hat{y}(\alpha,p),\alpha,p)\) with \((\alpha,p) \in \mathcal{D}\stackrel{def}{=} (\alpha _{0}-\delta,\alpha _{0}+\delta ) \times (p_{0}-\varepsilon,p_{0}+\varepsilon )\), the derivative of Q with respect to the first variable is given by
Identifying \(\partial \mathcal{D}\) with \({\mathbb{S}}^{1}\), since \((\hat{x}_{0},\alpha _{0},p_{0})\) is an isolated center, we can easily show that the mapping \(\mathrm{Id} - {(L + K)}^{-1}[K + D_{z}F(\hat{y}(\alpha,p),\alpha,p)]\) is an isomorphism of \(\mathbb{E}\) and that the mapping \(\varPsi : {\mathbb{S}}^{1} \rightarrow GL(\mathbb{E})\) defined by
is continuous.
The representation \(\mathbb{E}\) has the well-known isotypical decomposition \(\mathbb{E} =\bigoplus _{ k=0}^{\infty }\mathbb{E}_{k}\), where \(\mathbb{E}_{0}\mathop{\cong}{\mathbb{R}}^{n}\) and for each k ≥ 1, \(\mathbb{E}_{k}\) is spanned by \(\cos (kt)\varepsilon _{j}\) and \(\sin (kt)\varepsilon _{j}\), 1 ≤ j ≤ n, where \(\{\varepsilon _{1}, \ldots,\varepsilon _{n}\}\) is the standard basis of \({\mathbb{R}}^{n}\). So we have \(\varPsi (\alpha,p)\mathbb{E}_{k} \subseteq \mathbb{E}_{k}\). Let \(\varPsi _{k}(\alpha,p) = \varPsi (\alpha,p)\vert _{\mathbb{E}_{k}}\). It is not difficult to show that
Let
Then one can show, as in Erbe, Geba, Krawcewicz, and Wu [112], that \(\gamma _{k}(\hat{x}_{0},\alpha _{0},p_{0}) = n_{k}(\hat{x}_{0},\alpha _{0},p_{0})\), and therefore Theorems 6.9 and 6.10 are simply immediate consequences of Theorems 6.5 and 6.6 with \(M =\{ (\hat{x}_{0},\alpha _{0},p_{0}) \in {\mathbb{R}}^{n} \times \mathbb{R} \times \mathbb{R}_{+};F(\hat{x}_{0},\alpha _{0},p_{0}) = 0\}\). This completes the proof. □
For ease of applications, we describe below the local and global Hopf bifurcation theory for parameterized DDEs. Let X = C n, τ and consider the following functional differential equation:
with parameter \(\alpha \in \mathbb{R}\), \(F : X \times \mathbb{R} \rightarrow {\mathbb{R}}^{n}\) is completely continuous.
Identifying the subspace of X consisting of all constant mappings with \({\mathbb{R}}^{n}\), we obtain a mapping \(\hat{F} = F\vert _{{\mathbb{R}}^{n}\times \mathbb{R}} : {\mathbb{R}}^{n} \times \mathbb{R} \rightarrow {\mathbb{R}}^{n}\). We now describe conditions (GHB1)–(GHB6) in relatively simple form:
- (SGHB1):
-
\(\hat{F}\) is twice continuously differentiable.
Denote by \(\hat{x}_{0} \in X\) the constant mapping with the value \(x_{0} \in {\mathbb{R}}^{n}\). We call \((\hat{x}_{0},\alpha _{0})\) a stationary solution of (6.26) if \(\hat{F}(x_{0},\alpha _{0}) = 0\). We assume that:
- (SGHB2):
-
At each stationary solution \((\hat{x}_{0},\alpha _{0})\), the derivative of \(\hat{F}(x,\alpha )\) with respect to the first variable x, evaluated at \((\hat{x}_{0},\alpha _{0})\), is an isomorphism of \({\mathbb{R}}^{n}\).
Under (SGHB1)–(SGHB2), for each stationary solution \((\hat{x}_{0},\alpha _{0})\), there exist \(\varepsilon _{0} > 0\) and a continuously differentiable mapping \(y: B_{\varepsilon _{0}}(\alpha _{0}) \rightarrow {\mathbb{R}}^{n}\) such that \(\hat{F}(y(\alpha ),\alpha ) = 0\) for \(\alpha \in B_{\varepsilon _{0}}(\alpha _{0}) = (\alpha _{0} -\varepsilon _{0},\alpha _{0} +\varepsilon _{0})\).
We need the following smoothness condition:
- (SGHB3):
-
\(F(\varphi,\alpha )\) is differentiable with respect to \(\varphi\), and the n ×n complex matrix function \(\Delta _{(\hat{y}(\alpha ),\alpha )}(\lambda )\) is continuous in \((\alpha,\lambda ) \in B_{\varepsilon _{0}}(\alpha _{0}) \times \mathbb{C}\). Here, for each stationary solution \((\hat{x}_{0},\alpha _{0})\), we have \(\Delta _{(\hat{x}_{0},\alpha _{0})}(\lambda ) =\lambda \mathrm{ Id} - DF(\hat{x}_{0},\alpha _{0})(\mathrm{{e}}^{\lambda \cdot }\mathrm{Id})\), where \(DF(\hat{x}_{0},\alpha _{0})\) is the complexification of the derivative of \(F(\varphi,\alpha )\) with respect to \(\varphi\), evaluated at \((\hat{x}_{0},\alpha _{0})\).
For easy reference, we will again call \(\Delta _{(\hat{x}_{0},\alpha _{0})}(\lambda )\) the characteristic matrix and the zeros of \(\det \Delta _{(\hat{x}_{0},\alpha _{0})}(\lambda ) = 0\) the characteristic values of the stationary solution \((\hat{x}_{0},\alpha _{0})\). So (SGHB2) is equivalent to assuming that 0 is not a characteristic value of any stationary solution of (6.26).
The concepts of isolated centers and crossing numbers are now simplified as follows:
Definition 6.3.
A stationary solution \((\hat{x}_{0},\alpha _{0})\) is called a center if it has purely imaginary characteristic values ± iβ 0 for some positive β 0 > 0. A center \((\hat{x}_{0},\alpha _{0})\) is said to be isolated if it is the only center in some neighborhood of \((\hat{x}_{0},\alpha _{0})\).
Assume that \((\hat{x}_{0},\alpha _{0})\) is an isolated center. We assume that:
- (SGHB4):
-
There exist \(\varepsilon \in (0,\varepsilon _{0})\) and \(\delta \in (0,\varepsilon _{0})\) such that on \([\alpha _{0}-\delta,\alpha _{0}+\delta ] \times \partial \varOmega _{\varepsilon,p_{0}}\), \(\det \Delta _{(\hat{y}(\alpha ),\alpha )}(u +\mathrm{ i}\beta ) = 0\) if and only if α = α 0, u = 0, β = β 0, where \(\varOmega _{\varepsilon,\beta _{0}} =\{ (u,p) :\, 0 < u <\varepsilon,\beta _{0}-\varepsilon <\beta <\beta _{0}+\varepsilon \}\).
Let
Then (SGHB4) implies that \({H}^{\pm }(\hat{x}_{0},\alpha _{0},\beta _{0})\not =0\) on \(\partial \varOmega _{\varepsilon,\beta _{0}}\). Consequently, the integer
is well defined; it is called the first crossing number of \((\hat{x}_{0},\alpha _{0},\beta _{0})\).
The local Hopf bifurcation theory below shows that \(\gamma (\hat{x}_{0},\alpha _{0},\beta _{0})\not =0\) implies the existence of a local bifurcation of periodic solutions with periods near 2π ∕ β 0. More precisely, we have the following theorem:
Theorem 6.11.
Assume that (SGHB1)–(SGHB3) are satisfied, and that there exists an isolated center \((\hat{x}_{0},\alpha _{0})\) such that (SGHB4) holds and \(\gamma (\hat{x}_{0},\alpha _{0},\beta _{0})\not =0\) . Then there exists a sequence \((\alpha _{k},\beta _{k}) \in \mathbb{R} \times \mathbb{R}_{+}\) such that
-
(i)
\(\lim _{k\rightarrow \infty }(\alpha _{k},\beta _{k}) = (\alpha _{0},\beta _{0});\)
-
(ii)
at each α = α k, (6.26) has a nonconstant periodic solution x k (t) with a period \(\frac{2\pi } {\beta _{k}};\)
-
(iii)
\(\lim _{k\rightarrow \infty }x_{k}(t) =\hat{ x}_{0}\) , uniformly for \(t \in \mathbb{R}\).
The global Hopf bifurcation theorem can now be stated as follows:
- (SGHB5):
-
All centers of (6.26) are isolated and (SGHB4) holds for each center \((\hat{x}_{0},\alpha _{0})\) with the corresponding β 0.
- (SGHB6):
-
For each bounded set \(W \subseteq X \times \mathbb{R}\), there exists a constant l > 0 such that \(\vert F(\varphi,\alpha ) - F(\psi,\alpha )\vert \leq l\sup _{s\in R}\vert \varphi (s) -\psi (s)\vert \) for \((\varphi,\alpha ),(\psi,\alpha ) \in W\).
Theorem 6.12.
Set
Assume that \((\hat{x}_{0},\alpha _{0},\beta _{0}\) is an isolated center satisfying the conditions in Theorem 6.11 . Denote by \(C(\hat{x}_{0},\alpha _{0},\beta _{0})\) the connected component of \((\hat{x}_{0},\alpha _{0},\beta _{0})\) in Σ(F). Then either
-
(i)
\(C(\hat{x}_{0},\alpha _{0},\beta _{0})\) is unbounded, or
-
(ii)
\(C(\hat{x}_{0},\alpha _{0},\beta _{0})\) is bounded, \(C(\hat{x}_{0},\alpha _{0},\beta _{0}) \cap N(F)\) is finite, and
$$\displaystyle{ \sum _{(\hat{x},\alpha,\beta )\in C(\hat{x}_{0},\alpha _{0},\beta _{0})\cap N(F)}\gamma (\hat{x},\alpha,\beta ) = 0, }$$(6.27)where \(\gamma (\hat{x},\alpha,\beta )\) is the crossing number of \((\hat{x},\alpha,\beta )\).
6.7 Application to a Delayed Nicholson Blowflies Equation
6.7.1 The Nicholson Blowflies Equation
Gurney et al. [142] proposed the following simple-looking delay differential equation to explain the oscillatory behavior of the observed sheep blowfly Lucilia cuprina population in the experimental data collected by the Australian entomologist Nicholson [231]:
with f(N) = pNe − αN, where N(t) denotes the population of sexually mature adults at time t, p is the maximum possible per capita egg production rate, 1 ∕ α is the population size at which the whole population reproduces at its maximum rate. In the model, τ is the generation time, or the time from egg to sexually mature adult, and γ is the per capita mortality rate of adults. This model is now commonly called Nicholson’s blowflies equation. It was used by Oster and Ipatkchi [239] for the development of an insect population, and its modifications have been intensively studied in the literature of theoretical biology and delay differential equations. Notably, it has been shown that a unique positive equilibrium of the model is globally asymptotically stable (with respect to nonnegative and nontrivial initial conditions) for every τ ≥ 0, provided that 1 < p ∕ γ < e 2 (see, for example, [198]). In the case p ∕ γ > e 2, the positive equilibrium loses its local stability, and Hopf bifurcations occur at an unbounded sequence of critical values. In the next subsection, we introduce the work of Wei and Li [294] that uses the global Hopf bifurcation theorem coupled with Bendixson’s criterion for higher dimensional ordinary differential equations to establish the existence of periodic solutions when the delay τ is not necessarily near the local Hopf bifurcation values.
6.7.2 The Global Hopf Bifurcation Theorem of Wei–Li
In this subsection, we consider the equation
We introduce the theorem of Wei–Li [294] that shows that under the assumption p > γe2, as the delay τ increases, the positive equilibrium N ∗ loses its stability, a sequence of Hopf bifurcations occurs at N ∗ , and these periodic solutions persist for τ far away from these Hopf bifurcation values. Wei and Li established this theorem using a global Hopf bifurcation result (Theorem 6.12). A key step in establishing the global extension of the local Hopf branch at τ = τ 0 is to show that (6.28) has no nonconstant periodic solutions of period 4. This is accomplished by applying a higher-dimensional Bendixson criterion for ordinary differential equations due to Li and Muldowney [210].
The positive equilibrium \({N}^{{\ast}} = \frac{1} {a}\log \frac{p} {\gamma }\) of (6.28) exists if and only if a > 0 and p > γ. These relations are assumed throughout this section. Set \(N(t) = {N}^{{\ast}} + \frac{1} {a}y(t)\). Then x(t) satisfies
The linearization of (6.29) at y = 0 is
whose characteristic equation is
For τ = 0, the only root of (6.30) is λ = − aN ∗ < 0, since p > γ. For ω ≠ 0, iω is a root of (6.30) if and only if
Separating the real and imaginary parts, we obtain
which leads to
namely,
This is possible if and only if aN ∗ > 2, or equivalently, if p > γ e2.
For p > γ e2, let
k = 0, 1, 2, ⋯ . Set
Let \(\lambda _{k} =\alpha _{k}(\tau ) +\mathrm{ i}\,\omega _{k}(\tau )\) denote a root of (6.30) near τ = τ k such that \(\alpha _{k}(\tau _{k}) = 0,\,\omega _{k}(\tau _{k}) =\omega _{0}\). Obviously, α k ′(τ k ) > 0. Therefore, we have obtained that when γ < p ≤ γe2, all roots of the characteristic equation (6.30) have negative real parts; when p > γe2, (6.30) has a pair of simple imaginary roots ± i ω 0 at τ = τ k , k = 0, 1, 2, ⋯ . Furthermore, if τ ∈ [0, τ 0), then all roots of (6.30) have negative real part; if τ = τ 0, then all roots of (6.30) except ± i ω 0 have negative real part; and if τ ∈ (τ k , τ k + 1) for k = 0, 1, 2, ⋯ , then (6.30) has 2(k + 1) roots with positive real part. In particular, we have shown that under the condition p > γe2, N = N ∗ is asymptotically stable for τ ∈ [0, τ 0) and unstable for τ > τ 0. Furthermore, (6.28) undergoes a Hopf bifurcation at N ∗ when τ = τ k , for k = 0, 1, 2, ⋯ .
Let x(t) = y(τt). Then (6.29) becomes
Lemma 6.5.
All periodic solutions to (6.32) are uniformly bounded.
Proof.
Let x(t) be a nonconstant periodic solution to (6.32), and let \(x(t_{1}) = M,\,x(t_{2}) = m\) be its maximum and minimum, respectively. Then, x′(t 1) = x′(t 2) = 0, and by (6.32),
We claim that x(t 1 − 1) < 0 and x(t 2 − 1) > 0. In fact, if x(t 1 − 1) = 0, then (6.33) implies M = 0, and thus m < 0 and x(t 2 − 1) ≤ 0. Using (6.34), we know that x(t 2 − 1) < 0, and thus
which contradicts the fact that m is the minimum. If x(t 1 − 1) > 0, then by (6.33), we arrive at
a contradiction. Therefore, x(t 1 − 1) < 0. A similar argument shows that x(t 2 − 1) > 0. Therefore, we have m < 0 and M > 0. Again by (6.33) and (6.34), we have
Also by (6.33), we have
Here we have used the fact that \(x(t_{1} - 1) + a{N}^{{\ast}} > m + a{N}^{{\ast}} > 0\) and that \(x\mathrm{{e}}^{-x} <\mathrm{ {e}}^{-1}\) for x ≥ 0. Relations (6.35) and (6.36) imply uniform boundedness of the periodic solutions. □
Lemma 6.6.
Assume that \(\gamma \,\mathrm{{e}}^{2} < p < \sqrt{2}\,\gamma \,\mathrm{{e}}^{2}.\) Then (6.32) has no periodic solutions of period 4.
Proof.
Let x(t) be a periodic solution to (6.32) of period 4. Set \(u_{j}(t) = x(t - j + 1),\ j = 1,2,3,4.\) Then \(u(t) = (u_{1}(t),u_{2}(t),u_{3}(t),u_{4}(t))\) is a periodic solution of the following system of ordinary differential equations:
whose orbit belongs to the region
where \(\bar{m}\) and \(\bar{M}\) are a pair of uniform bounds for periodic solutions of (6.32) obtained in Lemma 6.5. To rule out 4-periodic solutions of (6.32), it suffices to prove the nonexistence of nonconstant periodic solutions of (6.37) in the region G. To do the latter, we use a general Bendixson’s criterion in higher dimensions developed in Li and Muldowney [210]. More specifically, we will apply Corollary 3.5 in [210]. The Jacobian matrix J = J(u) of (6.37), for \(u \in {\mathbb{R}}^{4},\) is
where
The second additive compound matrix J [2](u) of J(u) is (see [103] and [226])
Choose a vector norm in ℝ 6 as
Then with respect to this norm, the Lozinskiĭ measure μ(J [2](u) of the matrix J [2](u) is, see [73],
By Corollary 3.5 in [210], system (6.37) has no periodic orbits in G if μ(J [2](u)) < 0 for all u ∈ G. From (6.40), we see that μ(J [2](u)) < 0 if and only if
for u ∈ G.
To establish (6.41), we first use the assumption \(\mathrm{{e}}^{a{N}^{{\ast}} } = p/\gamma < \sqrt{2}\mathrm{{e}}^{2}\) to improve the lower bound m given in Lemma 6.5. In (6.34), we now have
Using the fact that the function xe − x is monotonically decreasing for x > 1 and that \(x(t_{2} - 1) + a{N}^{{\ast}} > 1,\) we have
Therefore, u ∈ G satisfies
For \(\delta = 2\sqrt{2}\mathrm{{e}}^{3-\sqrt{2}e} > 1,\) we can verify
From the graph of f(v), we know that f(v) has a global maximum \(\mathrm{{e}}^{a{N}^{{\ast}}-2 } =\mathrm{ {e}}^{-2}p/\gamma.\) Therefore, for u ∈ G,
and (6.41) is satisfied, completing the proof. □
Lemma 6.7.
Assume that γ e2 < p. Then (6.32) has no periodic solutions of period 1 or 2.
Proof.
First note that every nonconstant 1-periodic solution u(t) of (6.32) is also a nonconstant periodic solution of the ordinary differential equation
A simple phase-line analysis shows that (6.42) has no nonconstant periodic solutions.
As in the proof of Lemma 6.6, if u(t) is a periodic solution of (6.32) of period 2, then u 1(t) = u(t) and u 2(t) = u(t − 1) are periodic solutions of the system of ordinary differential equations
Let \((P(u_{1},u_{2}),\,Q(u_{1},u_{2}))\) denote the vector field of (6.43). Then
for all (u 1, u 2). Thus the classical Bendixson’s negative criterion implies that (6.43) has no nonconstant periodic solutions. □
Theorem 6.13.
Suppose that \(\gamma \mathrm{{e}}^{2} < p < \sqrt{2}\gamma \mathrm{{e}}^{2}\) holds. Then for each τ > τ k , k = 0,1,2,⋯ , (6.32) has at least k + 1 periodic solutions.
Proof.
First note that
satisfies hypotheses (SGHB1), (SGHB2), and (SGHB3) of Sect. 6.6, with
It can also be verified that (0, τ k ) are isolated centers with the corresponding imaginary characteristic values ± iτ k ω 0. We have shown that there exist ε > 0, δ > 0, and a smooth curve \(z :\, (\tau _{k}-\delta,\,\tau _{k}+\delta ) \rightarrow \, \mathbb{C}\) such that \(\Delta (z(\tau )) = 0,\,\vert z(\tau ) -\mathrm{ i}\tau _{k}\omega _{0}\vert <\epsilon\) for all \(\tau \in [\tau _{k}-\delta,\,\tau _{k}+\delta ],\) and
Set \(\beta _{k} =\tau _{k}\omega _{0}\) and let
Clearly, if | τ − τ k | ≤ δ and \((u,p) \in \partial \varOmega _{\epsilon }\) such that Δ 0, τ)(u + iβ) = 0, then τ = τ k , u = 0, and β = β k . This satisfies assumption (SGHB4) in Sect. 6.6. Moreover, if we put
then we have the cross number
By Theorem 6.12, we conclude that the connected component C(0, τ k , τ k ω 0) through \((0,\tau _{k},\tau _{k}\omega _{0})\) in Σ(F) is nonempty. Meanwhile, we have
and hence \(C(0,\tau _{k},\tau _{k}\omega _{0})\) is unbounded.
Lemma 6.5 implies that the projection of \(C(0,\tau _{k},\tau _{k}\omega _{0})\) onto the x-space is bounded. It can be verified using a phase-line analysis that when τ = 0, (6.32) has no nonconstant periodic solutions. Therefore, the projection of \(C(0,\tau _{k},\tau _{k}\omega _{0})\) onto the τ-space is bounded below. From the definitions of τ k and ω 0, we obtain
for k ≥ 0. Also, we know that sinω 0 τ k > 0 and cosω 0 τ k < 0, for k ≥ 0. Hence
Therefore
Applying Lemmas 6.6 and 6.7, we know that 2 < 2π ∕ β < 4 if \((x,\tau,\beta ) \in C(0,\tau _{0},\tau _{0}\omega _{0}),\) and that 1 ∕ (k + 1) < 2π ∕ β < 1 if \((x,\tau,\beta ) \in C(0,\tau _{k},\tau _{k}\omega _{0})\) for k ≥ 1. This shows that in order for \(C(0,\tau _{k},\tau _{k}\omega _{0})\) to be unbounded, its projection onto the τ-space must be unbounded. Consequently, the projection of \(C(0,\tau _{k},\tau _{k}\omega _{0})\) onto the τ-space includes [τ k , ∞). This shows that for each τ > τ k , (6.32) has k + 1 nonconstant periodic solutions, completing the proof of the theorem. □
Remark 6.1.
(i) From the proof of Theorem 6.13, we know that the first global Hopf branch contains periodic solutions of period between 2 and 4. These are the slowly oscillating periodic solutions. See [13, 60, 197, 291] for more details about the existence of slowly oscillating periodic solutions in delay differential equations. The τ k branches, for k ≥ 1, since the periods are less than 1, contain fast-oscillating periodic solutions.
-
(ii)
For k ≥ 1,
$$\displaystyle{ \frac{1} {k + 1}\, <\, \frac{2\pi } {\tau _{k}\omega _{0}}\, <\, 1}$$automatically holds. The bounds on the period 2π ∕ β for \((x,\tau,\beta )\in C(0,\tau _{k},\tau _{k}\omega _{0})\) hold without resort to Lemma 6.6. Thus, the global extension of the τ k -branch for k ≥ 1 can be proved without the restriction \(p < \sqrt{2}\gamma \mathrm{{e}}^{2}.\)
6.7.3 Nicholson’s Blowflies Equation Revisited: Onset and Termination of Nonlinear Oscillations
In [264], the authors reexamined the Nicholson’s blowflies model with natural death rate explicitly incorporated into the delay feedback, obtaining the following delay differential equation with a delay-dependent coefficient
where δ > 0 is the death rate of the immature population, and f(N) = pNe − αN. One can derive this, as was done in [82, 222], from a structured population model for u(t, a) (the population density at age a and time t) as follows:
with the stage-specific mortality rate
A simple application of the integration along characteristic lines leads to the model equation for the mature population \(N(t) =\int _{ \tau }^{\infty }u(t,a)da\) with the Ricker’s-type birth function f.
The additional term e − δτ is the probability of the immature population surviving τ time units before becoming mature. This addition, as shown in [264], leads to rather different dynamics for model (6.46): as the delay τ increases, the positive equilibrium loses its stability and undergoes local Hopf bifurcations at a finite even number of critical values, and as τ passes a critical threshold, the positive equilibrium regains its stability. In other words, as τ keeps increasing and passes another threshold value, the positive equilibrium disappears, and the species becomes extinct (the zero solution is globally asymptotically stable). Shu, Wang, and Wu [264] also observed the coexistence of multiple stable periodic solutions.
As we did in the last subsection, Shu, Wang, and Wu [264] considered the delay a bifurcation parameter and examined the onset and termination of Hopf bifurcations of periodic solutions from a positive equilibrium. They proved that the model has only a finite number of Hopf bifurcation values and that these branches of Hopf bifurcations are paired, so that the existence of periodic solutions with specific oscillation frequencies occurs only in bounded delay intervals. The bifurcation analysis then guided some numerical simulations to identify ranges of parameters for coexisting multiple attractive periodic solutions.
6.8 Rotating Waves and Circulant Matrices
We have noticed that a key step in applying the global Hopf bifurcation theory is to exclude the existence of nonconstant periodic solutions with a certain prescribed period, normally the integer multiplier of the delay if the delay is constant. A general approach outlined in [237] is as follows: If one assumes that y(t) is a periodic solution of a prototype equation x′(t) = f(x(t), x(t − τ)) for some scalar function f, of period mτ for a certain integer m, and defines u j (t) = y(t − (j − 1)τ)) for 1 ≤ j ≤ m, one then discovers that u(t) = (u 1(t), …, u m (t)) satisfies a cyclic system of ordinary differential equations u′(t) = g(u(t)), and we shall show that solutions of such a cyclic system satisfy lim t → ∞ | u(t) | = 0 or ∞, and the key step in proving the latter statement will be the construction of appropriate Lyapunov functions for the cyclic system. This will normally require the estimation of the spectral radius of a so-called circulant matrix. If we linearize this cyclic system at the trivial solution, we are led to a linear system with a real circulant matrix. Here and in what follows, an n ×n matrix is called circulant if its (i, j)-element is given by a j − i + 1 for n real numbers a 1, ⋯a n . This matrix will be written as \(A = \mbox{ circ}(a_{1},a_{2},\cdots a_{n})\). For such a matrix, it was shown in [237] that
In this section, we will demonstrate the use of the approach outlined by Nussbaum based on the above-mentioned spectral property of circulant matrices.
We consider the following partial NFDE:
where \(x \in {\mathbb{S}}^{1}\), a, d, τ are positive constants, \(g : \mathbb{R} \rightarrow \mathbb{R}\) is continuously differentiable with g(0) = 0, q ∈ (0, 1) is the bifurcation parameter. This partial NFDE can be obtained from the coupled lossless transmission line NFDE introduced in Sect. 5.9 by letting the number of coupled oscillators go to infinity.
We are interested in the Hopf bifurcation of rotating waves from the trivial solution. Rotating wave solutions are solutions that satisfy
where p > 0 is a constant.
Let \(y(t) = u(t\frac{p} {2\pi },0)\). Then using the spatiotemporal relation (6.48) of the rotating waves, we can show that u is a rotating wave if and only if y is a 2π-periodic solution of an NFDE with two parameters (q, p). This two-parameter NFDE is very much similar to (6.24), and a global Hopf bifurcation has been established (see [193] for details). Here we describe how Wu and Xia [306] applied this theory to establish the existence of rotating waves, and how this is related to circulant matrices.
Let g′(0) = − γ and assume that 0 < γ < a in what follows. The characteristic equation of (6.47) at the trivial solution takes the form
Letting λ = iβ in (6.49), we get
or equivalently,
It is easy to show that for a real number β > 0, the second equation of (6.50) has a solution q ∈ (0, 1) only if
Therefore, there are only finitely many k ≥ 1 such that (6.50) has a pair of purely imaginary solutions.
For each fixed k ≥ 1 such that dk 2 < γ, we can easily show graphically that there exists a sequence of positive numbers \(\beta _{k,1} <\beta _{k,2} < \cdots \) such that the first equation of (6.50) is satisfied by β k, j , j = 1, 2, …. Substituting this β k, j into the second equation of (6.50) gives
Therefore, we can conclude that the set {(q, p) ∈ (0, 1) ×(0, ∞); (6.49) has a solution i(2π ∕ p)m for some m ≥ 1} is discrete.
Let λ = λ(q) be a smooth curve of zeros of (6.49) such that \(\lambda (q_{k,j}) =\mathrm{ i}\beta _{k,j}\). Differentiating (6.49) with respect to q, we get
That is,
This leads to
From the definition of the crossing number in Sect. 6.6, we can see that this will be crucial in ruling out bounded connected components of rotating waves of (6.47).
For the sake of later application, let us look at the location of β 0 = β 1, 1. We assume that
Then β 0 is the first positive solution of
and hence iβ 0 is a solution of (6.49) with k = 1 and
Lemma 6.8.
If
then \(\pi /2\tau <\beta _{0} < \sqrt{(a +\gamma -d)(a -\gamma +d)}\) , and hence
In particular, if
then
In order to apply the global bifurcation theorem to establish the global existence of rotating waves, we need to obtain a priori bounds for rotating waves. Assume that u(t, x) is a rotating wave of (6.47) satisfying (6.48). Let \({[u(t_{0},x_{0}) - qu(t_{0}-\tau,x_{0})]}^{2}\) be the maximum value of [u(t, x) − qu(t − τ, x)]2 over \(\mathbb{R} \times {\mathbb{S}}^{1}\). Then
Without loss of generality, we may assume that \(u(t_{0},x_{0}) - qu(t_{0}-\tau,x_{0})\neq 0\). Therefore, from (6.47) it follows that
That is,
Note that
implies
Therefore, by (6.60), we obtain
If we assume that
then (6.62) implies the existence of Q = Q(2aq ∕ (1 − q)), so that
and hence from (6.61), it follows that
Summarizing the above discussion, we get the following.
Lemma 6.9.
If (6.63) is satisfied, then there exists a nondecreasing function Q : (0,∞) → (0,∞) such that every rotating wave u(t,x) of (6.47) satisfies |u(t,x)|≤ (1∕(1 − q)Q(2aq∕(1 − q)) for \(t \in \mathbb{R}\) and \(x \in {\mathbb{S}}^{1}\) . In particular, for fixed q ∗ ∈ (0,1), the set of rotating waves of (6.47) corresponding to q ∈ [0,q ∗ ) is uniformly bounded in the sup-norm.
Now we try to exclude nontrivial 4τ-periodic rotating waves. Assume that u(t, x) is a nontrivial rotating wave of (6.47) satisfying (6.48) with p = 4τ. Then
So, v(t) = u(t, 0) satisfies
\(t \in \mathbb{R}\). Let
Then
where we have the following circulant matrix:
Substituting (6.66) and (6.67) into (6.65), we get
Its similarity to the Liénard equation suggests a transformation that leads to an equivalent system,
and a related Lyapunov function,
The derivative of V along solutions of (6.68) is given by
We need the following lemma.
Lemma 6.10.
\(\sum _{i=1}^{4}z_{i}(Bz)_{i} \geq -(1 - q)(1 + {q}^{2})\sum _{i=1}^{4}z_{i}^{2},\quad z_{i} \in \mathbb{R},\quad 1 \leq i \leq 4\).
Proof.
Using the aforementioned spectral property of circulant matrices, we have \(\sum _{i=1}^{4}z_{i}(Bz)_{i} \geq \Gamma \sum _{i=1}^{4}z_{i}^{2},z_{i} \in \mathbb{R},\leq i \leq 4\), where
□
We also need to compute the eigenvalues of B T B. While this can be done directly, Wu and Xia [306] have presented an approach that can be extended to general circular matrices.
Lemma 6.11.
The minimal eigenvalue of B T B is \(\lambda _{\min }({B}^{T}B) = {(1 - {q}^{4})}^{2}/{(1 + q)}^{2}\) , and the maximal eigenvalue of B T B is \(\lambda _{\max }({B}^{T}B) = {(1 - {q}^{4})}^{2}/{(1 - q)}^{2}\).
Proof.
Let
It can be shown that v j is an eigenvector of B corresponding to the eigenvalue
and an eigenvector of B T corresponding to the eigenvector
Assume that \(x \in {\mathbb{C}}^{4}\) is an eigenvector of B T B corresponding to an eigenvalue \(\lambda \in \mathbb{C}\). Then \(x = a_{0}v_{0} + a_{1}v_{1} + a_{2}v_{2} + a_{3}v_{3}\), and B T Bx = λx is equivalent to
from which it follows that \(\lambda =\alpha _{j}\beta _{j}\) for some j = 0, 1, 2, 3. Therefore, all eigenvalues of B T B are given by
from which the conclusion follows. □
We also note the following result; see [306]
Lemma 6.12.
Assume that
Let x i (t), i = 1,…,4, be given by (6.66) . Then
We now return to the estimation of \(\dot{V }\). Using Lemma 6.11, we get
By Lemma 6.12, we have
Therefore, using Lemma 6.10, we get
Consequently, if we assume that
then \(\dot{V }\) is a strictly negative function of \((x_{1},\ldots,x_{4},y_{1},\ldots,y_{4})\) unless x i = y i = 0 for 1 ≤ i ≤ 4. Therefore, under assumptions (6.69), (6.70), (6.72), and (6.73), system (6.68) has no nontrivial periodic solution. This implies that system (6.47) has no nontrivial rotating wave of period 4τ. That is, we have proved the following.
Lemma 6.13.
Under assumptions (6.69) , (6.70), (6.72) , and (6.73), the partial neutral functional differential equation (6.47) has no nontrivial 4π-periodic rotating wave for q ∈ [0,1 −δ).
We can then use global bifurcation theory to obtain the following result, for which we refer to [306] for more details of the proof.
Theorem 6.14.
Assume that
-
(i)
\(g^{\prime}(0) = -\gamma,d <\gamma < a,\pi /2 < \sqrt{(a +\gamma -d)(a -\gamma +d)};\)
-
(ii)
\(\mathrm{inf}_{y\neq 0}g(y)/y > -a,\lim _{y\rightarrow \infty }g(y)/y = \infty; \)
-
(iii)
g(−y) = −g(y) for \(y \in \mathbb{R}\) and g(y)∕y is nondecreasing in y ∈ (0,∞);
-
(iv)
there exist constants δ ∈ (0,1) and K ≥ 0 such that
$$\displaystyle{\begin{array}{c} -K \leq g(x)/x\ \mbox{ for}\ x\neq 0, \\ \frac{1} {4}{a{}^{2}\delta }^{2} > \left [\frac{1} {d}{\left ( \frac{\pi } {2}\right )}^{2} + a\right ]K + 8a\left (\frac{1-\delta } {\delta } \right )\max \left \{K, \frac{4a} {\delta } \right \}, \end{array} }$$and
$$\displaystyle{q_{0}\stackrel{def}{=} \sqrt{\frac{\beta _{0 }^{2 } + {(a +\gamma -d)}^{2 } } {\beta _{0}^{2} + {(a -\gamma +d)}^{2}}} < 1-\delta,}$$where β 0 is the first solution in \(((\pi /2\tau ),\sqrt{(a +\gamma -d)(a -\gamma +d)})\) of the equation
$$\displaystyle{\tan (\beta \tau ) ={ \frac{2a\beta } {\beta }^{2} - (a +\gamma -d)(a -\gamma +d)}.}$$
Then for each q ∈ (q 0 ,1 −δ), system (6.47) has a rotating wave with a period less than 4. If, in addition, we assume
-
(iv)
$$\displaystyle{ \frac{\pi } {2} < \sqrt{(a +\gamma -d)(a -\gamma +d)} < \frac{\pi } {\tau },}$$
then for each q ∈ (q 0 ,1 −δ), system (6.47) has a slowly oscillating rotating wave, that is, a rotating wave with a period in (2τ,4τ).
6.9 State-Dependent DDEs
State-dependent DDEs arises from a number of applications such as electrodynamics, automatic and remote control, machine cutting, neutral networks, population biology, mathematical epidemiology, and economics. They describe the evolution of systems in which the rate of change depends on the history of the rate, and the delay depends on the system’s status in a complicated manner, such as by an explicit or implicit algebraic equation or a differential or integral equation.
Early results on the existence of periodic solutions for state-dependent DDEs include work by Smith [269] that considered bifurcations of periodic solutions from a stationary state for a system of integral equations with state-dependent delay, and work on the existence of periodic solutions by Nussbaum, Mallet-Paret, and Paraskevopoulos [215]. These studies address the aspect of global continuation of Hopf bifurcations of periodic solutions, especially the existence of periodic solutions in which the bifurcation parameter is away from the critical value where a local Hopf bifurcation is born. The work of Nussbaum et al. [215] focuses on important prototype classes of state-dependent delay differential equations with negative feedback and provides some detailed information on slowly oscillating periodic solutions. Here we introduce the work [170, 171] to provide a general tool and framework for studying the Hopf bifurcation problem, and in particular, the global continuation of local bifurcation of periodic solutions of the following parameterized state-dependent DDEs from an equivariant-degree point of view:
where \(x \in {\mathbb{R}}^{N},\,\tau \in \mathbb{R},\,t \in \mathbb{R}\) and \(\sigma \in \mathbb{R}\), \(f : {\mathbb{R}}^{N} \times {\mathbb{R}}^{N} \times \mathbb{R} \rightarrow {\mathbb{R}}^{N}\), and \(g : {\mathbb{R}}^{N} \times \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}\) are given maps. A stationary state of (6.74) with parameter σ is a vector \((x,\,\tau ) \in {\mathbb{R}}^{N} \times \mathbb{R}\) such that f(x, x, σ) = 0 and g(x, τ, σ) = 0.
The major problem in developing such a global Hopf bifurcation theory for the system of state-dependent DDEs (6.74) is that in the spaces of continuous periodic functions \({\mathcal{C}_{T}}{(\mathbb{R};\mathbb{R})^{N}}=\{ x\epsilon\,{{\mathcal{C}}(\mathbb{R};\mathbb{R})^{N}} :x(t+T)=x(t)for\, all\, t\epsilon\mathbb{R}\}\) and \(C_{T}(\mathbb{R}; \mathbb{R}) =\{\tau \epsilon \,\mathcal{C}(\mathbb{R}; \mathbb{R}) :\tau (t + T) =\tau (t)\mbox{ for all t} \,\epsilon \mathbb{R}\}\) with a fixed period T > 0, the composition operator
is generally not a C 1 (continuously differentiable) map with respect to τ in the supremum norm. This causes difficulty in formulating linearization at a stationary state, and such a linearization is usually necessary in the functional-analytic setting for Hopf bifurcation problems in which a topological index such as an \({\mathbb{S}}^{1}\)-equivariant degree can be calculated and applied to investigate the birth and continuation of periodic solutions bifurcating from a stationary state.
In [72], a system of auxiliary equations obtained through a formal linearization technique was used in the study of local stability of state-dependent DDEs in the space of continuously differentiable functions. This formal linearization technique is only heuristic and can be described in the following way: the state-dependent delay τ(t) in x(t − τ(t)) is first fixed at a given stationary state, and then the resulting nonlinear system with frozen constant delay is linearized. Other applications of systems of auxiliary equations obtained through a formal linearization process can be found in [26, 45, 156] and [157]. None of these results is sufficient for us to develop a global Hopf bifurcation theory based on the \({\mathbb{S}}^{1}\)-equivariant degree for state-dependent DDEs (6.74). However, the above-mentioned results strongly indicate that the system of auxiliary equations obtained through the heuristic technique of formal linearization can be used to detect the local Hopf bifurcation and to describe its global continuation for state-dependent DDEs.
In this section, we use the homotopy invariance property of the \({\mathbb{S}}^{1}\)-equivariant degree to relate the Hopf bifurcation problem of (6.74) to the change of stability of stationary states of the corresponding system of auxiliary equations obtained through formal linearization. As such, much of the effort has been dedicated to justifying that the detection of Hopf bifurcation can be achieved through the formal linearization technique: the state-dependent delay τ(t) in x(t − τ(t)) is first fixed at a given stationary state; then the resulting nonlinear system with frozen constant delay is linearized. This linearization technique is used in the functional-analytic setting that converts the Hopf bifurcation problem of system (6.74) to solving an operator equation (6.13) involving \({\mathbb{S}}^{1}\)-equivariant maps with two parameters, in the space of periodic functions with a fixed period. Implicitly used is the C 1-smoothness of the operator defined in Lemma 6.17 in the space \(\mathbb{E}\) (the space of periodic functions with fixed period 2π). The formal linearization leads to this operator naturally in the space of continuously differentiable periodic functions with period 2π, and the fact that this operator can be extended to a bounded operator in the space \(\mathbb{E}\) is essential in our homotopy argument. This technique of extending the linearized operator of a state-dependent delay differential equation from the space C 1 to the space C has previously been used in other contexts; see, for example, Mallet-Paret–Nussbaum–Paraskevopoulos [215], Krisztin [196], Walther [290], and the survey paper by Hartung–Krisztin–Walther–Wu [158].
6.9.1 Local Hopf Bifurcation
We turn to the Hopf bifurcation of (6.74), with its solution denoted by u(t) = (x(t), τ(t)). Denote by \(C(\mathbb{R};\, {\mathbb{R}}^{N})\) the normed space of continuous functions from \(\mathbb{R}\) to \({\mathbb{R}}^{N}\) equipped with the usual supremum norm \(\|x\| =\sup _{t\in \mathbb{R}}\vert x(t)\vert \) for \(x \in C(\mathbb{R};\, {\mathbb{R}}^{N})\), where | ⋅ | denotes the Euclidean norm. We also denote by \({C}^{1}(\mathbb{R};\, {\mathbb{R}}^{N})\) the normed space of continuously differentiable bounded functions from \(\mathbb{R}\) to \({\mathbb{R}}^{N}\) equipped with the usual C 1 norm
for \(x \in C(\mathbb{R};\, {\mathbb{R}}^{N}).\) For a stationary state (u 0, τ 0) of (6.74) with the parameter σ 0, we say that (u 0, σ 0) is a Hopf bifurcation point of system (6.74) if there exist a sequence \(\{(u_{k},\,\sigma _{k},\,T_{k})\}_{k=1}^{+\infty }\subseteq C(\mathbb{R}; {\mathbb{R}}^{N+1}) \times {\mathbb{R}}^{2}\) and T 0 > 0 such that
and \((u_{k},\,\sigma _{k})\) is a nonconstant T k -periodic solution of system (6.74).
We assume that:
- (SHB1):
-
The map f: \({\mathbb{R}}^{N} \times {\mathbb{R}}^{N} \times \mathbb{R} \ni (\theta _{1},\theta _{2},\sigma ) \rightarrow f(\theta _{1},\theta _{2},\sigma ) \in {\mathbb{R}}^{N}\) and the map g: \({\mathbb{R}}^{N} \times \mathbb{R} \times \mathbb{R} \ni (\gamma _{1},\,\gamma _{2},\,\sigma ) \rightarrow g(\gamma _{1},\,\gamma _{2},\,\sigma ) \in \mathbb{R}\) are C 2 (twice continuously differentiable).
- (SHB2):
-
There exists L 0 > 0 such that \(g(\gamma _{1},\,\gamma _{2},\,\sigma ) < \frac{L_{0}} {L_{0}+1}\) for \(\gamma _{1} \in {\mathbb{R}}^{N},\,\gamma _{2} \in \mathbb{R},\) \(\sigma \in \mathbb{R}\).
In what follows, we write \(\partial _{i}f = \frac{\partial } {\partial \theta _{i}}f\) for i = 1, 2, and similarly we define ∂ i g for i = 1, 2.
We shall study the Hopf bifurcation of (6.74) through its formal linearization. We assume that for a fixed \(\sigma _{0} \in \mathbb{R}\), \((x_{\sigma _{0}},\tau _{\sigma _{0}})\) (or abusing notation, \((x_{\sigma _{0}},\tau _{\sigma _{0}},\,\sigma _{0})\)) is a stationary state of (6.74). That is,
We also assume that
- (SHB3):
-
\(( \frac{\partial } {\partial \theta _{1}} + \frac{\partial } {\partial \theta _{2}} )f(\theta _{1},\,\theta _{2},\,\sigma )\vert _{\sigma =\sigma _{0},\,\theta _{1}=\theta _{2}=x_{\sigma _{0}}}\) is nonsingular and
$$\displaystyle{ \frac{\partial } {\partial \gamma _{2}}g(\gamma _{1},\,\gamma _{2},\,\sigma )\vert _{\sigma =\sigma _{0},\,\gamma _{1}=x_{\sigma _{ 0}},\,\gamma _{2}=\tau _{\sigma _{0}}}\neq 0.}$$
This assumption implies that there exist ε 0 > 0 and a C 1-smooth curve \((\sigma _{0} -\epsilon _{0},\,\sigma _{0} +\epsilon _{0}) \ni \sigma \mapsto (x_{\sigma },\,\tau _{\sigma }) \in {\mathbb{R}}^{N+1}\) such that (x σ , τ σ ) is the unique stationary state of (6.74) in a small neighborhood of \((x_{\sigma _{0}},\,\tau _{\sigma _{0}})\) for σ close to σ 0.
We now consider, for \(\sigma \in (\sigma _{0} -\epsilon _{0},\,\sigma _{0} +\epsilon _{0})\), the following formal linearization of system (6.74) at the stationary point \(\eta (\sigma ) = (x_{\sigma },\,z_{\sigma })\):
where
Then we obtain the following characteristic equation of the linear system corresponding to the inhomogeneous linear system (6.76):
where \(\Delta _{(x_{\sigma },\,\tau _{\sigma },\,\sigma )}(\omega )\) is an (N + 1) ×(N + 1) complex matrix defined by
A solution ω 0 to the characteristic equation (6.77) is called a characteristic value of the stationary state \((x_{\sigma _{0}},\,\tau _{\sigma _{0}},\,\sigma _{0})\). We call \((x_{\sigma _{0}},\,\tau _{\sigma _{0}},\,\sigma _{0})\) a nonsingular stationary state if and only if zero is not a characteristic value of \((x_{\sigma _{0}},\,\tau _{\sigma _{0}},\,\sigma _{0})\). Here \((x_{\sigma _{0}},\,\tau _{\sigma _{0}},\,\sigma _{0})\) is a center if the set of nonzero purely imaginary characteristic values of \((x_{\sigma _{0}},\,\tau _{\sigma _{0}},\,\sigma _{0})\) is nonempty and discrete. We call \((x_{\sigma _{0}},\,\tau _{\sigma _{0}},\,\sigma _{0})\) an isolated center if it is the only center in some neighborhood of \((x_{\sigma _{0}},\,\tau _{\sigma _{0}},\,\sigma _{0})\) in \({\mathbb{R}}^{N+1} \times \mathbb{R}\).
If \((x_{\sigma _{0}},\,\tau _{\sigma _{0}},\,\sigma _{0})\) is an isolated center of (6.76), then there exist β 0 > 0 and δ ∈ (0, ε 0) such that
for every \(\sigma \in (\sigma _{0}-\delta,\,\sigma _{0}) \cup (\sigma _{0},\,\sigma _{0}+\delta )\) and β ∈ (0, + ∞). Hence, we can choose constants \(\alpha _{0} =\alpha _{0}(\sigma _{0},\,\beta _{0}) > 0\) and \(\varepsilon =\varepsilon (\sigma _{0},\,\beta _{0}) > 0\) such that the closure of the set \(\varOmega \stackrel{def}{=} (0,\,\alpha _{0}) \times (\beta _{0}-\varepsilon,\,\beta _{0}+\varepsilon ) \subset {\mathbb{R}}^{2}\mathop{\cong}\mathbb{C}\) contains no other zero of \(\det \Delta _{(x_{\sigma _{ 0}},\,\tau _{\sigma _{0}},\,\sigma _{0})}(\cdot )\). The quantity p 0 = 2π ∕ β 0 is called the virtual period associated with the center \((x_{\sigma _{0}},\,\tau _{\sigma _{0}},\,\sigma _{0})\). We note that \(\det \Delta _{(x_{\sigma },\,\tau _{\sigma },\,\sigma )}(\omega )\) is analytic in ω and is continuous in σ. If δ > 0 is small enough, then there is no zero of \(\det \Delta _{(x_{\sigma _{ 0}\pm \delta },\,\tau _{\sigma _{0}\pm \delta },\,\sigma _{0}\pm \delta )}(\omega )\) in ∂Ω. So we can define the number
and the crossing number of \((x_{\sigma _{0}},\,\tau _{\sigma _{0}},\,\sigma _{0},\,\beta _{0})\) as
This crossing number counts the number of characteristic values (with multiplicities) escaping from the region Ω as α increases and crosses α 0. Define the function \(H : [\sigma _{0}-\delta,\,\sigma _{0}+\delta ] \times \overline{\varOmega } \rightarrow {\mathbb{R}}^{2} \simeq \mathbb{C}\) by
and
where Ψ H : \(\mathcal{D}(\sigma _{0},\,\beta _{0}) \rightarrow {\mathbb{R}}^{2} \simeq \mathbb{C}\) is defined by \(\varPsi _{H}(\sigma,\beta ) =\det \Delta _{(x_{\sigma },\,\tau _{\sigma },\,\sigma )}(\mathrm{i}\beta )\) and \(\mathcal{D}(\sigma _{0},\,\beta _{0}) = (\sigma _{0}-\delta,\,\sigma _{0}+\delta ) \times (\beta _{0} - c,\,\beta _{0} + c)\).
Let \(\mathbb{E}\stackrel{def}{=} C_{2\pi }(\mathbb{R};\, {\mathbb{R}}^{n})\) be the normed space of continuous 2π-periodic functions from \(\mathbb{R}\) to \({\mathbb{R}}^{n}\) equipped with the usual supremum norm. Then \({\mathbb{S}}^{1}\) acts on \(\mathbb{E}\) by argument shift. Namely, for \(\xi =\mathrm{ {e}}^{\mathrm{i}\nu } \in {\mathbb{S}}^{1}\), \(u \in \mathbb{E}\), \((\xi u)(t)\stackrel{def}{=} u(t+\nu )\). For the isotypical direct sum decomposition (6.12) of \(\mathbb{E}\), we see that \(\mathbb{E}_{0}\mathop{\cong}{\mathbb{R}}^{n}\) and for each k ≥ 1, \(\mathbb{E}_{k}\) is spanned by \(\cos (kt)\varepsilon _{j}\) and \(\sin (kt)\varepsilon _{j}\), 1 ≤ j ≤ n, where \(\{\varepsilon _{1}, \ldots,\varepsilon _{n}\}\) is the standard basis of \({\mathbb{R}}^{n}\). Therefore, \(\mathbb{E}_{k}\), k ≥ 0, are real 2n-dimensional and so satisfy (SD1) of Sect. 6.5. To formulate the Hopf bifurcation problem as a fixed-point problem in the space of continuous functions of period 2π, we normalize the period of the 2π ∕ β-periodic solution (x, τ) in (6.74) by (x(t), τ(t)) = (y(βt), z(βt)) and obtain
where u = (y, z)T and \((\sigma,\,\beta ) \in \mathcal{D}(\sigma _{0},\,\beta _{0})\), and
Correspondingly, (6.76) is transformed into
where \(\tilde{Q}\): \(\mathbb{E} \times \mathcal{D}(\sigma _{0},\,\beta _{0}) \rightarrow \mathbb{E}\) is defined by
and \((y_{\sigma },\,z_{\sigma }) =\eta (\sigma ) = (x_{\sigma },\,\tau _{\sigma })\).
Before we state and prove the local Hopf bifurcation theorem, we need some technical preparations. We denote by \(C_{2\pi }^{1}(\mathbb{R};\, {\mathbb{R}}^{N+1})\) the Banach space of 2π-periodic and continuously differentiable functions equipped with the C 1 norm
Lemma 6.14.
Let L: \(C_{2\pi }^{1}(\mathbb{R};\, {\mathbb{R}}^{N+1}) \rightarrow \mathbb{E}\) and K: \(\mathbb{E} \rightarrow {\mathbb{R}}^{N+1}\) be defined by
for \(t \in \mathbb{R}\) . Then L + K has a compact inverse \({(L + K)}^{-1} : \mathbb{E} \rightarrow \mathbb{E}\) , which is given by
This lemma can be found in [170] and is omitted here. It follows from Lemma 6.14 that \({(L + K)}^{-1} \circ [Q(\cdot,\alpha,\beta ) + K] : \mathbb{E} \rightarrow \mathbb{E}\) and \({(L + K)}^{-1} \circ [\tilde{Q}(\cdot,\alpha,\beta ) + K] : \mathbb{E} \rightarrow \mathbb{E}\) are completely continuous. That is, (SD2) and (SD4) are satisfied. Thus, finding a 2π ∕ β-periodic solution for the system (6.74) is equivalent to finding a solution of the following fixed-point problem:
where \((u,\,\sigma,\,\beta ) \in \mathbb{E} \times \mathbb{R} \times \mathbb{R}_{+}\). Define the maps \(\mathcal{F} : \mathbb{E} \times \mathbb{R} \times \mathbb{R}_{+} \rightarrow \mathbb{E}\) and \(\tilde{\mathcal{F}} : \mathbb{E} \times \mathbb{R} \times \mathbb{R}_{+} \rightarrow \mathbb{E}\) by
which are equivariant compact fields. Finding a 2π ∕ β-periodic solution of system (6.74) is equivalent to finding the solution of the problem
It is an easy exercise to verify the following results.
Lemma 6.15.
For \(\sigma \in \mathbb{R}\) and β > 0, the map \(Q(\cdot,\,\sigma,\,\beta ) : \mathbb{E} \rightarrow \mathbb{E}\) defined by (6.82) is continuous.
Lemma 6.16.
If system (6.76) has a nonconstant periodic solution with period T > 0, then there exists an integer \(m \geq 1,\,m \in \mathbb{N}\) such that ± i m 2π∕T are characteristic values of the stationary state \((x_{\sigma },\,\tau _{\sigma },\,\sigma )\).
Lemma 6.17.
Assume (SHB1)–(SHB3) hold. If \(B_{M}(u_{0},\,\sigma _{0},\,\beta _{0};r,\,\rho ) \subseteq \mathbb{E} \times {\mathbb{R}}^{2}\) is a special neighborhood of \(\tilde{\mathcal{F}}\) , where 0 < ρ < β 0 , then there exists r′∈ (0, r] such that the neighborhood
satisfies \(\dot{u}(t)\not\equiv Q(u,\sigma,\beta )\) for \((u,\,\sigma,\,\beta ) \in \overline{B_{M}(u_{0},\,\sigma _{0},\,\beta _{0};r^{\prime},\,\rho )}\) with u = (y,z) T ≠ η(σ) and \(\vert (\sigma,\,\beta ) - (\sigma _{0},\beta _{0})\vert =\rho\).
Proof.
Suppose not. Then for all 0 < r′ ≤ r, there exists (u, σ, β) such that \(0 <\| u -\eta (\sigma )\| < r^{\prime},\,\vert (\sigma,\,\beta ) - (\sigma _{0},\,\beta _{0})\vert =\rho\) and \(\dot{u}(t) = Q(u,\sigma,\beta )\) for \(t \in \mathbb{R}\). Then there exists a sequence of nonconstant periodic solutions \(\{(u_{k},\,\sigma _{k},\,\beta _{k}) = (y_{k},\,z_{k},\,\sigma _{k},\,\beta _{k})\}_{k=1}^{\infty }\) such that
and
Note that 0 < ρ < β 0 implies that β k ≥ β 0 − ρ > 0 for every \(k \in \mathbb{N}\). Also, since the sequence \(\{(\sigma _{k},\,\beta _{k})\}_{k=1}^{\infty }\) belongs to a bounded neighborhood of (σ 0, β 0) in \({\mathbb{R}}^{2}\), there exists a subsequence, denoted by \(\{(\sigma _{k},\,\beta _{k})\}_{k=1}^{\infty }\), that converges to \({(\sigma }^{{\ast}}{,\,\beta }^{{\ast}})\), so that \(\vert {(\sigma }^{{\ast}}{,\,\beta }^{{\ast}}) - (\sigma _{0},\,\beta _{0})\vert =\rho\) and β ∗ > 0. Without loss of generality, we denote this sequence by \(\{(\sigma _{k},\beta _{k})\}_{k=1}^{\infty }\). Our strategy here is to show that the system
has a nonconstant periodic solution, which contradicts the assumption that \(u_{0} = {(y_{\sigma _{0}},\,z_{\sigma _{0}})}^{T}\) is the only center of (6.83) in \(\overline{B_{M}(u_{0},\,\sigma _{0},\,\beta _{0};r,\,\rho )}\).
Put
Then we have
where
for all j = 1, 2. We claim that there exists a convergent subsequence of \(\{v_{k}\}_{k=1}^{+\infty }\). Indeed, by (6.85), we know that \(\{(z_{k},\,\beta _{k})\}_{k=1}^{+\infty }\) is uniformly bounded in \(C(\mathbb{R}; \mathbb{R}) \times \mathbb{R}\), and hence \(\lim _{t\rightarrow +\infty }[t -\beta _{k}z_{k}(t)] = +\infty \). Then, we have
Recall that \(\partial _{i}f{(\sigma }^{{\ast}})\) and \(\partial _{i}g{(\sigma }^{{\ast}})\), i = 1, 2, are defined in (6.76). By (6.85), we know that \((y_{\sigma _{k}} + s(y_{k}(t) - y_{\sigma _{k}}),y_{\sigma _{k}} + s(y_{k}(t - z_{k}(t)) - y_{\sigma _{k}}),\,\sigma _{k})\) converges to the stationary state \((x_{{\sigma }^{{\ast}}},\,\tau _{{\sigma }^{{\ast}}}{,\,\sigma }^{{\ast}})\) in \(C(\mathbb{R}; \mathbb{R}) \times \mathbb{R}\) uniformly for all s ∈ [0, 1]. By (SHB1), we know that f(θ 1, θ 2, σ) is C 2 in (θ 1, θ 2) and \(\partial _{1}f(\theta _{1},\,\theta _{2},\,\sigma )\) is C 1 in σ. Also, by (6.85), the sequence \(\{(u_{k},\,\beta _{k},\,\sigma _{k})\}_{k=1}^{+\infty }\) is uniformly bounded in \(C(\mathbb{R}; {\mathbb{R}}^{N+1}) \times {\mathbb{R}}^{2}\). Then we obtain
uniformly for s ∈ [0, 1], j = 1, 2. It is clear from (6.89) that \(\|\partial _{j}f_{k}(\sigma _{k},\,s)\|\) and \(\|\partial _{j}g_{k}(\sigma _{k},\,s)\|\) (j = 1, 2) are all uniformly bounded for all \(k \in \mathbb{N}\) and s ∈ [0, 1]. Then it follows from (6.88) that there exists a constant \(\tilde{L}_{2} > 0\) such that \(\|\dot{v}_{k}\| <\tilde{ L}_{2}\) for every \(k \in \mathbb{N}\). By the Arzelà–Ascoli theorem, there exists a convergent subsequence \(\{v_{k_{j}}\}_{j=1}^{+\infty }\) of \(\{v_{k}\}_{k=1}^{+\infty }\). That is, there exists \({v}^{{\ast}}\in \{ v \in V :\| v\| = 1\}\) such that
By the integral mean value theorem, we have
Therefore, it follows from (6.90) and (6.92) that
It follows from (6.85), (6.89), (6.90), and (6.93) that the right-hand side of (6.88) converges uniformly to the right-hand side of (6.87). Therefore, v ∗ is differentiable and satisfies (6.87). Moreover, we have
Since by (SHB3), the matrix
is nonsingular, v = 0 is the only constant solution of (6.87). Also, we have \({v}^{{\ast}}\in \{ v \in V :\| v\| = 1\}\), \(\|{v}^{{\ast}}\|\neq 0\). Therefore, \(({v}^{{\ast}}(t){,\,\sigma }^{{\ast}}{,\,\beta }^{{\ast}})\) is a nonconstant periodic solution of the linear equation (6.87). Then by Lemma 6.16, \((\eta {(\sigma }^{{\ast}}){,\,\sigma }^{{\ast}}{,\,\beta }^{{\ast}})\) is also a center of (6.83) in \(\overline{B_{M}(u_{0},\,\sigma _{0},\,\beta _{0};r,\,\rho )}\). This contradicts the assumption that \(B_{M}(u_{0},\,\sigma _{0},\,\beta _{0};r,\,\rho )\) is a special neighborhood of (6.82). This completes the proof. □
As preparation for the proof of the local Hopf bifurcation theorem, we need the following lemma.
Lemma 6.18.
Assume that (SHB1)–(SHB3) hold. If \(\mathcal{U} = B_{M}(u_{0},\,\sigma _{0},\,\beta _{0};r,\,\rho )\) \(\subseteq \mathbb{E} \times {\mathbb{R}}^{2}\) is a special neighborhood of \(\tilde{\mathcal{F}}\) with 0 < ρ < β 0 , then there exists r′∈ (0, r] such that \(\mathcal{F}_{\theta } = (\mathcal{F},\,\theta )\) and \(\tilde{\mathcal{F}}_{\theta } = (\tilde{\mathcal{F}},\,\theta )\) are homotopic on \(\overline{B_{M}(u_{0},\,\sigma _{0},\,\beta _{0};r^{\prime},\,\rho )}\) , where θ is a completing function defined on \(\overline{B_{M}(u_{0},\,\sigma _{0},\,\beta _{0};r^{\prime},\,\rho )}\).
Proof.
Since \(\mathcal{U} = B_{M}(u_{0},\,\sigma _{0},\,\beta _{0};r,\,\rho )\) \(\subseteq \mathbb{E} \times {\mathbb{R}}^{2}\) is a special neighborhood of \(\tilde{\mathcal{F}}\) with 0 < ρ < β 0, then by Lemma 6.17, both \(\mathcal{F}_{\theta } = (\mathcal{F},\,\theta )\) and \(\tilde{\mathcal{F}}_{\theta } = (\mathcal{F},\,\theta )\) are \(\mathcal{U}\)-admissible.
Suppose that the conclusion is not true. Then for every r′ ∈ (0, r], \(\mathcal{F}_{\theta } = (\mathcal{F},\,\theta )\) and \(\tilde{\mathcal{F}}_{\theta } = (\mathcal{F},\,\theta )\) are not homotopic on \(\overline{B_{M}(u_{0},\,\sigma _{0},\,\beta _{0};r^{\prime},\,\rho )}\). That is, every homotopy map between \(\mathcal{F}_{\theta }\) and \(\tilde{\mathcal{F}}_{\theta }\) has a zero on the boundary of \(\overline{B_{M}(u_{0},\,\sigma _{0},\,\beta _{0};r^{\prime},\,\rho )}\). In particular, the linear homotopy \(h(\cdot,\,\alpha )\stackrel{def}{=} \alpha \mathcal{F}_{\theta } + (1-\alpha )\tilde{\mathcal{F}}_{\theta } = (\alpha \mathcal{F} + (1-\alpha )\tilde{\mathcal{F}},\theta )\) has a zero on the boundary of \(\overline{B_{M}(u_{0},\,\sigma _{0},\,\beta _{0};r^{\prime},\,\rho )}\), where α ∈ [0, 1].
Note that θ(u, σ, β) < 0 if \(\|u -\eta (\sigma )\| = r^{\prime}\). Then there exist (u, σ, β) and α ∈ [0, 1] such that \(\|u -\eta (\sigma )\| < r^{\prime},\,\vert (\sigma,\,\beta ) - (\sigma _{0},\,\beta _{0})\vert =\rho\) and
Since r′ > 0 is arbitrary in the interval (0, r], there exists a nonconstant sequence \(\{(y_{k},\,z_{k},\,\sigma _{k},\,\beta _{k},\,\alpha _{k})\}_{k=1}^{\infty }\) of solutions of (6.94) such that
and
Note that 0 < ρ < β 0 implies that β k ≥ β 0 − ρ > 0 for every \(k \in \mathbb{N}\). From (6.95), we know that \(\{(\sigma _{k},\,\beta _{k},\,\alpha _{k})\}_{k=1}^{\infty }\) belongs to a compact subset of \({\mathbb{R}}^{3}\). Therefore, there exist a convergent subsequence, denoted still by \(\{(\sigma _{k},\,\beta _{k},\,\alpha _{k})\}_{k=1}^{\infty }\) without loss of generality, and \({(\sigma }^{{\ast}}{,\,\beta }^{{\ast}}{,\,\alpha }^{{\ast}}) \in {\mathbb{R}}^{3}\) such that \({\beta }^{{\ast}}\geq \beta _{0}-\rho > 0\), α ∗ ∈ [0, 1] and
Similarly to the proof of Lemma 6.17, we can show that the system
with \(\partial _{i}f{(\sigma }^{{\ast}}),\,\partial _{i}g{(\sigma }^{{\ast}})\), i = 1, 2, defined after (6.76), has a nonconstant periodic solution, which contradicts the assumption that \(B_{M}(u_{0},\,\sigma _{0},\,\beta _{0};r,\,\rho )\) is a special neighborhood that contains an isolated center of (6.83). This completes the proof. □
Now we are able to state and prove the local Hopf bifurcation theorem.
Theorem 6.15.
Assume that (SHB1)–(SHB3) hold. Let \((x_{\sigma _{0}},\,\tau _{\sigma _{0}},\,\sigma _{0})\) be an isolated center of system (6.76) . If the crossing number defined by (6.80) satisfies
then there exists a bifurcation of nonconstant periodic solutions of (6.74) near \((x_{\sigma _{0}},\tau _{\sigma _{0}},\,\sigma _{0})\) . More precisely, there exists a sequence \(\{(x_{n},\,\tau _{n},\,\sigma _{n},\beta _{n})\}\) such that σ n →σ 0 , β n →β 0 as n →∞, and \(\lim _{n\rightarrow \infty }\|x_{n} - x_{\sigma _{0}}\| = 0\), \(\lim _{n\rightarrow \infty }\|\tau _{n} -\tau _{\sigma _{0}}\| = 0\) , where
is a nonconstant 2π∕β n -periodic solution of system (6.74).
Proof.
By (SHB1), we know that the linear operator \(\tilde{Q}\) is continuous. By Lemma 6.15, we know that Q( ⋅, σ, β): \(\mathbb{E} \rightarrow \mathbb{E}\) is continuous. Moreover, as stated before, by Lemma 6.14, (SD2) and (SD4) are satisfied. Since \((u_{0},\,\sigma _{0})\stackrel{def}{=} (x_{\sigma _{0}},\,\tau _{\sigma _{0}},\,\sigma _{0})\) is an isolated center of system (6.76) with a purely imaginary characteristic value iβ 0, β 0 > 0, \((u_{0},\,\sigma _{0},\,\beta _{0}) \in \mathbb{E} \times \mathbb{R} \times (0,\,+\infty )\) is an isolated \(\mathbb{E}\)-singular point of \(\tilde{\mathcal{F}}\). One can define the following two-dimensional submanifold \(M \subset {\mathbb{E}}^{G} \times \mathbb{R} \times (0,\,+\infty )\) by
such that the point \((\eta (\sigma _{0}),\,\sigma _{0},\,\beta _{0})) = (u_{0},\,\sigma _{0},\,\beta _{0})\) is the only \(\mathbb{E}\)-singular point of \(\tilde{\mathcal{F}}\) in M, which is the set of trivial solutions to the system (6.76); it satisfies assumption (SD3).
Moreover, \((u_{0},\,\sigma _{0},\,\beta _{0}) \in \mathbb{E} \times \mathbb{R} \times (0,\,+\infty )\) is an isolated \(\mathbb{E}\)-singular point of \(\tilde{\mathcal{F}}\). That is, for ρ > 0 sufficiently small, the linear operator \(D_{u}\tilde{\mathcal{F}}(\eta (\sigma ),\,\sigma,\,\beta ) : \mathbb{E} \rightarrow \mathbb{E}\) with | (σ, β) − (σ 0, β 0) | < ρ is not an isomorphism only if (σ, β) = (σ 0, β 0). Then by the implicit function theorem, there exists r > 0 such that for all \((u,\,\sigma,\,\beta ) \in \mathbb{E} \times \mathbb{R} \times (0,\,+\infty )\) with \(\vert (\sigma,\,\beta ) - (\sigma _{0},\,\beta _{0})\vert =\rho\) and \(0 <\| u -\eta (\sigma )\| \leq r\), we have \(\tilde{\mathcal{F}}(u,\,\sigma,\,\beta )\neq 0\). Then the set \(B_{M}(u_{0},\,\sigma _{0},\,\beta _{0};r,\rho )\) defined by
is a special neighborhood for \(\tilde{\mathcal{F}}\). By Lemma 6.17, there exists a special neighborhood \(\mathcal{U} = B_{M}(u_{0},\,\sigma _{0},\,\beta _{0};r^{\prime},\rho )\) such that \(\mathcal{F}\) and \(\tilde{\mathcal{F}}\) are nonzero for \((u,\,\sigma,\,\beta ) \in \overline{B_{M}(u_{0},\,\sigma _{0},\,\beta _{0};r^{\prime},\rho )}\) with u ≠ η(σ) and \(\vert (\sigma,\,\beta ) - (\sigma _{0},\,\beta _{0})\vert =\rho\). That is, (SD5) is satisfied.
Let θ be a completing function on \(\mathcal{U}\). It follows from Lemma 6.18 that \((\mathcal{F},\,\theta )\) is homotopic to \((\tilde{\mathcal{F}},\,\theta )\) on \(\mathcal{U}\).
For \((\sigma,\,\beta ) \in \mathcal{D}(\sigma _{0},\,\beta _{0})\), we denote by Ψ(σ, β) the map \(D_{u}\tilde{\mathcal{F}}(u(\sigma ),\,\sigma,\,\beta ) : \mathbb{E} \rightarrow \mathbb{E}\). It is easy to see that \(\varPsi (\sigma,\,\beta )(\mathbb{E}_{k}) \subset \mathbb{E}_{k}\) for all k = 0, 1, 2, ⋯ . Define \(\varPsi _{k} : \mathcal{D}(\sigma _{0},\,\beta _{0}) \rightarrow L(\mathbb{E}_{k},\, \mathbb{E}_{k})\) by
Thus, the matrix representation [Ψ k ] of Ψ k (σ, β) \(\{\mathrm{{e}}^{\mathrm{i}k\cdot }\epsilon _{j}\}_{j=1}^{N+1}\) is given by
For the application of Theorem 6.7, we now show that there exists some \(k \in \mathbb{Z}\), k ≥ 1, such that \(\epsilon _{0}\mu _{k}(u(\sigma _{0}),\,\sigma _{0},\beta _{0}) =\epsilon _{0}\deg _{B}(\det _{\mathbb{C}}[\varPsi _{k}])\neq 0\), where \(\epsilon _{0} =\mathrm{ sgn}\det \varPsi _{0}(\sigma,\,\beta )\) for \((\sigma,\beta ) \in \mathcal{D}(\sigma _{0},\,\beta _{0})\). For a constant map \(v_{0} \in \mathbb{E}_{0}\),
Then by (SHB3), we have ε 0 ≠ 0, and therefore (SD6) is satisfied. In view of (6.81), we have
which by Theorem 6.7, implies that \((u(\sigma _{0}),\,\sigma _{0},\,\beta _{0})\) is a bifurcation point of the system (6.82). Consequently, there exists a sequence of nonconstant periodic solutions \((u_{n},\,\sigma _{n},\,\beta _{n}) = (x_{n},\,\tau _{n},\,\sigma _{n},\,\beta _{n})\) such that σ n → σ 0, β n → β 0 as n → ∞, and (x n (t), τ n (t)) is a 2π ∕ β n -periodic solution of (6.74) such that \(\lim _{n\rightarrow +\infty }\|(x_{n},\,\tau _{n}) - (x_{\sigma _{0}},\,\tau _{\sigma _{0}})\| = 0\). □
Remark 6.2.
A local Hopf bifurcation theory for FDEs with state-dependent delays was developed by Eichmann [86], where the existence of a local Hopf bifurcation is guaranteed by a transversality condition. This transversality implies that the crossing number defined by (6.80) is not zero, and hence the existence of a local Hopf bifurcation is also established in Theorem 6.15. Note that even in the case of a constant delay, one can have nontrivial crossing number while the transversality condition is not satisfied. Note also that the work of Eichmann gives more information about the local Hopf bifurcation such as smoothness of the bifurcation curve with respect to the parameter.
6.9.2 Global Bifurcation
To use Theorem 6.8 to describe the maximal continuation of bifurcated periodic solutions with large amplitudes when the bifurcation parameter σ is far away from the bifurcation value, we need to prove that there is a lower bound for the periods of periodic solutions of system (6.74).
Lemma 6.19 (Vidossich [287]).
Let X be a Banach space and \(v : \mathbb{R} \rightarrow X\) a p-periodic function with the following properties:
-
(i)
\(v \in L_{loc}^{1}(\mathbb{R},\,X)\);
-
(ii)
there exists \(U \in {L}^{1}([0,\, \frac{\mathsf{p}} {2}]; \mathbb{R}_{+})\) such that |v(t) − v(s)|≤ U(t − s) for almost every (in the sense of the Lebesgue measure) \(s,\,t \in \mathbb{R}\) such that s ≤ t, \(t - s \leq \frac{\mathsf{p}} {2};\)
-
(iii)
\(\int _{0}^{\mathsf{p}}v(t)\,dt = 0\).
Then
We make the following assumption on system (6.74):
- (SHB4):
-
There exist constants L f > 0, L g > 0 such that
$$\displaystyle\begin{array}{rcl} & & \vert f(\theta _{1},\,\theta _{2},\,\sigma ) - f(\overline{\theta }_{1},\,\overline{\theta }_{2},\,\sigma )\vert \leq L_{f}(\vert \theta _{1} -\overline{\theta }_{1}\vert + \vert \theta _{2} -\overline{\theta }_{2}\vert ) {}\\ & & \vert g(\gamma _{1},\,\gamma _{2},\,\sigma ) - g(\overline{\gamma }_{1},\,\overline{\gamma }_{2},\,\sigma )\vert \leq L_{g}(\vert \gamma _{1} -\overline{\gamma }_{1}\vert + \vert \gamma _{2} -\overline{\gamma }_{2}\vert ) {}\\ \end{array}$$for every \(\theta _{1},\,\theta _{2},\,\overline{\theta }_{1},\,\overline{\theta }_{2},\,\gamma _{1},\,\overline{\gamma }_{1} \in {\mathbb{R}}^{N},\,\gamma _{2},\,\overline{\gamma }_{2} \in \mathbb{R},\,\sigma \in \mathbb{R}\).
Lemma 6.20.
Assume that system (6.74) satisfies the assumption (SHB4) . If u = (x, τ) is a nonconstant periodic solution of (1.1), then the minimal period of u satisfies
Moreover, suppose g(x, τ, σ) satisfies
- (SHB5):
-
for every \(\sigma \in \mathbb{R}\) , there exists L 0 > 0 such that − L 0 ≤ g(x, τ, σ) < 1 for all \((x,\,\tau ) \in {\mathbb{R}}^{N+1}\).
Then the minimal period p of u satisfies
Proof.
Let \(v(t) =\dot{ u}(t)\). Then \(\int _{0}^{\mathsf{p}}v(t)dt = 0\), since u(t) is a p-periodic solution. For s ≤ t, by (SHB4) and the integral mean value theorem, we have
Let
Then by Lemma 6.19, we obtain
Therefore,
Moreover, if − L 0 ≤ g(x(t), τ(t), σ) < 1, then
and hence
□
The following result was first established by Mallet-Paret and Yorke [216] for ordinary differential equations and was extended to neutral equations by Wu [301].
Lemma 6.21.
Suppose that system (6.74) satisfies (SHB1)–(SHB2) and (SHB4)–(SHB5) . Assume further that there exists a sequence of real numbers \(\{\sigma _{k}\}_{k=1}^{\infty }\) such that:
-
(i)
For each k, system (6.74) with σ = σ k has a nonconstant periodic solution \(u_{k} = (x_{k},\,\tau _{k}) \in C(\mathbb{R}; {\mathbb{R}}^{N+1})\) with minimal period T k > 0;
-
(ii)
\(\lim \limits _{k\rightarrow \infty }\sigma _{k} =\sigma _{0} \in \mathbb{R}\), \(\lim \limits _{k\rightarrow \infty }T_{k} = T_{0} < \infty \) , and \(\lim \limits _{k\rightarrow \infty }\|u_{k} - u_{0}\| = 0\) , where \(u_{0} : \mathbb{R} \rightarrow {\mathbb{R}}^{N+1}\) is a constant map with the value \((x_{0},\,\tau _{0})\).
Then (u 0 , σ 0 ) is a stationary state of (6.74) , and there exists \(m \geq 1,\,m \in \mathbb{N}\) such that \(\pm \mathrm{i}m\,2\pi /T_{0}\) are the roots of the characteristic equation (6.77) with σ = σ 0
Proof.
By Lemma 6.20, we conclude that \(T_{k} \geq \frac{4} {\max \{L_{0},1\}+2(L_{f}+L_{g})}\) and therefore \(T_{0} \geq \frac{4} {\max \{L_{0},1\}+2(L_{f}+L_{g})} > 0\).
Now we show that (u 0, σ 0) is a stationary state of (6.74). Since by (ii), \(\lim \limits _{k\rightarrow \infty }\sigma _{k} =\sigma _{0}\) and \(\lim \limits _{k\rightarrow \infty }\|u_{k} - u_{0}\| = 0\), we have only to show that the derivatives \(\{\dot{u}_{k}\}_{k=1}^{+\infty }\) converge uniformly to the right-hand side of system (6.74). That is,
as k → + ∞. Note that we have used \(f(x_{k},x_{k}(\cdot -\tau _{k}),\sigma _{k})\) to denote the function \(f(x_{k}(\cdot ),x_{k}(\cdot -\tau _{k}),\sigma _{k})\). In fact, it follows from (SHB1) and assumption (ii) that \(\lim \limits _{k\rightarrow \infty }\|g(x_{k},\,\tau _{k},\sigma _{k}) - g(x_{0},\,\tau _{0},\sigma _{0})\| = 0\). Moreover, by the integral mean value theorem, we have \(\|f(x_{k},\,x_{k}(\cdot -\tau _{k}),\sigma _{k}) - f(x_{0},\,x_{0},\sigma _{0})\| \rightarrow 0\) as k → + ∞. This completes the proof of (6.97). Therefore, \((u_{0},\,\sigma _{0}) = (x_{\sigma _{0}},\,\tau _{\sigma _{0}},\,\sigma _{0})\) is the stationary state of (6.74) with σ = σ 0.
Next, we show that the linear system
has a nonconstant periodic solution.
For ρ ∈ (0, 1), define
Then \(\|v_{k}\| = 1\), and \(v_{k}(t)\stackrel{def}{=} (y_{k}(t),\,z_{k}(t))\) satisfies
where
Using the integral mean value theorem, we can show that | δ 1 k (t) | → 0, | δ 2 k (t) | → 0 as k → + ∞ uniformly for \(t \in \mathbb{R}\). This, together with the fact that \(\|v_{k}\| = 1\), implies that there exists \(\tilde{L}_{6} > 0\) such that \(\|\dot{v}_{k}\| \leq \tilde{ L}_{6}\) for all \(k \in \mathbb{N}\). Also, by assumption (ii), the set of periods \(\{T_{k}\}_{k=1}^{+\infty }\) is bounded. Then by the Arzelà–Ascoli theorem, \(\{v_{k}\}_{k=1}^{+\infty }\) has a convergent subsequence, denoted by \(\{v_{k_{j}}\}_{j=1}^{+\infty }\). Let
Then v ρ is a periodic solution of (6.98) with period T 0. Since \(\|v_{k}\| = 1\) and the average value of each v k is zero, the same is true for v ρ . So v ρ is a nonconstant T 0-periodic solution of (6.98). Then by Lemma 6.16, there exists \(m \geq 1,\,m \in \mathbb{N}\), such that ± im 2π ∕ T 0 are characteristic values of (6.77). This completes the proof. □
Now we can describe the relation between 2π ∕ β k and the minimal period of u k in Theorem 6.15.
Theorem 6.16.
Assume that (SHB1)–(SHB5) hold. In Theorem 6.15 , every limit point of the minimal period of \(u_{k} = (x_{k},\,\tau _{k})\) as k → +∞ is contained in the set
Moreover, if ± i m nβ 0 are not characteristic values of (u 0 , σ 0 ) for any integers \(m,\,n \in \mathbb{N}\) such that m n > 1, then 2π∕β k is the minimal period of u k (t) and 2π∕β k → 2π∕β 0 as k →∞.
Proof.
Let T k denote the minimal period of u k (t). Then there exists a positive integer n k such that \(2\pi /\beta _{k} = n_{k}T_{k}\). Since \(T_{k} \leq 2\pi /\beta _{k} \rightarrow 2\pi /\beta _{0}\) as k → ∞, there exist a subsequence \(\{T_{k_{j}}\}_{j=1}^{\infty }\) and T 0 such that \(T_{0} =\lim _{j\rightarrow \infty }T_{k_{j}}\). Since \(2\pi /\beta _{k_{j}} \rightarrow 2\pi /\beta _{0}\), \(T_{k_{j}} \rightarrow T_{0}\) as j → ∞, \(n_{k_{j}}\) is identical to a constant n for k large enough. Therefore, \(2\pi /\beta _{0} = nT_{0}\). Thus \(T_{k_{j}} \rightarrow 2\pi /(n\beta _{0})\) as j → ∞. By Lemma 6.21, \(\pm \mathrm{i}m\,2\pi /T_{0} = \pm \mathrm{i}m\,n\beta _{0}\) are characteristic values of (u 0, σ 0) for some \(m \geq 1,\,m \in \mathbb{N}\).
Moreover, if ± i m nβ 0 are not characteristic values of (u 0, σ 0) for any integers \(m \in \mathbb{N}\) and \(n \in \mathbb{N}\) with mn > 1, then m = n = 1. Therefore, for k large enough, \(n_{k_{j}} = 1\) and \(2\pi /\beta _{k} = T_{k}\) is the minimal period of u k (t) and 2π ∕ β k → 2π ∕ β 0 as k → ∞. This completes the proof. □
The following lemma shows that we can locate all possible Hopf bifurcation points of system (6.74) with state-dependent delay at the centers of its corresponding formal linearization.
Lemma 6.22.
Assume that (SHB1)–(SHB3) hold. If (u 0 , σ 0 ) is a Hopf bifurcation point of system (6.74) , then it is a center of (6.76).
Proof.
If (u 0, σ 0) is a Hopf bifurcation point of system (6.74), then there exist a sequence \(\{(u_{k},\,\sigma _{k},\,T_{k})\}_{k=1}^{+\infty }\subseteq C(\mathbb{R}; {\mathbb{R}}^{N+1}) \times {\mathbb{R}}^{2}\) and T 0 ≥ 0 such that \(\lim _{k\rightarrow +\infty }\|(u_{k},\,\sigma _{k},\,T_{k}) - (u_{0},\,\sigma _{0},\,T_{0})\| = 0\), where (u k , σ k ) is a nonconstant T k -periodic solution of system (6.74). Using a similar argument to that in the proof of Lemma 6.17, we see that the system
has a nonconstant periodic solution v ∗ . Therefore, \(({v}^{{\ast}} + u_{0},\,\sigma _{0})\) is a nonconstant periodic solution of (6.76). Then, by Lemma 6.16, (u 0, σ 0) is a center of (6.76). □
Now we are able to consider the global Hopf bifurcation problem of system (6.74). Letting \((x(t),\tau (t)) = (y(\frac{2\pi } {\mathsf{p}} t),\,z(\frac{2\pi } {\mathsf{p}} t))\), we can reformulate the problem as a problem of finding 2π-periodic solutions to the following equation:
where u(t) = (y(t), z(t)). Accordingly, the formal linearization (6.76) becomes
Using the same notation as in the proof of Theorem 6.15, we can define \(\mathcal{N}_{0}(u,\,\sigma,\,\mathsf{p}) = Q(u,\,\sigma,\,2\pi /\mathsf{p}),\tilde{\mathcal{N}}_{0}(u,\,\sigma,\,\mathsf{p}) =\tilde{ Q}(u,\,\sigma,\,2\pi /\mathsf{p})\). Then the system
is equivalent to (6.100), and
is equivalent to (6.101). Let \(\mathcal{S}\) denote the closure of the set of all nontrivial periodic solutions of system (6.102) in the space \(\mathbb{E} \times \mathbb{R} \times \mathbb{R}_{+}\), where \(\mathbb{R}_{+}\) is the set of all nonnegative real numbers. It follows from Lemma 6.20 that the constant solution (u 0, σ 0, 0) does not belong to this set. Consequently, we can assume that problem (6.102) is well posed on the whole space \(\mathbb{E} \times {\mathbb{R}}^{2}\), in the sense that if \(\mathcal{S}\) exists in \(\mathbb{E} \times {\mathbb{R}}^{2}\), then it must be contained in \(\mathbb{E} \times \mathbb{R} \times \mathbb{R}_{+}\).
On the other hand, assume that (SHB3) holds at every center of (6.103). Then from the proof of Theorem 6.15, we know that the assumptions (SHB1–SHB3) are sufficient for the systems (6.102) and (6.103) to satisfy the conditions (SD1)–(SD6). Also, under the same assumptions, Lemma 6.22 implies (SD7), and Lemma 6.18 implies (SD8). Then by Theorem 6.8, we obtain the following global Hopf bifurcation theorem for system (6.102) with state-dependent delay.
Theorem 6.17.
Suppose that system (6.74) satisfies (SHB1)–(SHB5) and (SHB3) holds at every center of (6.103) . Assume that all the centers of (6.103) are isolated. Let \(\mathcal{M}\) be the set of trivial periodic solutions of (6.102) and suppose that \(\mathcal{M}\) is complete. If \((u_{0},\,\sigma _{0},\,\mathsf{p}_{0}) \in \mathcal{M}\) is a bifurcation point, then either the connected component \(C(u_{0},\,\sigma _{0},\,\mathsf{p}_{0})\) of \((u_{0},\,\sigma _{0},\,\mathsf{p}_{0})\) in \(\mathcal{S}\) is unbounded, or
where \(\mathsf{p}_{i} \in \mathbb{R}_{+}\), \((u_{i},\,\sigma _{i},\,\mathsf{p}_{i}) \in \mathcal{M}\) , i = 0, 1, 2,⋯ ,q. Moreover, in the latter case, we have
where \(\gamma (u_{i},\,\sigma _{i},\,2\pi /\mathsf{p}_{i})\) is the crossing number of \((u_{i},\,\sigma _{i},\,\mathsf{p}_{i})\) defined by (6.80) and \(\epsilon _{i} = \mbox{ $\mathrm{sgn}$}\det \left [\begin{array}{cc} \partial _{1}f(\sigma _{i}) + \partial _{2}f(\sigma _{i})& 0 \\ \partial _{1}g(\sigma _{i}) &\partial _{2}\,g(\sigma _{i}) \end{array} \right ].\)
Definition 6.4.
Let \(\mathcal{C}\) be a connected component of the closure of all nonconstant periodic solutions of (6.74) in the Fuller space \(C(\mathbb{R}; {\mathbb{R}}^{N+1}) \times {\mathbb{R}}^{2}\). We call \(\mathcal{C}\) a continuum of slowly oscillating periodic solutions if for every \((x,\,\tau,\,\sigma,\,p) \in \mathcal{C}\), there exists \(t_{0} \in \mathbb{R}\) such that p > τ(t 0) > 0. Similarly, we call \(\mathcal{C}\) a continuum of rapidly oscillating periodic solutions if for every \((x,\,\tau,\,\sigma,\,p) \in \mathcal{C}\), there exists \(t_{0} \in \mathbb{R}\) such that 0 < p < τ(t 0).
Theorem 6.17 shows that for a given trivial solution \(({x}^{{\ast}}{,\,\tau }^{{\ast}}{,\,\sigma }^{{\ast}})\) with virtual period p ∗ , either the connected component \(C({x}^{{\ast}}{,\,\tau }^{{\ast}}{,\,\sigma }^{{\ast}},\,{p}^{{\ast}})\) has finitely many bifurcation points with the sum of \({\mathbb{S}}^{1}\)-equivariant degrees being zero or \(C({x}^{{\ast}}{,\,\tau }^{{\ast}}{,\,\sigma }^{{\ast}},\,{p}^{{\ast}})\) is unbounded in the Fuller space \(C(\mathbb{R}; {\mathbb{R}}^{N+1}) \times {\mathbb{R}}^{2}\). Therefore, if global persistence of periodic solutions when the parameter is far away from the local Hopf bifurcation value σ ∗ is desired, we should find conditions to ensure that the connected component \(C({x}^{{\ast}}{,\,\tau }^{{\ast}}{,\,\sigma }^{{\ast}},\,{p}^{{\ast}})\) of Hopf bifurcation is unbounded in the Fuller space \(C(\mathbb{R}; {\mathbb{R}}^{N+1}) \times {\mathbb{R}}^{2}\) and \(C({x}^{{\ast}}{,\,\tau }^{{\ast}}{,\,\sigma }^{{\ast}},\,{p}^{{\ast}})\) will not blow up to infinity at any given σ in the norm of the Fuller space \(C(\mathbb{R};\, {\mathbb{R}}^{N+1}) \times {\mathbb{R}}^{2}\). That is, there exists a continuous function \(M : \mathbb{R} \ni \sigma \rightarrow M(\sigma ) > 0\) such that for every \((x,\tau,\sigma,\,p) \in C({x}^{{\ast}}{,\,\tau }^{{\ast}}{,\,\sigma }^{{\ast}},\,{p}^{{\ast}})\), we have
To achieve this goal, we shall give some sufficient geometric conditions ensuring the uniform boundedness of all possible periodic solutions (x, τ, σ) of (6.74), that is, we show that there exists a continuous function \(M_{1} : \mathbb{R} \ni \sigma \rightarrow M_{1}(\sigma ) > 0\) such that for every \((x,\tau,\sigma,\,p) \in C({x}^{{\ast}}{,\,\tau }^{{\ast}}{,\,\sigma }^{{\ast}},\,{p}^{{\ast}})\), we have
Then we seek a continuous function \(M_{2} : \mathbb{R} \ni \sigma \rightarrow M_{2}(\sigma ) > 0\) such that for every \((x,\tau,\sigma,\,p) \in C({x}^{{\ast}}{,\,\tau }^{{\ast}}{,\,\sigma }^{{\ast}},\,{p}^{{\ast}})\), we have
6.9.3 Uniform Bounds for Periods of Periodic Solutions in a Connected Component
In order, by way of contradiction, to exclude certain values of the period of the periodic solutions in a given connected component, we need some analytic properties of an interval map under the following assumptions:
- (SHB6):
-
For every \((\sigma,\,\tau ) \in {\mathbb{R}}^{2}\), \(\frac{\partial g} {\partial \tau } (x_{\sigma },\,\tau,\,\sigma )\neq 0\).
- (SHB7):
-
\(\frac{\partial g} {\partial x}(x,\,\tau,\,\sigma )f(x,\,x,\,\sigma )\neq 0\) for \((x,\,\tau,\,\sigma ) \in {\mathbb{R}}^{N+1} \times \mathbb{R}\) such that x ≠ x σ and g(x, τ, σ) = 0.
In this section, to avoid notational complications, we use superscripts to denote function compositions, e.g., l j(t) denotes the jth composition of l evaluated at time t.
The following result can be found in [171].
Lemma 6.23.
Suppose that (6.74) satisfies (SHB1) – (SHB2) and (SHB6) – (SHB7) and (x, τ, σ 0 ) is a nonconstant periodic solution of (6.74) . If (x, τ) is τ(t 0 )-periodic and if \(\tau (t_{0})\neq \tau _{\sigma _{0}}\) , then the function l(t) = t −τ(t) + τ(t 0 ) defined on \([t_{0},\,t_{0} +\tau (t_{0})]\) satisfies the following properties:
-
(a)
l(t) is a self-mapping on \([t_{0},\,t_{0} +\tau (t_{0})]\).
-
(b)
l(t) has only finitely many fixed points \(\{t_{i}\}_{i=1}^{n}\) in \([t_{0},\,t_{0} +\tau (t_{0})]\) with t i < t i+1 for every i ∈{ 1, 2, ⋯ , n − 1}.
-
(c)
For every \(t \in (t_{i},\,t_{i+1}) \subseteq [t_{0},\,t_{0} +\tau (t_{0})]\),
$$\displaystyle{\lim _{j\rightarrow +\infty }{l}^{j}(t) =\left \{\begin{array}{lc} t_{i}, &\mbox{ if there exists}\ \bar{t} \in [t_{i},\,t_{i+1}]\ \mbox{ such that}\ \bar{t} > l(\bar{t}), \\ t_{i+1},&\mbox{ if there exists}\ \bar{t} \in [t_{i},\,t_{i+1}]\ \mbox{ such that}\ \bar{t} < l(\bar{t}). \end{array} \right.}$$ -
(d)>
Let \(\{t_{i_{k}}\}_{k=1}^{k_{0}} \subseteq \{ t_{i}\}_{i=1}^{n}\) be all the fixed points such that \(\lim _{j\rightarrow +\infty }{l}^{j}(t) = t_{i_{k}}\) for every \(t \in [t_{i_{k}},\,t_{i_{k}+1})\) . Then for δ > 0 small enough,
$$\displaystyle\begin{array}{rcl} & & \lim _{j\rightarrow +\infty }\sup _{t\in [t_{i_{ k}},\,t_{i_{k}+1}-\delta ]}\vert {l}^{j}(t) - t_{ i_{k}}\vert = 0, {}\\ & & \lim _{j\rightarrow +\infty }\sup _{t\in [t_{i}+\delta,\,t_{i+1}],\,t_{i}\in \{t_{1},\,t_{2},\,\cdots \,,t_{n}\}\setminus \{t_{i_{ k}}\}_{k=1}^{k_{0}}}\vert {l}^{j}(t) - t_{ i+1}\vert = 0. {}\\ \end{array}$$ -
(e)
Let h(t) = t −τ(t). Then \({l}^{j}(t) = {h}^{j}(t) + j\tau (t_{0})\) for every \(t \in [t_{0},\,t_{0} +\tau (t_{0})]\) and \(j \in \mathbb{N};\)
-
(f)
\({h}^{j}(t +\tau (t_{0})) = {h}^{j}(t) +\tau (t_{0})\) for all \(t \in \mathbb{R}\) and \(j \in \mathbb{N}\).
Recall that \(C({x}^{{\ast}}{,\,\tau }^{{\ast}}{,\,\sigma }^{{\ast}},\,{p}^{{\ast}})\) denotes the connected component of the closure of all the nonconstant periodic solutions of system (6.74) bifurcated at \(({x}^{{\ast}}{,\,\tau }^{{\ast}}{,\,\sigma }^{{\ast}},\,{p}^{{\ast}})\) in the Fuller space \(C(\mathbb{R};\, {\mathbb{R}}^{N+1}) \times {\mathbb{R}}^{2}\). We hope to exclude, for each periodic solution \((x_{0},\tau _{0},\,\sigma _{0},\,p_{0})\), certain values of the period. To be specific, we find an open interval I and a small open neighborhood \(U \ni (x_{0},\tau _{0},\,\sigma _{0},\,p_{0})\) such that every \((x,\tau,\,\sigma,\,p) \in U \cap C({x}^{{\ast}}{,\,\tau }^{{\ast}}{,\,\sigma }^{{\ast}},\,{p}^{{\ast}})\) satisfies τ(t) ≠ mp for all t ∈ I and \(m \in \mathbb{N}\). Then we will glue up these local exclusions to a global upper bound for the period along the rescaled (by period normalization) connected component \(C({y}^{{\ast}},\,{z}^{{\ast}}{,\,\sigma }^{{\ast}},\,{p}^{{\ast}})\).
We first consider the periods of the solutions in a neighborhood of a periodic solution that does not assume a certain period.
Lemma 6.24.
If a solution \((x_{0},\,\tau _{0},\,\sigma _{0},\,p_{0}) \in C({x}^{{\ast}}{,\,\tau }^{{\ast}}{,\,\sigma }^{{\ast}},\,{p}^{{\ast}})\) satisfies \(\tau _{0}(t_{0})\neq mp_{0}\) for some \(t_{0} \in \mathbb{R}\) and for all \(m \in \mathbb{N}\) , then there exist an open neighborhood I ∋ t 0 and an open neighborhood \(U \ni (x_{0},\,\tau _{0},\,\sigma _{0},\,p_{0})\) in \(C(\mathbb{R};\, {\mathbb{R}}^{N+1}) \times {\mathbb{R}}^{2}\) such that every solution \((x,\,\tau,\,\sigma,\,p) \in U \cap C({x}^{{\ast}}{,\,\tau }^{{\ast}}{,\,\sigma }^{{\ast}},\,{p}^{{\ast}})\) satisfies τ(t) ≠ mp for all \(m \in \mathbb{N}\) and t∈I.
Proof.
By way of contradiction, we suppose that for every open interval I ∋ t 0 and every open neighborhood \(U \ni (x_{0},\,\tau _{0},\,\sigma _{0},\,p_{0})\) in \(C(\mathbb{R};\, {\mathbb{R}}^{N+1}) \times {\mathbb{R}}^{2}\), there exist t ∈ I, \(m \in \mathbb{N}\), and a periodic solution \((x,\,\tau,\,\sigma,\,p) \in U \cap C({x}^{{\ast}}{,\,\tau }^{{\ast}}{,\,\sigma }^{{\ast}},\,{p}^{{\ast}})\) such that τ(t) = mp. Then there exist sequences \(\{(x_{k},\,\tau _{k},\,\sigma _{k},\,p_{k},\,t_{k})\}_{k=1}^{+\infty }\subseteq U \cap C({x}^{{\ast}}{,\,\tau }^{{\ast}}{,\,\sigma }^{{\ast}},\,{p}^{{\ast}})\) and \(\{m_{k} :\, m_{k} \in \mathbb{N}\}_{k=1}^{+\infty }\) such that
Without loss of generality, we assume \(m_{k} \rightarrow m_{0} \in \mathbb{N}\) as k → + ∞ (otherwise, we take a subsequence). Then it follows from (6.107), (SHB2), and (SHB5) that
Therefore, we have \(\tau _{0}(t_{0}) = m_{0}p_{0}\), which is a contradiction to the assumption. □
We note that for a nonconstant periodic solution (x, τ, σ) of system (6.74), it is allowed that τ(t) assume its stationary value τ σ , or even τ(t) = τ σ for all \(t \in \mathbb{R}\). Ruling out these cases turns out to be crucial for us to exclude certain values of periods of the periodic solutions.
Now we consider the periods of the periodic solutions in a neighborhood of a given nonconstant periodic solution in the Fuller space for which the delay τ-component is not equal to the corresponding stationary value at some time t. We need the following condition:
- (SHB8):
-
-
(i)
f(0, 0, σ) = 0 for all \(\sigma \in \mathbb{R}\);
-
(ii)
xf(x, x, σ) is positive (or negative) if f(x, x, σ) ≠ 0.
-
(i)
Theorem 6.18.
Suppose that system (6.74) satisfies (SHB6) – (SHB8) . Let \((x_{0},\,\tau _{0},\) σ 0 , p 0 ) be a nonconstant periodic solution in \(C({x}^{{\ast}}{,\,\tau }^{{\ast}}{,\,\sigma }^{{\ast}},\,{p}^{{\ast}})\) . If \(\tau _{0}(t_{0})\neq \tau _{\sigma _{0}}\) for some t 0 , then there exist an open interval I and an open neighborhood U of \((x_{0},\,\tau _{0},\,\sigma _{0},\,p_{0})\) in \(C(\mathbb{R};\, {\mathbb{R}}^{N+1}) \times {\mathbb{R}}^{2}\) such that every solution (x, τ, σ, p) in \(U \cap C({x}^{{\ast}}{,\,\tau }^{{\ast}}{,\,\sigma }^{{\ast}},\,{p}^{{\ast}})\) satisfies τ(t) ≠ mp for all \(m \in \mathbb{N}\) and t ∈ I.
Proof.
We first show that there exist an open neighborhood U of \((x_{0},\,\tau _{0},\,\sigma _{0},\,p_{0})\) and an open neighborhood I 0 of t 0 such that \(\tau (t)\neq \tau _{\sigma _{0}}\) for every \((x,\,\tau,\,\sigma,\,p) \in U \cap C({x}^{{\ast}}{,\,\tau }^{{\ast}}{,\,\sigma }^{{\ast}},\,{p}^{{\ast}})\) and t ∈ I 0.
By way of contradiction, suppose that for every neighborhood \(\tilde{I}\) of t 0 and neighborhood U of \((x_{0},\,\tau _{0},\,\sigma _{0},\,p_{0})\), there exist \(t \in \tilde{ I}\) and a nonconstant solution \((x,\,\tau,\,\sigma,\,p) \in U \cap C({x}^{{\ast}}{,\,\tau }^{{\ast}}{,\,\sigma }^{{\ast}},\,{p}^{{\ast}})\) such that \(\tau (t) =\tau _{\sigma _{0}}\). Then there exist a sequence of periodic solutions \(\{(x_{k},\,\tau _{k},\,\sigma _{k},\,p_{k})\}_{k=1}^{+\infty }\) and \(\{t_{k}\}_{k=1}^{+\infty }\) such that
This, together with assumption (SHB2), implies that
Therefore, we have
This is a contradiction to the assumption that \(\tau _{0}(t_{0})\neq \tau _{\sigma _{0}}\), and hence the claim is proved.
If \((x_{0},\,\tau _{0},\,\sigma _{0},\,p_{0})\) satisfies \(\tau _{0}(t_{0})\neq mp_{0}\) for all \(m \in \mathbb{N}\), then the existence of I and U is followed from Lemma 6.24. Otherwise, \((x_{0},\,\tau _{0},\,\sigma _{0},\,p_{0})\) is τ 0(t 0)-periodic. Let \(\Gamma _{\sigma _{0}}\) be the nonempty solution set of the equation f(x, x, σ 0) = 0 for \(x \in {\mathbb{R}}^{N}\). Then by (SHB6), for every \(x \in \Gamma _{\sigma _{0}}\), \(\tau _{\sigma _{ 0}}\) is the unique solution of g(x, τ, σ 0) = 0 for \(\tau \in \mathbb{R}\). Now we distinguish two cases:
Case 1. \(x_{0}(t_{0}) = x_{\sigma _{0}}\) for some \(x_{\sigma _{0}} \in \Gamma _{\sigma _{0}}\). Since \(\tau _{0}(t_{0})\neq \tau _{\sigma _{0}}\), by system (6.74) and by (SHB6), we have
Without loss of generality, we suppose \(\dot{\tau }_{0}(t) > 0\) for t in some open neighborhood of t 0. Then, by the continuity and local monotonicity of τ 0(t), there exists δ > 0 small enough that
where p min > 0 is the minimal period of (x 0, τ 0). Then \(\tau _{0}(t)\neq m\,p_{\min }\) for every \(m \in \mathbb{N}\). Therefore, (x 0, τ 0) is not τ 0(t)-periodic for all t ∈ (t 0, t 0 + δ). So we have τ 0(t) ≠ mp 0 for all t ∈ (t 0, t 0 + δ) and \(m \in \mathbb{N}\).
By Lemma 6.24, for every \({t}^{{\ast}}\in (t_{0},\,t_{0}+\delta )\), there exist an open interval I of t ∗ and an open neighborhood U of \((x_{0},\,\tau _{0},\,\sigma _{0},\,p_{0})\) in \(C(\mathbb{R};\, {\mathbb{R}}^{N+1}) \times {\mathbb{R}}^{2}\) such that every solution (x, τ, σ, p) in \(U \cap C({x}^{{\ast}}{,\,\tau }^{{\ast}}{,\,\sigma }^{{\ast}},\,{p}^{{\ast}})\) satisfies τ(t) ≠ mp for all \(m \in \mathbb{N}\) and t ∈ I.
Case 2. \(x_{0}(t_{0})\neq x_{\sigma }\) for every \(x_{\sigma } \in \Gamma _{\sigma _{0}}\). By Lemma 6.23 (c), there are finitely many fixed points \(\{t_{i}\}_{i=1}^{n}\) of \(l(t) = t -\tau _{0}(t) +\tau _{0}(t_{0})\) in \([t_{0},\,t_{0} +\tau _{0}(t_{0})]\) that are in ascending order (we assume in the proof that all the sequences of the fixed points of l are in ascending order). And we can let the subsequence \(\{t_{i_{k}}\}_{k=1}^{k_{0}} \subseteq \{ t_{i}\}_{i=1}^{n}\) be all the fixed points such that \(\lim _{j\rightarrow +\infty }{l}^{j}(t) = t_{i_{k}}\) for every \(t \in [t_{i_{k}},\,t_{i_{k}+1})\). Note that \(\tau _{0}(t_{i}) =\tau _{0}(t_{0})\) and \(\tau _{0}(t_{0})\neq \tau _{\sigma _{0}}\) implies that \(\tau _{0}(t_{i})\neq \tau _{\sigma _{0}}\) for all \(i \in \{ 1,\,2,\,\cdots \,,\,n\}\). If \(x_{0}(t_{i_{0}}) = x_{\sigma _{0}}\) for some \(i_{0} \in \{ 1,\,2,\,\cdots \,,\,n\}\) and for some \(x_{\sigma _{0}} \in \Gamma _{\sigma _{0}}\). Then the conclusion follows by Case 1 with t 0 replaced by \(t_{i_{0}}\).
Now we exclude that \(x_{0}(t_{i})\neq x_{\sigma }\) for every i ∈ { 0, 1, 2, ⋯ , n} and for every \(x_{\sigma } \in \Gamma _{\sigma _{0}}\). Assume that the contrary is true. We want to obtain a contradiction under the assumption that (x 0, τ 0) is τ 0(t 0)-periodic.
For δ > 0 small enough, we consider the following compact subset I δ of \([t_{0},\,t_{0} +\tau _{0}(t_{0})]\):
Note that for each interval [t i , t i + 1], only one of the endpoints is the limit of lim j → + ∞ l j(t) for every \(t \in (t_{i},\,t_{i+1})\). Note also that when δ goes to zero, I δ goes to \([t_{0},\,t_{0} +\tau _{0}(t_{0})]\) in the sense of Lebesgue measure.
Now for δ > 0 small enough, we introduce the following piecewise constant function χ(t) on the compact subset I δ of \([t_{0},\,t_{0} +\tau _{0}(t_{0})]\):
Since the number of intervals with endpoints the fixed points of l(t) is finite, it is clear from Lemma 6.23 (d) that
Note that (x(t), τ(t)) is a periodic solution of system (6.74). There exists \(\tilde{M} > 0\) such that \(\vert \dot{x}(t)\vert \leq \tilde{ M}\) for every \(t \in [t_{0},\,t_{0} +\tau (t_{0})]\). Let I i with \(i \in \{ 1,\,2,\,\cdots \,,n\}\) be the subinterval of I δ that is either [t i − 1, t i − δ] or \([t_{i-1}+\delta,\,t_{i}]\). Then we have χ(t) = t i − 1 or χ(t) = t i for t ∈ I i , and hence we have
Since \(x_{0}(t_{i})\neq x_{\sigma }\) for every i ∈ { 0, 1, 2, ⋯ , n} and for every \(x_{\sigma } \in \Gamma _{\sigma _{0}}\), by (6.111), we have
By (6.110), for every ε > 0, there exists N 0 > 0 large enough that
Let \((x_{j}(t),\,\tau _{j}(t)) = (x_{0}({h}^{j}(t)),\,\tau _{0}({h}^{j}(t))\) for j = 0, 1, 2, ⋯ , where we define h 0(t) = t. Then by Lemma 6.23 (e), we have \((x_{j}(t),\,\tau _{j}(t)) = (x_{0}({l}^{j}(t)),\,\tau _{0}({l}^{j}(t))\). Note that I δ is composed of finitely many subintervals. By applying the integral mean value theorem to each subinterval of I δ and by (6.113), we have for every j > N 0 that
Differentiating x j (t) for j = 1, 2, ⋯ , we can obtain from system (6.74) that
Since g(x, τ, σ) < 1, we have
Also by (ii) of (SHB8), xf(x, x, σ 0) > 0 as long as \(x\not\in \Gamma _{\sigma _{0}}\). Then by (6.112) we have
for every t ∈ I δ . By (6.114), (6.117), and by the continuity of f, it follows that there exists N 1 > N 0 such that
Therefore, for every t ∈ I δ and j > N 1, by (6.115), (6.116), and (6.118), we have
Since δ > 0 is arbitrary and I δ goes to I in measure as δ → 0, by the continuity of \(x_{j} \cdot \dot{ x}_{j}\), we have \(x_{j}(t) \cdot \dot{ x}_{j}(t) \geq 0\) for every t ∈ I and j > N 1. By (6.119), we know that \(x_{j} \cdot \dot{ x}_{j}\not\equiv 0\) on I with j > N 1. Therefore, \(x_{j} \cdot x_{j}\) is a nonconstant increasing continuous function. But this is impossible, since x j ⋅x j is continuous and periodic. This completes the proof. □
We now consider the periods of nonconstant periodic solutions, where the delay coincides with the corresponding stationary value for every \(t \in \mathbb{R}\).
Lemma 6.25.
Suppose system (6.74) satisfies (SHB7) . Let (x, τ, σ, p) be a nonconstant p-periodic solution of system (6.74) . If τ(t) = τ σ for every \(t \in \mathbb{R}\) , then (x, τ, σ, p) is not τ σ -periodic.
Proof.
Suppose, by way of contradiction, that (x, τ, σ, p) is τ σ -periodic. If τ(t) = τ σ for every \(t \in \mathbb{R}\), then we have
It follows from (6.120) that
Then by (SHB7) and (6.121), x(t) = x σ for every \(t \in \mathbb{R}\). Thus, (x, τ, σ, p) is a constant periodic solution of (6.74). This is a contradiction. □
We now formulate our next assumption:
- (SHB9):
-
For every Hopf bifurcation point \((x,\,\tau,\,\sigma,\,p) \in C({x}^{{\ast}}{,\,\tau }^{{\ast}}{,\,\sigma }^{{\ast}},\,{p}^{{\ast}})\), mp ≠ τ for every \(m \in \mathbb{N}\).
Theorem 6.19.
Assume that system (6.74) satisfies (SHB6)–(SHB9) . Then for every solution \((x_{0},\,\tau _{0},\,\sigma _{0},\,p_{0}) \in C({x}^{{\ast}}{,\,\tau }^{{\ast}}{,\,\sigma }^{{\ast}},\,{p}^{{\ast}})\) , there exist an open interval I and an open neighborhood \(U \ni (x_{0},\,\tau _{0},\,\sigma _{0},\,p_{0})\) such that every solution
satisfies τ(t) ≠ mp for all \(m \in \mathbb{N}\) and t ∈ I.
Proof.
For a given \(\sigma _{0} \in \mathbb{R}\), if \((x_{0},\,\tau _{0},\,\sigma _{0},\,p_{0}) \in C({x}^{{\ast}}{,\,\tau }^{{\ast}}{,\,\sigma }^{{\ast}},\,{p}^{{\ast}})\) is a constant periodic solution, then it is a Hopf bifurcation point of system (6.74) (See Lemma 6.21). Thus the existence of an open interval I and an open neighborhood \(U \ni (x_{0},\,\tau _{0},\,\sigma _{0},\,p_{0})\) follows immediately from (SHB9) and Lemma 6.24.
If \((x_{0},\,\tau _{0},\,\sigma _{0},\,p_{0}) \in C({x}^{{\ast}}{,\,\tau }^{{\ast}}{,\,\sigma }^{{\ast}},\,{p}^{{\ast}})\) is a nonconstant periodic solution and \(\tau _{0}(t) =\tau _{\sigma _{0}}\) for all \(t \in \mathbb{R}\), then by Lemma 6.25, \((x_{0},\,\tau _{0},\,\sigma _{0},\,p_{0})\) is not \(\tau _{\sigma _{0}}\)-periodic. The conclusion is implied by Lemma 6.24.
If \((x_{0},\,\tau _{0},\,\sigma _{0},\,p_{0})\) is a nonconstant periodic solution and \(\tau _{0}(t)\neq \tau _{\sigma _{0}}\) for some \(t \in \mathbb{R}\), then the conclusion follows from Theorem 6.18. □
We now start the process that uses the local exclusion of periods developed above to construct a uniform upper bound for periods of solutions in the Fuller space. To achieve this goal, we need to “glue” the local exclusion of periods along the connected component. Now we shall show that (6.106) is valid, provided that (6.105) holds.
Theorem 6.20.
Let \(C({y}^{{\ast}},\,{z}^{{\ast}}{,\,\sigma }^{{\ast}},\,{p}^{{\ast}})\) be a connected component of the closure of all the nonconstant periodic solutions of system (6.100) , bifurcated from \(({y}^{{\ast}},\,{z}^{{\ast}}{,\,\sigma }^{{\ast}},\,{p}^{{\ast}})\) in the Fuller space \(C(\mathbb{R}/2\pi ;\, {\mathbb{R}}^{N+1}) \times {\mathbb{R}}^{2}\) . Suppose that system (6.74) satisfies (SHB6)–(SHB9) . Then for every \((y_{0},z_{0},\sigma _{0},p_{0})\in C({y}^{{\ast}},{z}^{{\ast}}{,\sigma }^{{\ast}},{p}^{{\ast}})\) , there exist an open interval I and an open neighborhood \(U \ni (y_{0},\,z_{0},\,\sigma _{0},\,p_{0})\) such that mp ≠ z(t) for every solution \((y,\,z,\,\sigma,\,p) \in U \cap C({y}^{{\ast}},\,{z}^{{\ast}}{,\,\sigma }^{{\ast}},\,{p}^{{\ast}})\), \(m \in \mathbb{N}\) and t ∈ I.
Proof.
Note that p > 0 for every solution (y, z, σ, p) in \(C({y}^{{\ast}},\,{z}^{{\ast}}{,\,\sigma }^{{\ast}},\,{p}^{{\ast}})\). We show that the mapping
is continuous, where \(C({x}^{{\ast}}{,\,\tau }^{{\ast}}{,\,\sigma }^{{\ast}},\,{p}^{{\ast}}) \subseteq C(\mathbb{R}; {\mathbb{R}}^{N+1}) \times {\mathbb{R}}^{2}\). Indeed, if
then we have
where | ⋅ | C denotes the supremum norm in either \(C(\mathbb{R}/2\pi ;\, {\mathbb{R}}^{N})\) or \(C(\mathbb{R}/2\pi ;\, \mathbb{R})\). Therefore, \(C({x}^{{\ast}}{,\,\tau }^{{\ast}}{,\,\sigma }^{{\ast}},\,{p}^{{\ast}})\) is a connected component of periodic solutions of (6.74).
Let \((x_{0},\,\tau _{0},\,\sigma _{0},\,p_{0}) =\iota (y_{0},\,z_{0},\,\sigma _{0},\,p_{0}) \in C({x}^{{\ast}}{,\,\tau }^{{\ast}}{,\,\sigma }^{{\ast}},\,{p}^{{\ast}})\). Then by Theorem 6.19, there exist an open interval I′ and an open neighborhood \(U^{\prime} \ni (x_{0},\,\tau _{0},\,\sigma _{0},\,p_{0})\) such that every solution \((x,\,\tau,\,\sigma,\,p) \in U^{\prime} \cap C({x}^{{\ast}}{,\,\tau }^{{\ast}}{,\,\sigma }^{{\ast}},\,{p}^{{\ast}})\) satisfies τ(t) ≠ mp for all \(m \in \mathbb{N}\) and t ∈ I′.
Since ι is continuous, we can choose an open set \(U \subseteq C(\mathbb{R}/2\pi ;\, {\mathbb{R}}^{N+1}) \times {\mathbb{R}}^{2}\) small enough that \((y_{0},\,z_{0},\,\sigma _{0},\,p_{0}) \in U {\subseteq \iota }^{-1}(U^{\prime})\) and the open set
is nonempty. Then by the definition of ι, mp ≠ z(t) for every \((y,\,z,\,\sigma,\,p) \in U \cap C({y}^{{\ast}},\,{z}^{{\ast}}{,\,\sigma }^{{\ast}},\,{p}^{{\ast}})\), \(m \in \mathbb{N}\), and t ∈ I. □
Lemma 6.26 (The generalized intermediate value theorem [227]).
Let f : X → Y be a continuous map from a connected space X to a linearly ordered set Y with order topology. If a,b ∈ X and y ∈ Y lies between f(a) and f(b), then there exists x ∈ X such that f(x) = y.
Definition 6.5.
Let \(C({y}^{{\ast}},\,{z}^{{\ast}}{,\,\sigma }^{{\ast}},\,{p}^{{\ast}})\) be a connected component of the closure of all the nonconstant periodic solutions of system (6.100), bifurcated from \(({y}^{{\ast}},\,{z}^{{\ast}}{,\,\sigma }^{{\ast}},\,{p}^{{\ast}})\) in the Fuller space \(C(\mathbb{R}/2\pi ;\, {\mathbb{R}}^{N+1}) \times {\mathbb{R}}^{2}\). Let \(I \subset \mathbb{R}\) be an interval and U a subset in \(C({y}^{{\ast}},\,{z}^{{\ast}}{,\,\sigma }^{{\ast}},\,{p}^{{\ast}})\). We call \(I \times (U \cap C({y}^{{\ast}},\,{z}^{{\ast}}{,\,\sigma }^{{\ast}},\,{p}^{{\ast}}))\) a delay-period disparity set if every solution
satisfies mp ≠ z(t) for every t ∈ I and \(m \in \mathbb{N}\). We call \(I \times (U \cap C({y}^{{\ast}},\,{z}^{{\ast}}{,\,\sigma }^{{\ast}},\,{p}^{{\ast}}))\) a delay-period disparity set at \((t_{0},\,y_{0},\,z_{0},\,\sigma _{0},\,p_{0})\) if \((t_{0},\,y_{0},\,z_{0},\,\sigma _{0},\,p_{0}) \in I \times (U \cap C({y}^{{\ast}},\,{z}^{{\ast}}{,\,\sigma }^{{\ast}},\,{p}^{{\ast}}))\).
In the remainder of this subsection, the following assumption is sometimes needed:
- (SHB10):
-
Every periodic solution (x, τ, σ) of (6.74) satisfies τ(t) > 0 for every \(t \in \mathbb{R}\).
Lemma 6.27.
Suppose that system (6.74) satisfies (SHB6)–(SHB7) and (x, τ, σ) is a nonconstant periodic solution. If
-
(i)
τ≢τ σ and there exists \(t_{0} \in \mathbb{R}\) such that \(\tau (t_{0}) =\tau _{\sigma }\) , and
-
(ii)
(x, τ) is τ σ -periodic,
then there exists \(t_{1} \in \mathbb{R}\) such that τ(t 1 ) > τ σ
Proof.
We prove the result by contradiction. Suppose that
Then since \(\tau \not\equiv \tau _{\sigma }\), there exists \({t}^{{\ast}}\in \mathbb{R}\) such that \(\tau ({t}^{{\ast}}) <\tau _{\sigma }\). We can choose a maximal interval \([a,\,b] \subset \mathbb{R}\) that contains t ∗ in the sense that
If \(\dot{\tau }(a)\neq 0\) or \(\dot{\tau }(b)\neq 0\), then it follows from the local monotonicity of τ(t) (at a or b) that there exists \(t_{1} \in \mathbb{R}\) in some neighborhood of a or b such that \(\tau (t_{1}) >\tau _{\sigma }\). This is a contradiction to (6.123).
If \(\dot{\tau }(a) =\dot{\tau } (b) = 0\), then we have
We distinguish the following two cases:
Case 1. x(a) ≠ x σ or x(b) ≠ x σ . Without loss of generality, we suppose x(a) ≠ x σ . Then by (ii), we have
It follows from (SHB7), (6.126), and (6.127) that \(\ddot{\tau }(a)\neq 0\). Therefore, we have that \(\dot{\tau }(t)\) is strictly monotonic in some neighborhood of a. Hence there exists \(t_{1} \in \mathbb{R}\) such that \(\tau (t_{1}) >\tau _{\sigma }\). This is also a contradiction to (6.123).
Case 2. x(a) = x(b) = x σ . By (S5), we have \(\frac{\partial g} {\partial \tau } (x_{\sigma },\,\tau _{\sigma },\,\sigma )\neq 0\). Without loss of generality, we assume that
Then by (6.124), (6.126), (6.128), and the continuity of x(t) and τ(t), we can choose ε > 0 small enough that
Therefore, we have τ(a) < τ(a + ε). That is, there exists t 1 = a + ε such that \(\tau (a) =\tau _{\sigma } <\tau (t_{1})\). This is a contradiction to (6.123). The proof is complete. □
Lemma 6.28.
Suppose that (6.74) satisfies (SHB6)–(SHB10) . Let \(C({y}^{{\ast}},\,{z}^{{\ast}}{,\,\sigma }^{{\ast}},\,{p}^{{\ast}})\) be a connected component of the closure of all the nonconstant periodic solutions of system (6.100) , bifurcated from \(({y}^{{\ast}},\,{z}^{{\ast}}{,\,\sigma }^{{\ast}},\,{p}^{{\ast}})\) in the Fuller space \(C(\mathbb{R}/2\pi ;\, {\mathbb{R}}^{N+1}) \times {\mathbb{R}}^{2}\) . Let \(I \subset \mathbb{R}\) be an open interval and \(\bar{v}\stackrel{def}{=} (\bar{y},\,\bar{z},\,\bar{\sigma },\,\bar{p}) \in C({y}^{{\ast}},\,{z}^{{\ast}}{,\,\sigma }^{{\ast}},\,{p}^{{\ast}})\) . If there is no delay-period disparity set at \((t,\,\bar{u})\) for any t ∈ I, then
-
(i)
there exists \(m \in \mathbb{N}\) such that \(m\,\bar{p} =\bar{ z}(t) = z_{\bar{\sigma }}\) for every t ∈ I;
-
(ii)
\(\bar{v}\) is a nonconstant solution with \(\bar{z}(t) = z_{\bar{\sigma }}\) for every t ∈ I;
-
(iii)
there exist an open interval \(I^{\prime} \subseteq \mathbb{R}\) and an open neighborhood U′ of \(\bar{v}\) such that \(I^{\prime} \times (U^{\prime} \cap C({y}^{{\ast}},{z}^{{\ast}}{,\sigma }^{{\ast}},\,{p}^{{\ast}}))\) is a delay-period disparity set with \(\bar{v} \in U^{\prime} \cap C({y}^{{\ast}},{z}^{{\ast}}{,\sigma }^{{\ast}},{p}^{{\ast}})\) , and the inequality \(z_{\bar{\sigma }} <\bar{ z}(t)\) holds for every t ∈ I′.
Proof.
-
(i)
By Definition 6.5, for every t ∈ I, there exists \(m \in \mathbb{N}\) such that \(\bar{z}(t) = m\,\bar{p}\). Note that \(\bar{z}(t)\) is continuous, \(\bar{z}(t) = m\,\bar{p}\) for every t ∈ I. Then for every t ∈ I, we have
$$\displaystyle{ \dot{\bar{y}}(t) = \frac{\bar{p}} {2\pi }f(\bar{y}(t),\,\bar{y}(t),\,\bar{\sigma }), }$$(6.130)$$\displaystyle{ \dot{\bar{z}}(t) = \frac{\bar{p}} {2\pi }g(\bar{y}(t),\,m\bar{p},\,\bar{\sigma }) = 0. }$$(6.131)By (6.131), we have
$$\displaystyle{ \ddot{\bar{z}}(t) = \frac{\bar{{p}}^{2}} {{4\pi }^{2}} \frac{\partial g} {\partial x}(\bar{y}(t),\,m\bar{p}),\,\bar{\sigma }) \cdot f(\bar{y}(t),\,\bar{y}(t),\,\bar{\sigma }) = 0. }$$(6.132)By (SHB7), (6.131), and (6.132), we have \(\bar{y}(t) = y_{\bar{\sigma }}\) on I. Hence by (SHB6) and by (6.131), we have \(\bar{z}(t) = z_{\bar{\sigma }} = m\bar{p}\) on I. This finishes the proof of (i).
-
(ii)
Note that the stationary solutions of (6.74) and (6.100) are equal. That is, \((x_{\sigma },\,\tau _{\sigma }) = (y_{\sigma },\,z_{\sigma })\) for every \(\sigma \in \mathbb{R}\).
If \(\bar{v}\) is a constant solution, then by (i) we have \(\bar{z}(t) = z_{\bar{\sigma }} = m\bar{p}\) and \(\bar{y}(t) = y_{\bar{\sigma }}\) for all \(t \in \mathbb{R}\). Then \((y_{\bar{\sigma }},\,z_{\bar{\sigma }},\,\bar{\sigma },\,\bar{p})\) is a bifurcation point in \(C({y}^{{\ast}},{z}^{{\ast}}{,\sigma }^{{\ast}},\,{p}^{{\ast}})\) that satisfies \(z_{\bar{\sigma }} = m\bar{p}\) for some \(m \in \mathbb{N}\). This contradicts assumption (SHB9). So \(\bar{v}\) is a nonconstant solution with \(\bar{z}(t) = z_{\bar{\sigma }}\) for all t ∈ I.
-
(iii)
Now we show that there exists \(t_{0} \in \mathbb{R}\) such that \(\bar{z}(t_{0})\neq z_{\bar{\sigma }}\). If not, that is, if \(\bar{z}(t) = z_{\bar{\sigma }}\) for all \(t \in \mathbb{R}\), then
$$\displaystyle{(\bar{x}(\cdot ),\,\bar{\tau }(\cdot ),\,\bar{\sigma })\stackrel{def}{=} (\bar{y}(\frac{2\pi } {\bar{p}}\cdot ),\,\bar{z}(\frac{2\pi } {\bar{p}}\cdot ),\,\bar{\sigma }) = (\bar{y}(\frac{2\pi } {\bar{p}}\cdot ),\,z_{\bar{\sigma }},\,\bar{\sigma }) = (\bar{y}(\frac{2\pi } {\bar{p}}\cdot ),\,\tau _{\bar{\sigma }},\,\bar{\sigma })}$$is a solution of (6.74). Then by Lemma 6.25, \((\bar{x},\,\bar{\tau })\) is not \(\tau _{\bar{\sigma }}\)-periodic. Then we have \(m\bar{p}\neq z_{\bar{\sigma }}\) for every \(m \in \mathbb{N}\). This is a contradiction to (i).
Therefore, there exists \(t_{0} \in \mathbb{R}\) such that \(\bar{z}(t_{0})\neq z_{\bar{\sigma }}\). That is, \(\bar{\tau }(\frac{\bar{p}} {2\pi }t_{0})\neq \tau _{\bar{\sigma }}\). Note that by (i), \((\bar{x},\,\bar{\tau })\) is \(\tau _{\bar{\sigma }}\)-periodic and \(\bar{\tau }(t) =\tau _{\bar{\sigma }}\) on \(\frac{\bar{p}} {2\pi }I\). Then by Lemma 6.27, there exists \(t_{1} \in \mathbb{R}\) such that
By the continuity of \(\bar{\tau }\) and by (6.133), there exists a finite interval (a, b) ∋ t 1 such that for every t ∈ (a, b),
We claim that there exists t 0 ∈ (a, b) such that \(\bar{v}\) is not \(\bar{\tau }(t_{0})\)-periodic. Indeed, if not, then \(\bar{v}\) would be \(\bar{\tau }(t)\)-periodic for every t ∈ (a, b). Then by the continuity of \(\bar{\tau }\) and by (6.134), there would exist \(t_{1},\,t_{2} \in (a,\,b)\) and an interval \((\bar{\tau }(t_{1}),\,\bar{\tau }(t_{2}))\) with \(\bar{\tau }(t_{2}) >\bar{\tau } (t_{1})\), so that \(\bar{\tau }\) would be p-periodic for all \(p \in (\bar{\tau }(t_{1}),\,\bar{\tau }(t_{2}))\). Hence \(\bar{v}\) would be a constant solution. This is a contradiction to (ii), and the claim is proved.
Then we have \(\bar{\tau }(t_{0})\neq m\bar{p}\) for all \(m \in \mathbb{N}\). By Lemma 6.24, there exist an open interval I 1 ∋ t 0 and an open neighborhood \(U_{1} \ni (\bar{x},\,\bar{\tau },\,\bar{\sigma },\,\bar{p})\) such that every solution (x, τ, σ, p) of (6.74) in \(U_{1} \cap C({x}^{{\ast}}{,\,\tau }^{{\ast}}{,\,\sigma }^{{\ast}},\,{p}^{{\ast}})\) satisfies τ(t) ≠ mp for all \(m \in \mathbb{N}\) and t ∈ I 1. Note that \(\bar{\tau }\) is continuous at t = t 0. We can therefore choose I 1 small enough that (6.134) holds for all t ∈ I 1.
Let ι be the continuous mapping defined by (6.122). Then we can choose an open set \(U^{\prime} \subseteq C(\mathbb{R}/2\pi ;\, {\mathbb{R}}^{N+1}) \times {\mathbb{R}}^{2}\) small enough that \(\bar{v} \in U^{\prime} {\subseteq \iota }^{-1}(U_{1})\) and
is nonempty. It follows from the definition of ι that mp ≠ z(t) for every solution \((y,\,z,\,\sigma,\,p) \in U^{\prime} \cap C({y}^{{\ast}},\,{z}^{{\ast}}{,\,\sigma }^{{\ast}},\,{p}^{{\ast}})\), \(m \in \mathbb{N}\), and t ∈ I′. In particular, noting that (6.134) holds for all t ∈ I 1 and \(I^{\prime} \subseteq \frac{\bar{p}} {2\pi }I_{1}\), we have
for every t ∈ I′. This completes the proof. □
Now we are able to state our main result.
Theorem 6.21.
Let \(C({y}^{{\ast}},\,{z}^{{\ast}}{,\,\sigma }^{{\ast}},\,{p}^{{\ast}})\) be a connected component of the closure of all the nonconstant periodic solutions of system (6.100) , bifurcated from \(({y}^{{\ast}},\,{z}^{{\ast}}{,\,\sigma }^{{\ast}},\,{p}^{{\ast}})\) in the Fuller space \(C(\mathbb{R}/2\pi ;\, {\mathbb{R}}^{N+1}) \times {\mathbb{R}}^{2}\) . Suppose that (6.74) satisfies (SHB6)–(SHB10) . If \({p}^{{\ast}} < {z}^{{\ast}}\) , then for every \((y,\,z,\,\sigma,\,p) \in C({y}^{{\ast}},\,{z}^{{\ast}}{,\,\sigma }^{{\ast}},\,{p}^{{\ast}})\) , p < z(t) for some \(t \in \mathbb{R}\).
Proof.
By Theorem 6.20 and (SHB9), there exist an open interval \({I}^{{\ast}}\subseteq \mathbb{R}\) and an open set U ∗ in \(C(\mathbb{R}/2\pi ;\, {\mathbb{R}}^{N+1}) \times {\mathbb{R}}^{2}\) such that \({I}^{{\ast}}\times ({U}^{{\ast}}\cap C({y}^{{\ast}},\,{z}^{{\ast}}{,\,\sigma }^{{\ast}},\,{p}^{{\ast}}))\) is a delay-period disparity set with \(({y}^{{\ast}},\,{z}^{{\ast}}{,\,\sigma }^{{\ast}},\,{p}^{{\ast}}) \in {U}^{{\ast}}\).
Let \({A}^{{\ast}}\ni ({y}^{{\ast}},\,{z}^{{\ast}}{,\,\sigma }^{{\ast}},\,{p}^{{\ast}})\) be a connected component of \(({U}^{{\ast}}\cap C({y}^{{\ast}},\,{z}^{{\ast}}{,\,\sigma }^{{\ast}},\,{p}^{{\ast}}))\). Then \({I}^{{\ast}}\times {A}^{{\ast}}\) is connected in \(\mathbb{R}\times C(\mathbb{R}/2\pi ; {\mathbb{R}}^{N+1})\times {\mathbb{R}}^{2}\). Define \(S : \mathbb{R}\times C(\mathbb{R}/2\pi ; {\mathbb{R}}^{N+1})\times \) \({\mathbb{R}}^{2} \rightarrow \mathbb{R}\) by
Note that we have \({p}^{{\ast}} < {z}^{{\ast}}\). Then it follows that \(S(t,\,{y}^{{\ast}},\,{z}^{{\ast}}{,\,\sigma }^{{\ast}},\,{p}^{{\ast}}) = {p}^{{\ast}}- {z}^{{\ast}} < 0\). Note that S is continuous. By Lemma 6.26, we have
for every \((t,\,y,\,z,\,\sigma,\,p) \in {I}^{{\ast}}\times {A}^{{\ast}}\), for otherwise, there would exist \((t_{0},\,y_{0},\,z_{0},\,\sigma _{0},\,p_{0})\) ∈ I ∗ ×A ∗ such that \(p_{0} = z_{0}(t_{0})\), which contradicts the fact that \({I}^{{\ast}}\times {A}^{{\ast}}\) is a subset of the forbidden range of delay \({I}^{{\ast}}\times ({U}^{{\ast}}\cap C({y}^{{\ast}},\,{z}^{{\ast}}{,\,\sigma }^{{\ast}},\,{p}^{{\ast}}))\).
Now we show that there exists a sequence of connected subsets of \(C({y}^{{\ast}},\,{z}^{{\ast}}{,\,\sigma }^{{\ast}},\,{p}^{{\ast}})\), denoted by \(\{A_{n}\}_{n=1}^{n_{0}}\), \(n_{0} \in \mathbb{N}\) or n 0 = + ∞, that satisfies
-
(i)
\({A}^{{\ast}}\subseteq A_{1} \subset A_{2} \subset \cdots \subset A_{n_{0}}\) and \(\cup _{n=1}^{n_{0}}A_{n} = C({y}^{{\ast}},\,{z}^{{\ast}}{,\,\sigma }^{{\ast}},\,{p}^{{\ast}})\);
-
(ii)
for every (y, z, σ, p) ∈ A n with n ∈ { 1, 2, ⋯ , n 0}, p < z(t) at some \(t \in \mathbb{R}\).
Let \(A_{1}\stackrel{def}{=} {A}^{{\ast}}\). If \(A_{1} = C({y}^{{\ast}},\,{z}^{{\ast}}{,\,\sigma }^{{\ast}},\,{p}^{{\ast}}))\), then we are done by (6.136). If not, since the only sets that are both closed and open in the connected topological space \(C({y}^{{\ast}},\,{z}^{{\ast}}{,\,\sigma }^{{\ast}},\,{p}^{{\ast}})\) are the empty set and the connected component \(C({y}^{{\ast}},\,{z}^{{\ast}}{,\,\sigma }^{{\ast}},\,{p}^{{\ast}})\) itself, \(A_{1} \ni ({y}^{{\ast}},\,{z}^{{\ast}}{,\,\sigma }^{{\ast}},\,{p}^{{\ast}})\) is not both closed and open. Then the boundary of A 1 in the sense of the relative topology induced by \(C({y}^{{\ast}},\,{z}^{{\ast}}{,\,\sigma }^{{\ast}},\,{p}^{{\ast}})\) is nonempty. That is,
Let \(\bar{v} = (\bar{y},\,\bar{z},\,\bar{\sigma },\,\bar{p}) \in \partial A_{1}\). If there exist \(t_{1} \in I_{1}\stackrel{def}{=} {I}^{{\ast}}\) and a delay-period disparity set \(I^{\prime} \times (U^{\prime} \cap C({y}^{{\ast}},\,{z}^{{\ast}}{,\,\sigma }^{{\ast}},\,{p}^{{\ast}}))\) such that \((t_{1},\,\bar{v}) \in \bar{ I^{\prime}} \times (U^{\prime} \cap C({y}^{{\ast}},\,{z}^{{\ast}}{,\,\sigma }^{{\ast}},\,{p}^{{\ast}})\), and if \(A_{\bar{v}} \ni \bar{ v}\) is the connected component of \(U^{\prime} \cap C({y}^{{\ast}},\,{z}^{{\ast}}{,\,\sigma }^{{\ast}},\,{p}^{{\ast}})\), then it is clear that \(A_{1} \cup A_{\bar{v}}\) is connected. Since A 1 is closed, we have \(\bar{p} <\bar{ z}(t_{1})\). Then by Lemma 6.26, we have
If for every t ∈ I 1, there is no delay-period disparity set at \((t,\,\bar{u})\), then by Lemma 6.28, there exists a delay-period disparity set \(I{\prime\prime} \times (U{\prime\prime} \cap C({y}^{{\ast}},\,{z}^{{\ast}}{,\,\sigma }^{{\ast}},\,{p}^{{\ast}})\) with \(\bar{v} \in U^{\prime} \cap C({y}^{{\ast}},\,{z}^{{\ast}}{,\,\sigma }^{{\ast}},\,{p}^{{\ast}})\) and
Let \(A_{\bar{v}} \ni \bar{ v}\) be the connected component of \(U{\prime\prime} \cap C({y}^{{\ast}},\,{z}^{{\ast}}{,\,\sigma }^{{\ast}},\,{p}^{{\ast}})\). It is clear that \(A_{1} \cup A_{\bar{v}}\) is connected. Then by (6.139) and Lemma 6.26,
By (6.138) and (6.140), we know that if \(\bar{v} \in \partial A_{1}\), then there exists a delay-period disparity set \(\tilde{I} \times (\tilde{U} \cap C({y}^{{\ast}},\,{z}^{{\ast}}{,\,\sigma }^{{\ast}},\,{p}^{{\ast}}))\) with \(A_{\bar{v}} \ni \bar{ v}\) the connected component of \(\tilde{U} \cap C({y}^{{\ast}},\,{z}^{{\ast}}{,\,\sigma }^{{\ast}},\,{p}^{{\ast}})\) such that
For every \(\bar{v} \in \partial A_{1}\), we find a \(A_{\bar{v}}\) satisfying (6.141). Then we define
It follows from (6.136), (6.138), and (6.140) that for every (y, z, σ, p) ∈ A 2, p < z(t) for some \(t \in \mathbb{R}\). Note that for every \(\bar{v} \in \partial A_{1}\), \(A_{1} \cup A_{\bar{v}}\) is connected. Therefore, A 2 is connected.
Note that the existence of A 2 depends only on the fact that ∂A 1 ≠ ∅, in the sense of the relative topology induced by \(C({y}^{{\ast}},\,{z}^{{\ast}}{,\,\sigma }^{{\ast}},\,{p}^{{\ast}})\). Beginning with n = 1, we can always recursively construct a connected subset for each n ≥ 1, \(n \in \mathbb{N}\), with ∂A n ≠ ∅,
satisfying that for every (y, z, σ, p) ∈ A n + 1,
where \(I_{n} \times (U_{n} \cap C({y}^{{\ast}},\,{z}^{{\ast}}{,\,\sigma }^{{\ast}},\,{p}^{{\ast}}))\) is a delay-period disparity set at \((t,\,\bar{v}) \in I_{n} \times \partial A_{n}\) and \(A_{\bar{v}}\) is the connected component of U n .
If the construction in (6.142) stops at some \(n_{0} \in \mathbb{N}\) with \(\partial A_{n_{0}} = \varnothing \), then \(A_{n_{0}} = C({y}^{{\ast}},\,{z}^{{\ast}}{,\,\sigma }^{{\ast}},\,{p}^{{\ast}})\), and we are done. If not, then n 0 = + ∞, and we obtain a sequence of sets \(\{A_{n}\}_{n=1}^{+\infty }\) that is a totally ordered family of sets with respect to the set inclusion relation ⊆ . Note that \(\cup _{n=1}^{+\infty }A_{n}\) is the upper bound of \(\{A_{n}\}_{n=1}^{+\infty }\). Then by Zorn’s lemma, there exists a maximal element A ∞ for the sequence \(\{A_{n}\}_{n=1}^{+\infty }\).
Now we show that ∂A ∞ = ∅, in the sense of the relative topology induced by \(C({y}^{{\ast}},\,{z}^{{\ast}}{,\,\sigma }^{{\ast}},\,{p}^{{\ast}})\). Suppose not. Then there exist \(\bar{v} \in \partial A_{\infty }\) and \(A_{\bar{v}}\), which is the connected component of U ∞ , where \(I_{\infty }\times (U_{\infty }\times C({y}^{{\ast}},\,{z}^{{\ast}}{,\,\sigma }^{{\ast}},\,{p}^{{\ast}}))\) is a delay-period disparity set at \((t,\,\bar{v}) \in I_{\infty }\times \partial A_{\infty }\). We distinguish two cases:
Case 1. \(A_{\bar{v}} \setminus A_{\infty } = \varnothing \) for all \(\bar{v} \in \partial A_{\infty }\). Then A ∞ is a connected component of \(C({y}^{{\ast}},\,{z}^{{\ast}}{,\,\sigma }^{{\ast}},\,{p}^{{\ast}})\). Recall that \(C({y}^{{\ast}},\,{z}^{{\ast}}{,\,\sigma }^{{\ast}},\,{p}^{{\ast}})\) itself is a connected component of the closure of all the nonconstant periodic solutions of system (6.100). So we have \(A_{\infty } = C({y}^{{\ast}},\,{z}^{{\ast}}{,\,\sigma }^{{\ast}},\,{p}^{{\ast}})\). That is, ∂A ∞ = ∅. This is a contradiction.
Case 2. \(A_{\bar{v}} \setminus A_{\infty }\neq \varnothing \). But this means that \(A_{\infty }\subset A_{\infty }\cup A_{\bar{v}}\), which contradicts the maximality of A ∞ .
These contradictions show that ∂A ∞ = ∅, and hence \(A_{\infty } = C({y}^{{\ast}},\,{z}^{{\ast}}{,\,\sigma }^{{\ast}},\,{p}^{{\ast}})\). Therefore, (6.143) holds for all \((y,\,z,\,\sigma,\,p) \in C({y}^{{\ast}},\,{z}^{{\ast}}{,\,\sigma }^{{\ast}},\,{p}^{{\ast}})\). This completes the proof. □
Theorem 6.22.
Let \(C({y}^{{\ast}},\,{z}^{{\ast}}{,\,\sigma }^{{\ast}},\,{p}^{{\ast}})\) be a connected component of the closure of all the nonconstant periodic solutions of system (6.100) , bifurcated at \(({y}^{{\ast}},\,{z}^{{\ast}}{,\,\sigma }^{{\ast}},\,{p}^{{\ast}})\) in the Fuller space \(C(\mathbb{R}/2\pi ;\, {\mathbb{R}}^{N+1}) \times {\mathbb{R}}^{2}\) . Suppose that (6.74) satisfies (S5)–(S9). If there exists a continuous function \(M_{1} : \mathbb{R} \ni \sigma \rightarrow M_{1}(\sigma ) > 0\) such that for every \((y,\,z,\sigma,\,p) \in C({y}^{{\ast}},\,{z}^{{\ast}}{,\,\sigma }^{{\ast}},\,{p}^{{\ast}})\) , we have
then \({p}^{{\ast}} < {z}^{{\ast}}\) implies that p < M 1 (σ) for every \((y,\,z,\,\sigma,\,p) \in C({y}^{{\ast}},\,{z}^{{\ast}}{,\,\sigma }^{{\ast}},\,{p}^{{\ast}})\).
Proof.
By Theorem 6.21, we have, for every \((y,\,z,\,\sigma,\,p) \in C({y}^{{\ast}},\,{z}^{{\ast}}{,\,\sigma }^{{\ast}},\,{p}^{{\ast}})\), that p < z(t) for some \(t \in \mathbb{R}\). Then by (6.144), we have p < M 1(σ). □
6.9.4 Uniform Boundedness of Periodic Solutions
We refer to [254] for the concepts of balanced, convex, and absorbing subsets and the Minkowski functional.
Lemma 6.29.
Let G be a convex absorbing subset of a locally convex linear topological space X that defines a Minkowski functional \(p_{G} : X \rightarrow \mathbb{R}\) with \(p_{G}(x) =\inf \{\alpha > 0 {:\alpha }^{-1}x\stackrel{def}{=} x/\alpha \in G\}\) . For each γ > 0, define
Then x ∈ ∂G γ if and only if p G (x) = γ.
Proof.
It is clear that G γ = γG. By linearity, the Minkowski functional \(p_{{G}^{\gamma }} : X \rightarrow \mathbb{R}\) determined by G γ is well defined. By (6.145) and by the definition of Minkowski functional, we have
□
Lemma 6.30.
Let G 1 and G 2 be convex absorbing subsets of locally convex linear topological spaces X 1 and X 2 , respectively. Let the Minkowski functionals associated with G 1 and G 2 be \(p_{G_{1}}(x)\) and \(p_{G_{2}}(\tau )\) , respectively. Then the Minkowski functional defined by G = G 1 × G 2 exists and satisfies
Proof.
The existence of p G (x, τ) is clear from the definition of a Minkowski functional. Let A = {α : x ∕ α ∈ G 1}, B = {α : τ ∕ α ∈ G 2}. Then it is clear that infA ∩ B ≥ infA and infA ∩ B ≥ infB. It follows that \(\inf A \cap B \geq \max \{\inf A,\,\inf B\}\), that is,
On the other hand, if \(\alpha _{A} =\inf A \geq \alpha _{B} =\inf B\), since G 1 and G 2 are absorbing, we have for every ε > 0 that α A + ε ∈ A, α A + ε ∈ B. Therefore, infA ∩ B ≤ α A + ε. Similarly, if α A = infA ≤ α B = infB, we have infA ∩ B ≤ α B + ε. Hence we obtain \(\inf A \cap B \leq \max \{\alpha _{A},\,\alpha _{B}\}+\epsilon\). By the arbitrariness of ε > 0, we get infA ∩ B ≤ max{α A , α B }, that is,
By (6.146) and (6.147), we have
This completes the proof. □
An immediate corollary of Lemmas 6.29 and 6.30 is the following.
Corollary 6.1.
Let G 1 and G 2 be convex absorbing subsets of locally convex linear topological spaces X 1 and X 2 , respectively. Let \(p_{G_{1}}(x)\) and \(p_{G_{2}}(\tau )\) be the Minkowski functionals associated with G 1 and G 2 , respectively. Let G = G 1 × G 2 , and for every γ > 0, define
Then \({G}^{\gamma } = G_{1}^{\gamma } \times G_{2}^{\gamma }\) and \(\bar{{G}}^{\gamma } =\bar{ G}_{1}^{\gamma } \times \bar{ G}_{2}^{\gamma }.\)
In this section, we use “ ⋅” to denote the usual inner product of a Euclidean space, and we use G c and D c to denote the complementary sets of G and D, respectively.
We can now state and prove the geometric conditions for uniform boundedness of the periodic solutions of (6.74) with σ ∈ Σ, where \(\Sigma \subseteq \mathbb{R}\) is a given subset.
Theorem 6.23.
Suppose that \(G_{1} \subset {\mathbb{R}}^{N}\) and \(G_{2} \subset \mathbb{R}\) are bounded, balanced, convex, and absorbing open subsets with associated Minkowski functionals \(p_{G_{1}}(x)\) and \(p_{G_{2}}(\tau )\) . Let \(G = G_{1} \times G_{2}\) and \((\overline{x},\,\overline{\tau }) = \frac{1} {p_{G}(x,\,\tau )}(x,\,\tau ) \in \partial G\) for (x, τ) ≠ 0. Assume that there exists a vector-valued function \(N : \partial G \setminus (\partial G_{1} \times \partial G_{2}) \rightarrow {\mathbb{R}}^{N+1} \setminus \{ 0\}\) satisfying
-
(i):
\(\overline{G} \subseteq U_{1} \cup U_{2}\) , where
$$\displaystyle\begin{array}{rcl} U_{1}& =& \bigcap \limits _{(x,\,\tau )\in \partial G\setminus (\partial G_{1}\times \partial G_{2})}\{(u,\,v) : N(x,\,\tau ) \cdot (u - x,\,v-\tau ) \leq 0\}; {}\\ U_{2}& =& \bigcap _{(x,\,\tau )\in \partial G_{1}\times \partial G_{2}}\{(u,\,v) : x \cdot (u - x) \leq 0,\,\tau \cdot (v-\tau ) \leq 0\}; {}\\ \end{array}$$ -
(ii):
\(N(\overline{x},\,\overline{\tau }) \cdot (f(x,\,\tilde{x},\,\sigma ),\,g(x,\,\tau,\,\sigma ))\) is positive (or negative) for all (x, τ) ∈ G c with \((\overline{x},\,\overline{\tau })\not\in \partial G_{1} \times \partial G_{2}\) , and all \((\tilde{x},\,\tau ) \in {\mathbb{R}}^{N} \times \mathbb{R}\) with \(p_{G}(\tilde{x},\,\tau ) \leq p_{G}(x,\,\tau )\) and σ ∈ Σ;
-
(iii):
\(x \cdot f(x,\,\tilde{x},\,\sigma )\) and τ ⋅ g(x, τ, σ) are both positive (or negative) for all (x, τ) ∈ G c with \((\overline{x},\,\overline{\tau }) \in \partial G_{1} \times \partial G_{2}\) , and all \((\tilde{x},\,\tau ) \in {\mathbb{R}}^{N} \times \mathbb{R}\) with \(p_{G}(\tilde{x},\,\tau ) \leq p_{G}(x,\,\tau )\) and σ ∈ Σ.
Then the range of all the periodic solutions of (6.74) with σ ∈ Σ is contained in G.
Remark 6.3.
The prototype of the vector-valued function N(x, τ) is the (outer or inner) normal of G, which is not defined on ∂G 1 ×∂G 2. If G is a rectangle in a planar space, ∂G 1 ×∂G 2 are the four corner points of G. Conditions (ii)–(iii) of Theorem 6.23 require that the vector field determined by the right-hand side of system (6.74) have positive (or negative) inner product with respect to the normal of a given rectangle G, where the vector field is evaluated at \((x,\,\tau ) \in {\mathbb{R}}^{N+1}\), which satisfies (x, τ) ∈ G c and \(p_{G}(\tilde{x},\,\tau ) \leq p_{G}(x,\,\tau )\).
Proof.
Letting (x, τ)(t) = (y, z)(βt) with a normalization parameter β > 0, we only need to consider the 2π-periodic solutions of the following system:
where \(x \in {\mathbb{R}}^{N}\) and \(\tau \in \mathbb{R}\). It is clear that if (x(t), τ(t)) and (y(t), z(t)) are solutions of (6.74) and (6.148), respectively, then (x(t), τ(t)) ∈ G for all \(t \in \mathbb{R}\) if and only if (y(t), z(t)) ∈ G for all \(t \in \mathbb{R}\).
For simplicity, we denote y(t − βz(t)) by \(\tilde{y}(t)\) for each solution (y(t), z(t)) of (6.148). Let \((\overline{y},\,\overline{z})\) be the positive constant multiple of (y, z) such that \((\overline{y},\,\overline{z}) \in \partial G\). That is, for every \((y,\,z) \in {\mathbb{R}}^{N+1} \setminus \{ 0\}\), there exists \((\bar{y},\,\bar{z}) \in \partial G\) such that \((y,\,z) = p_{G}(y,\,z)(\bar{y},\,\bar{z})\).
Suppose there exists a 2π-periodic solution of (6.148) such that \((y(t_{0}),\,z(t_{0}))\not\in G\) for some t 0 ∈ [0, 2π] and define the map \(\gamma : \mathbb{R} \ni t \rightarrow p_{G}(y(t),\,z(t)) \in \mathbb{R}\). Since \({\mathbb{R}}^{N+1} \ni (y,\,z)\mapsto p_{G}(y,\,z) \in \mathbb{R}\) and \(\mathbb{R} \ni t\mapsto (y(t),\,z(t)) \in {\mathbb{R}}^{N+1}\) are continuous, the map γ : t → p G (y(t), z(t)) is continuous and there exist γ ∗ ≥ 1 and t ∗ ∈ [0, 2π] such that
Then by Lemma 6.29 and (6.149), we have \((y({t}^{{\ast}}),\,z({t}^{{\ast}})) \in \partial {G}^{{\gamma }^{{\ast}} }\) and \({G}^{\gamma (t)} \subseteq {G}^{{\gamma }^{{\ast}} }\) for all \(t \in \mathbb{R}\). Therefore, by Corollary 6.1, \((y(t),\,z(t)) \in \bar{ {G}}^{{\gamma }^{{\ast}} } =\bar{ G}_{1}^{{\gamma }^{{\ast}} }\times \bar{ G}_{2}^{{\gamma }^{{\ast}} }\) for all t ∈ [0, 2π]. In particular, by the periodicity of (y(t), z(t)), we obtain \((y(t -\beta z(t)),\,z(t)) \in \bar{ {G}}^{{\gamma }^{{\ast}} }\) for all t ∈ [0, 2π] and β > 0. Therefore, we have
We first suppose that \((\bar{y}({t}^{{\ast}}),\,\bar{z}({t}^{{\ast}})) = \frac{1} {p_{G}(y({t}^{{\ast}}),\,z({t}^{{\ast}}))}(y({t}^{{\ast}}),\,z({t}^{{\ast}})) \in U_{ 1}\). Then by (6.149), (6.150), and assumption (ii), we have (we use the positivity assumption in the proof; the proof is similar if we use the negativity assumption; see Remark 6.4 for details)
Let us write
and choose h > 0 small enough that
for \({t}^{{\ast}}\leq t < {t}^{{\ast}} + h\). Then by (6.148), (6.152), and (6.153), we have
Now we distinguish the following two cases in order to deduce contradictions:
Case 1. If \((y({t}^{{\ast}} + h),\,z({t}^{{\ast}} + h)) \in \bar{ G}\), then \({{\gamma }^{{\ast}}}^{-1}(y({t}^{{\ast}} + h),\,z({t}^{{\ast}} + h)) \in \bar{ G}\), since γ ∗ ≥ 1. Also, we have \((y({t}^{{\ast}}),\,z({t}^{{\ast}})) = {(\gamma }^{{\ast}}\bar{y}({t}^{{\ast}}){,\,\gamma }^{{\ast}}\bar{z}({t}^{{\ast}}))\) with \((\bar{y}({t}^{{\ast}}),\,\bar{z}({t}^{{\ast}})) \in \partial G\). Then by assumption (i), we have
On the other hand, we have by (6.154),
which contradicts (6.155).
Case 2. If \((y({t}^{{\ast}} + h),\,z({t}^{{\ast}} + h))\not\in \bar{G}\), then by (6.149), we have
Also, we have \((y({t}^{{\ast}} + h),\,z({t}^{{\ast}} + h)) =\gamma _{h}(\bar{y}({t}^{{\ast}} + h),\,\bar{z}({t}^{{\ast}} + h))\) with \((\bar{y}({t}^{{\ast}} + h),\,\bar{z}({t}^{{\ast}} + h)) \in \partial G\). By the convexity of \(\bar{G}\) and by the inequality \(\gamma _{h}{/\gamma }^{{\ast}}\leq 1\), we have
Then by assumption (i), we have
On the other hand, we have by (6.154),
which contradicts (6.158).
Second, we suppose that \((\bar{y}({t}^{{\ast}}),\,\bar{z}({t}^{{\ast}})) = \frac{1} {p_{G}(y({t}^{{\ast}}),\,z({t}^{{\ast}}))}(y({t}^{{\ast}}),\,z({t}^{{\ast}})) \in U_{ 2}\). By assumption (iii), we have
Therefore, we can choose h > 0 small enough that for \({t}^{{\ast}}\leq t < {t}^{{\ast}} + h\),
Then by (6.148), (6.152), and (6.161), we have
We distinguish the following two cases in order to deduce contradictions:
Case 1′. If \((y({t}^{{\ast}} + h),\,z({t}^{{\ast}} + h)) \in \bar{ G}\), then \({{\gamma }^{{\ast}}}^{-1}(y({t}^{{\ast}} + h),\,z({t}^{{\ast}} + h)) \in \bar{ G}\), since γ ∗ ≥ 1. Also, we have \((y({t}^{{\ast}}),\,z({t}^{{\ast}})) = {(\gamma }^{{\ast}}\bar{y}({t}^{{\ast}}){,\,\gamma }^{{\ast}}\bar{z}({t}^{{\ast}}))\) with \((\bar{y}({t}^{{\ast}}),\,\bar{z}({t}^{{\ast}})) \in \partial G\). Then by assumption (i), we have
On the other hand, we have by (6.162),
which contradicts (6.163).
Case 2′. If \((y({t}^{{\ast}} + h),\,z({t}^{{\ast}} + h))\not\in \bar{G}\), then by (6.157) and the convexity of \(\bar{G}\), we have
where \(\gamma _{h} = p_{G}(y({t}^{{\ast}} + h),\,z({t}^{{\ast}} + h))\). Then by assumption (i), we have
On the other hand, we have by (6.162),
which contradicts (6.165). Therefore, contradictions are obtained in all cases, and the proof is complete. □
Remark 6.4.
If we use < 0 instead of > 0 in the inequality (6.151), we need to change (6.152) to be the difference between \((y({t}^{{\ast}}),\,z({t}^{{\ast}}))\) and \((y({t}^{{\ast}}- h),\,z({t}^{{\ast}}- h))\). That is,
Then the rest of the proof is similar.
Corollary 6.2.
Suppose that \(G_{1} \subset {\mathbb{R}}^{N}\) and \(G_{2} \subset \mathbb{R}\) are bounded, balanced, convex, and absorbing open subsets that define the Minkowski functionals \(p_{G_{1}}(x)\) and \(p_{G_{2}}(\tau )\) . Suppose \(N : \partial G \setminus (\partial G_{1} \times \partial G_{2}) \rightarrow {\mathbb{R}}^{N+1} \setminus \{ 0\}\) is the outer normal of G. Fix σ ∈ Σ and let G = G 1 × G 2 and
Then the range of all the periodic solutions of (6.74) are contained in G if either of the following conditions holds:
- (H1):
-
\(F_{\max}(x,\,\sigma)<0\) for every \(x \in G_{1}^{c}\) and τ ⋅ g(x, τ) < 0 for every \(\tau \in G_{2}^{c},\,x \in {\mathbb{R}}^{N}\).
- (H2):
-
\(F_{\min}(x,\,\sigma)>0\) for every \(x \in G_{1}^{c}\) and τ ⋅ g(x, τ) > 0 for every \(\tau \in G_{2}^{c},\,x \in {\mathbb{R}}^{N}\).
Proof.
We prove the conclusions by applying Theorem 6.23. By Corollary 6.1, there exist Minkowski functionals p G (x, τ), \(p_{G_{1}}(x)\), and \(p_{G_{2}}(\tau )\) defined on \({\mathbb{R}}^{N} \times \mathbb{R}\), \({\mathbb{R}}^{N}\), and \(\mathbb{R}\), respectively. For every (x, τ) ∈ G c, let \((\bar{x},\,\bar{\tau }) = (x,\,\tau )/p_{G}(x,\,\tau ) \in \partial G\). Recall that \(N : \partial G \setminus (\partial G_{1} \times \partial G_{2}) \rightarrow {\mathbb{R}}^{N+1} \setminus \{ 0\}\) is the outer normal of the convex set G. Then condition (i) of Theorem 6.23 is satisfied.
Suppose (H1) holds. Then we have
For every (x, τ) ∈ G c with \(p_{G}(\tilde{x},\,\tau ) \leq p_{G}(x,\,\tau )\), let \((\bar{x},\,\bar{\tau }) = (x,\,\tau )/p_{G}(x,\,\tau ) \in \partial G.\) Note that \(\partial G = (G_{1} \times \partial G_{2}) \cup (\partial G_{1} \times G_{2}) \cup (\partial G_{1} \times \partial G_{2})\). We distinguish the following three cases:
Case 1: If \((\bar{x},\,\bar{\tau }) \in G_{1} \times \partial G_{2}\), then \(N(\bar{x},\,\bar{\tau }) = (0,\,\tau )/p_{G}(x,\,\tau )\neq 0\) is an outer normal of G. We claim that τ ∈ G 2 c holds.
Indeed, since \(\bar{x} \in G_{1}\), we have \(p_{G_{1}}(\bar{x}) = p_{G_{1}}(x/p_{G}(x,\,\tau )) < 1\). Therefore, \(p_{G_{1}}(x) < p_{G}(x,\,\tau )\). By Lemma 6.30, we know that \(p_{G}(x,\,\tau ) =\max \{ p_{G_{1}}(x),\,p_{G_{2}}(\tau )\}\). Then we have \(p_{G_{1}}(x) < p_{G_{2}}(\tau )\) and \(p_{G}(x,\,\tau ) = p_{G_{2}}(\tau ) > 1\). Then by Lemma 6.29, we have \(\tau \in G_{2}^{c}\).
Then by (6.168), we have
Case 2: If \((\bar{x},\,\bar{\tau }) \in \partial G_{1} \times G_{2}\), then \(N(\bar{x},\,\bar{\tau }) = (x,\,0)/p_{G}(x,\,\tau )\neq 0\) is an outer normal of G. We claim that \(x \in G_{1}^{c}\) and \(p_{G_{1}}(\tilde{x}) \leq p_{G_{1}}(x)\).
Indeed, since \(\bar{\tau }\in G_{2}\), we have \(p_{G_{2}}(\bar{\tau }) = p_{G_{2}}(\tau /p_{G}(x,\,\tau )) < 1\). Therefore, \(p_{G_{2}}(\tau ) < p_{G}(x,\,\tau )\). By Lemma 6.30, we know that \(p_{G}(x,\,\tau ) =\max \{ p_{G_{1}}(x),\,p_{G_{2}}(\tau )\}\). Then we have \(p_{G_{2}}(\tau ) < p_{G_{1}}(x)\) and \(p_{G}(x,\,\tau ) = p_{G_{1}}(x) > 1\). Then by Lemma 6.29, we have \(x \in G_{1}^{c}\). Moreover, it follows again by Lemma 6.30 that \(p_{G}(\tilde{x},\,\tau ) \leq p_{G}(x,\,\tau )\) implies \(p_{G_{1}}(\tilde{x}) \leq p_{G_{1}}(x)\). This proves the claim.
By (6.167), we have
From Case 1 and Case 2, we know that \(N(\bar{x},\,\bar{\tau }) \cdot (f(x,\tilde{x},\,\sigma ),\,g(x,\,\tau,\,\sigma ))\) is negative definite for all (x, τ) ∈ G c and σ ∈ Σ with \((\overline{x},\,\overline{\tau })\not\in \partial G_{1} \times \partial G_{2}\), and all \((\tilde{x},\,\tau ) \in {\mathbb{R}}^{N} \times \mathbb{R}\) with \(p_{G}(\tilde{x},\,\tau ) \leq p_{G}(x,\,\tau )\). That is, condition (ii) of Theorem 6.23 is satisfied.
Case 3: If \((\bar{x},\,\bar{\tau }) \in \partial G_{1} \times \partial G_{2}\), we claim that \((x,\,\tau ) \in G_{1}^{c} \times G_{2}^{c}\) and \(p_{G_{1}}(\tilde{x}) = p_{G_{1}}(x)\) hold.
Indeed, since \((\bar{x},\,\bar{\tau }) \in \partial G_{1} \times \partial G_{2}\), we have \(p_{G_{1}}(\bar{x}) = p_{G_{1}}(x/p_{G}(x,\,\tau )) = 1\) and \(p_{G_{2}}(\bar{\tau }) = p_{G_{2}}(\tau /p_{G}(x,\,\tau )) = 1\). Therefore, \(p_{G}(x,\,\tau ) = p_{G_{1}}(x) = p_{G_{2}}(\tau )\). Since (x, τ) ∈ G c, we have \(p_{G_{1}}(x) = p_{G_{2}}(\tau ) = p_{G}(x,\,\tau ) > 1\). Then by Lemma 6.29, we have \((x,\,\tau ) \in G_{1}^{c} \times G_{2}^{c}\). Moreover, it follows again by Lemma 6.30 that \(p_{G}(\tilde{x},\,\tau ) \leq p_{G}(x,\,\tau )\) implies \(p_{G_{1}}(\tilde{x}) \leq p_{G_{1}}(x)\). This proves the claim.
Then by (6.167) and (6.168), we have
From Case 3, we know that \(x \cdot f(x,\,\widetilde{x},\,\sigma )\) and τ ⋅g(x, τ, σ) are both negative definite for all (x, τ) ∈ G c and σ ∈ Σ with \((\overline{x},\,\overline{\tau }) \in \partial G_{1} \times \partial G_{2}\), and all \((\tilde{x},\,\tau ) \in {\mathbb{R}}^{N} \times \mathbb{R}\) with \(p_{G}(\tilde{x},\,\tau ) \leq p_{G}(x,\,\tau )\). That is, condition (iii) of Theorem 6.23 is satisfied.
It follows from Theorem 6.23 that the range of all the periodic solutions of (6.74) with σ ∈ Σ is contained in G. Similarly, if (H2) holds, we can obtain from Theorem 6.23 the same conclusion. This completes the proof. □
6.9.5 Global Continuation of Rapidly Oscillating Periodic Solutions: An Example
In this section, we illustrate the general results in the previous subsections by applying them to the study of the global continua of rapidly oscillating periodic solutions for the following differential equations with state-dependent delay:
where tanh(τ) = (e2τ − 1) ∕ (e2τ + 1) and μ > 0 is a constant. We make the following assumptions:
- (α 1):
-
\(b,\,h : \mathbb{R} \rightarrow \mathbb{R}\) are C 2 functions with b′(0) = − 1;
- (α 2):
-
There exist h 0 < h 1 in (1 ∕ 2, 1) such that \(h_{1} > h(x) > h_{0}\) for all \(x \in \mathbb{R}\);
- (α 3):
-
b is decreasing on \(\mathbb{R}\);
- (α 4):
-
xb(x) < 0 for x ≠ 0, and there exists a continuous function \(M : \mathbb{R} \ni \sigma \rightarrow M(\sigma ) \in (0,\,+\infty )\) such that
$$\displaystyle{\frac{b(x)} {x} > -{\frac{\mu } {\sigma }^{2}}}$$for every \(x \in \mathbb{R}\) with | x | ≥ M(σ);
- (α 5):
-
There exists M 0 > 0 such that | b′(x) | < M 0 for every \(x \in \mathbb{R}\);
- (α 6):
-
h′(x) = 0 only if x satisfies \(-\mu x {+\sigma }^{2}b(x) = 0\).
Remark 6.5.
We use tanh(τ) just for the sake of simplicity. Other types of functions can be used with minor changes in our arguments below.
We start with the uniform boundedness of periodic solutions (x(t), τ(t)) of (6.169).
Lemma 6.31.
Assume that (α 1 )–(α 4 ) hold. Then the range of every periodic solution (x, τ) of (6.169) with \(\sigma \in \mathbb{R}\) is contained in
Proof.
If σ = 0, the only periodic solution is \(\left (0,\,-\frac{\ln (2h(0)-1)} {2} \right )\), which is contained in Ω 1. Now we assume that σ ≠ 0. If x > 0, then by assumptions (α 3) and (α 4), we have
for every \(x \in \mathbb{R}\) with x ≥ M(σ). It follows that
Similarly, we have
Thus,
It is clear from (α 2) that for all \(x \in \mathbb{R}\),
To obtain an upper bound for τ, where (x, τ) is a periodic solution of (6.169), we introduce the following change of variable:
Then system (6.169) is transformed to
By (α 2) and the monotonicity of tanhτ, we have, for every \(z\not\in \left (\frac{\ln (2h_{0}-1)} {4},\,\,\right.\) \(\left.-\frac{\ln (2h_{0}-1)} {4} \right )\) and for all \(x \in \mathbb{R}\),
Thus it follows from Corollary 6.2, (6.170), and (6.173) that the range of all the periodic solutions (x, z) of (6.172) is contained in \((-M(\sigma ),\,M(\sigma )) \times \left (\frac{\ln (2h_{0}-1)} {4},\,\,-\frac{\ln (2h_{0}-1)} {4} \right )\). Then by (6.171), all periodic solutions (x, τ) of (6.169) with σ ≠ 0 are contained in Ω 1. The proof is complete. □
Now we consider the global Hopf bifurcation problem of system (6.169) under the assumptions (α 1)–(α 6). By (α 4), (x, τ) = (0, τ ∗ ) is the only stationary solution of (6.169), where \({\tau }^{{\ast}} = -\frac{1} {2}\ln (2h(0) - 1) > 0\). Freezing the state-dependent delay τ(t) at τ ∗ for the term x(t − τ(t)) of (6.169) and linearizing the resulting system with constant delay at the stationary solution (0, τ ∗ ), we obtain the following formal linearization of system (6.169):
where
In the following, we regard σ as the bifurcation parameter. We obtain the characteristic equation of the linear system corresponding to (6.174):
Since the zero of λ + q = 0 is − q, which is real, Hopf bifurcation points are related to zeros of only the first factor \((\lambda +\mu {+\sigma }^{2}\mathrm{{e}}^{{-\tau }^{{\ast}}\lambda })\). To locate local Hopf bifurcation points, we let λ = iβ, β > 0, in \(\lambda +\mu {+\sigma }^{2}\mathrm{{e}}^{{-\tau }^{{\ast}}\lambda } = 0\) and express the resulting equation in terms of its real and imaginary parts as
It is easy to verify the following lemma.
Lemma 6.32.
(i) All the positive solutions of (6.177) can be represented by an infinite sequence \(\{\beta _{n}\}_{n=1}^{+\infty }\) that satisfies \(0 <\beta _{1} <\beta _{2} < \cdots <\beta _{n} < \cdots \), \(\lim _{n\rightarrow +\infty }\beta _{n} = +\infty \) , and
-
(ii)
± i β n are characteristic values of the stationary solution (0, τ ∗ , σ n ), where
$$\displaystyle{\sigma _{n} = \pm {(\beta _{n}^{2} {+\mu }^{2})}^{1/4}.}$$If σ ≠ σ n , then the stationary solution (0, τ ∗ , σ) has no purely imaginary characteristic value.
-
(iii)
Let \(\lambda _{n}(\sigma ) = u_{n}(\sigma ) +\mathrm{ i}v_{n}(\sigma )\) be the root of (6.176)for σ close to σ n such that \(u_{n}(\sigma _{n}) +\mathrm{ i}v_{n}(\sigma _{n}) =\mathrm{ i}\beta _{n}\) . Then
$$\displaystyle{u^{\prime}_{n}(\sigma )_{\,\sigma =\sigma _{n}} = \frac{2} {\sigma _{n}} \frac{{{(\mu }^{2} +\beta _{ n}^{2})\tau }^{{\ast}}+\mu } {{(1 {+\mu \tau }^{{\ast}})}^{2} + {(\beta {_{n}\tau }^{{\ast}})}^{2}}.}$$
Now we are able to state our main results.
Theorem 6.24.
Assume that (α 1 )–(α 6 ) hold. Let \(\beta _{n} \in \left (\frac{(4n-3)\pi } {{2\tau }^{{\ast}}},\, \frac{(4n-2)\pi } {{2\tau }^{{\ast}}} \right ),\) n ≥ 1, be as given in (i) of Lemma 6.32 . Let \(\sigma _{n} = \pm {{(\mu }^{2} +\beta _{n})}^{1/4}\) for n ≥ 1. Then:
-
(a)
There exists an unbounded connected component \(C\left (0{,\,\tau }^{{\ast}},\,\sigma _{n},\, \frac{2\pi } {\beta _{n}}\right )\) of the closure of all the nonconstant periodic solutions of system (6.169) , bifurcated from \((0{,\,\tau }^{{\ast}},\,\sigma _{n},\, \frac{2\pi } {\beta _{n}})\) in the Fuller space where σ satisfies sgn (σ n )σ > 0.
-
(b)
\((0{,\,\tau }^{{\ast}},\,\sigma _{1},\, \frac{2\pi } {\beta _{1}} )\not\in C\left (0{,\,\tau }^{{\ast}},\,\sigma _{n},\, \frac{2\pi } {\beta _{n}}\right )\) for every n ≥ 2.
-
(c)
For every n ≥ 2, the projection of \(C\left (0{,\,\tau }^{{\ast}},\,\sigma _{n},\, \frac{2\pi } {\beta _{n}}\right )\) onto the parameter space \(\mathbb{R}\) is unbounded in (0, +∞) if σ n > 0 and is unbounded in (−∞, 0) if σ n < 0.
Proof.
(a) We apply Theorem 6.15. We first verify assumptions (SHB1)–(SHB3) and (SHB5). It is clear that (α 2) and (α 1) imply (SHB1), (SHB2), and (SHB5). Let us check (SHB3). Indeed, noticing that \(\sigma _{n} = \pm {{(\mu }^{2} +\beta _{ n}^{2})}^{1/4}\), b′(0) = − 1, and β n > 0, we have
Also, it follows from \({\tau }^{{\ast}} = -\frac{\ln (2h(0)-1)} {2}\) that
Therefore, condition (SHB3) is satisfied by system (6.169).
We note from Lemma 6.32 (i), (ii), and (iii) that every center (including those with σ < 0) of system (6.174) is isolated. We now calculate the crossing number of \((0{,\,\tau }^{{\ast}},\,\sigma _{n},\,\beta _{n})\). Let \(u_{n}(\sigma ) +\mathrm{ i}v_{n}(\sigma )\) be the characteristic value of (6.174) such that \(u_{n}(\sigma _{n}) +\mathrm{ i}v_{n}(\sigma _{n}) =\mathrm{ i}\beta _{n}\). By (iv) of Lemma 6.32, we have
That is, \(\frac{d} {d\sigma }u_{n}(\sigma )_{\,\sigma =\sigma _{n}}\) has the same sign as σ n , since τ ∗ > 0 and μ > 0. We note from (6.80) that the crossing number \(\gamma (0{,\,\tau }^{{\ast}},\,\sigma _{n},\, \frac{2\pi } {\beta _{n}})\) counts the difference, as σ varies from σ n − to σ n + , of the number of imaginary characteristic values with positive real parts in a small neighborhood of iβ n in the complex plane, where \(\sigma _{n}^{-} <\sigma _{n} <\sigma _{ n}^{+}\) are numbers in a small neighborhood of σ n . Then by (6.180), the crossing number of the isolated center \((0{,\,\tau }^{{\ast}},\,\sigma _{n},\, \frac{2\pi } {\beta _{n}})\) in the Fuller space \(C(\mathbb{R};\, {\mathbb{R}}^{2}) \times {\mathbb{R}}^{2}\) satisfies
Then by Theorem 6.15, there exists a connected component \(C\left (0{,\,\tau }^{{\ast}},\,\sigma _{n},\, \frac{2\pi } {\beta _{n}}\right )\) of the closure of all the nonconstant periodic solutions of system (6.169), bifurcated from the stationary solution \((0{,\,\tau }^{{\ast}},\,\sigma _{n},\, \frac{2\pi } {\beta _{n}})\) in the Fuller space. Note that there is no nonconstant periodic solution for the system (6.169) if σ = 0, since in this case, x satisfies a scalar ordinary differential equation. Moreover, there is no bifurcation point at σ = 0. Therefore, \(C\left (0{,\,\tau }^{{\ast}},\,\sigma _{n},\, \frac{2\pi } {\beta _{n}}\right )\) is located in the Fuller space where σ satisfies sgn(σ n )σ > 0.
To prove the unboundedness of \(C\left (0{,\,\tau }^{{\ast}},\,\sigma _{n},\, \frac{2\pi } {\beta _{n}}\right )\) in the Fuller space, we apply the global Hopf bifurcation Theorem 6.17 to exclude the case that there are finitely many bifurcation points in \(C\left (0{,\,\tau }^{{\ast}},\,\sigma _{n},\, \frac{2\pi } {\beta _{n}}\right )\).
Now we suppose there are finitely many bifurcation points \(\{(0{,\,\tau }^{{\ast}},\,\sigma _{n_{j}},\, \frac{2\pi } {\beta _{n_{j}}})\}_{j=1}^{q}\), \(q \in \mathbb{N}\), in \(C\left (0{,\,\tau }^{{\ast}},\,\sigma _{n},\, \frac{2\pi } {\beta _{n}}\right )\). We know that \(C\left (0{,\,\tau }^{{\ast}},\,\sigma _{n},\, \frac{2\pi } {\beta _{n}}\right )\) is located in the Fuller space where σ satisfies sgn(σ n )σ > 0. Then the bifurcation points \(\{(0{,\,\tau }^{{\ast}},\,\sigma _{n_{j}},\, \frac{2\pi } {\beta _{n_{j}}})\}_{j=1}^{q}\) satisfy \(\mathrm{sgn}(\sigma _{n})\sigma _{n_{j}} > 0\) for all j ∈ { 1, 2, ⋯ , q}.
Let \(\epsilon _{n_{j}}\) be the value of
evaluated at \((\theta _{1},\,\theta _{2},\,\sigma ) = (0,\,0,\,\sigma _{n_{j}})\) and \((\gamma _{1},\,\gamma _{2},\,\sigma ) = (0{,\,\tau }^{{\ast}},\,\sigma _{n_{j}})\), where
Then by (6.178) and (6.179), we have
By (6.181) and (6.182), we have
Note that (α 5) and (α 6) implies (SHB4). Then by Theorem 6.17, (6.183) is a contradiction. The unboundedness of \(C\left (0{,\,\tau }^{{\ast}},\,\sigma _{n},\, \frac{2\pi } {\beta _{n}}\right )\) follows.
(b) In order to verify assumption (SHB7), we claim that the virtual period p n of every bifurcation point \((0{,\,\tau }^{{\ast}},\,\sigma _{n},\,2\pi /\beta _{n})\) satisfies
Suppose that there exist \(m_{0},\,n_{0} \in \mathbb{N}\) such that \(m_{0}p_{n_{0}} = m_{0} \cdot 2\pi /\beta _{n_{0}} {=\tau }^{{\ast}}\). We note that
Then we have
This is a contradiction, and the claim is proved.
We note that by (6.185), a sufficient condition for \(p_{n} = \frac{2\pi } {\beta _{n}} {<\tau }^{{\ast}}\), is that \(\frac{2\pi } {\beta _{n}} < {4\tau }^{{\ast}}/(4n - 3) {<\tau }^{{\ast}}\), that is, n ≥ 7 ∕ 4. Therefore, every \((0{,\,\tau }^{{\ast}},\,\sigma _{n},\,p_{n})\) with n ≥ 2 is a bifurcation point of system (6.169) satisfying
For the bifurcation point \((0{,\,\tau }^{{\ast}},\,\sigma _{1},\,p_{1})\), we can conclude from (6.185) that
We want to obtain the uniform boundedness of the period in \(C(0{,\,\tau }^{{\ast}},\,\sigma _{n},\, \frac{2\pi } {\beta _{n}})\) with n ≥ 2. We only need to check the conditions (SHB6)–(SHB10) for applying Theorems 6.21 and 6.22.
It is clear that (α 4), (6.184), and (6.179) imply (SHB8), (SHB9), and (SHB6), respectively. Also we conclude from (SHB2), (SHB4), and Lemma 6.20 that
for every \((x,\,\tau,\,\sigma,\,p) \in C(0{,\,\tau }^{{\ast}},\,\sigma _{n},\, \frac{2\pi } {\beta _{n}})\). Also, by Lemma 6.31, we have
for every \(t \in \mathbb{R}\), and hence (SHB10) is satisfied. To check (SHB7), we let
Then by (α 1), (α 4), and (α 6), the solutions of (6.190) are stationary solutions of (6.169). This verifies (SHB7).
Therefore, we can use Theorems 6.21, 6.22, (6.186), (6.188), and (6.189) to conclude that there exists some \(t \in \mathbb{R}\) such that
for every \((x,\,\tau,\,\sigma,\,p) \in C(0{,\,\tau }^{{\ast}},\,\sigma _{n},\, \frac{2\pi } {\beta _{n}})\) with n ≥ 2. Then by (6.187) and (6.191), we know that \((0{,\,\tau }^{{\ast}},\,\sigma _{1},\, \frac{2\pi } {\beta _{1}} )\not\in C\left (0{,\,\tau }^{{\ast}},\,\sigma _{n},\, \frac{2\pi } {\beta _{n}}\right )\) for every n ≥ 2. This proves (b).
(c) Let Σ be the projection of \(C\left (0{,\,\tau }^{{\ast}},\,\sigma _{n},\, \frac{2\pi } {\beta _{n}}\right )\) on the σ-parameter space \(\mathbb{R}\). By (a), we know that Σ ⊆ (0, + ∞) if σ n > 0 and Σ ⊆ ( − ∞, 0) if σ n < 0. By Lemma 6.31, we know that for every σ ∈ Σ, there exists a constant M n (σ) > 0 such that
where (x, τ, σ, p) is the solution associated with σ in \(C\left (0{,\,\tau }^{{\ast}},\,\sigma _{n},\, \frac{2\pi } {\beta _{n}}\right )\) and \(M_{n} : \mathbb{R} \ni \sigma \rightarrow M_{n}(\sigma ) \in (0,\,+\infty )\) is a continuous function on \(\mathbb{R}\).
We know from (6.191) that the projection of \(C\left (0{,\,\tau }^{{\ast}},\,\sigma _{n},\, \frac{2\pi } {\beta _{n}}\right )\) on the p-parameter space \(\mathbb{R}\) is bounded. If Σ is bounded, then it follows from (a) that the projection of \(C\left (0{,\,\tau }^{{\ast}},\,\sigma _{n},\, \frac{2\pi } {\beta _{n}}\right )\) on the (x, τ)-space \(C(\mathbb{R};\, {\mathbb{R}}^{N+1})\) must be unbounded in the supremum norm. But by the continuity of M n on \(\mathbb{R}\) and by (6.192), the projection of \(C\left (0{,\,\tau }^{{\ast}},\,\sigma _{n},\, \frac{2\pi } {\beta _{n}}\right )\) on the (x, τ)-space \(C(\mathbb{R};\, {\mathbb{R}}^{N+1})\) is uniformly bounded with respect to σ ∈ Σ. This is a contradiction, and the proof is complete. □
We conclude by noting that the global continuation of slowly oscillating periodic solutions is addressed in [172].
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Guo, S., Wu, J. (2013). Degree Theory. In: Bifurcation Theory of Functional Differential Equations. Applied Mathematical Sciences, vol 184. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-6992-6_6
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