Abstract
A diffeomorphism of a plane into itself with a fixed hyperbolic point and a non-transversal point homoclinic to it is studied. There are various ways of touching stable and unstable manifolds at the homoclinic point. Periodic points whose trajectories do not leave the neighborhood of the trajectory of a homoclinic point are divided into many types. Periodic points of the same type are called n-pass periodic points if their trajectories have n turns lying outside a sufficiently small neighborhood of the hyperbolic point. Diffeomorphisms of the plane with a non-transversal homoclinic point were previously analyzed in the studies of Sh. Newhouse, L.P. Shil’nikov, and B.F. Ivanov, where it was assumed that this point is a tangency point of finite order. In these papers, it was shown that infinite sets of stable two-pass and three-pass periodic points can lie in a neighborhood of a homoclinic point. The presence of such sets depends on the properties of the hyperbolic point. In this paper, we assume that a homoclinic point is not a point with a finite-order tangency of a stable and an unstable manifold. It is shown that, for any fixed natural number n, the neighborhood of a non-transversal homoclinic point can contain an infinite set of stable n-pass periodic points with characteristic exponents bounded away from zero.
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1 INTRODUCTION
In the paper, we study diffeomorphism of a plane into itself with a fixed hyperbolic point and a non-transversal point homoclinic to it. It is assumed that the homoclinic point is not a point of finite-order tangency. The main goal of this study is to show that, under certain conditions, an arbitrarily small neighborhood of a homoclinic point contains an infinite set of stable multi-pass periodic points with characteristic exponents bounded away from zero. In [1], an example is given for a two-dimensional diffeomorphism with an infinite set of periodic points whose trajectories lie in a bounded set of the plane, and their characteristic exponents are bounded away from zero. It is known that in this case, under small perturbations, the diffeomorphism remains to have arbitrarily many stable periodic points with characteristic exponents bounded away from zero.
The point that lies at the intersection of stable and unstable manifolds of a hyperbolic point is called a homoclinic point. If these manifolds are tangent to each other at the homoclinic point, it is called a non-transversal homoclinic point. A periodic point whose trajectory does not leave the neighborhood of the trajectory of the homoclinic point but has n (n > 3) turns lying outside a sufficiently small neighborhood of the hyperbolic point is called a multi-pass or n-pass periodic point.
In [2–6], a neighborhood of a non-transversal homoclinic point was studied. The homoclinic point was assumed to be a point with a finite-order tangency of stable and unstable manifolds.
Let f be self-diffeomorphism of the plane with a hyperbolic fixed point at the origin and a non-transversal point homoclinic to it, and let λ and μ be the eigenvalues of matrix Df(0), where 0 < λ < 1 < μ. Let
In [2–6], it was assumed that θ > 1 and it was shown that there exists an unbounded set Θ such that, for θ ∈ Θ, a neighborhood of a non-transversal homoclinic point contains infinite sets of two-pass or three-pass stable periodic points; however, it follows from [3] that there exists unbounded set Θ1 such that, for θ ∈ Θ1, a neighborhood of a non-transversal homoclinic point contains no stable two-pass periodic points.
The present paper is a continuation of [7, 8]. In these papers, it is assumed that a non-transversal homoclinic point is not a point of finite-order tangency of stable and unstable manifolds. Sufficient conditions are given under which infinite sets of single-pass or two-pass stable periodic points with characteristic exponents bounded away from zero exist in a neighborhood of the homoclinic point. In this paper, we give sufficient conditions in order that, for any θ > n, where n is a natural number (n > 3), the neighborhood of the homoclinic point contains an infinite set of n-pass stable periodic points with characteristic exponents bounded away from zero.
2 BASIC DEFINITIONS AND NOTATION
Let f be a C1 self-diffeomorphism of the plane with a hyperbolic fixed point at the origin: f : \({{\mathbb{R}}^{2}}\) → \({{\mathbb{R}}^{2}}\), f(0) = 0. Suppose that
where 0 < λ < 1 < μ.
Fix a natural number n ≥ 4. Suppose that
As usual, let W s(0) and W u(0) be stable and unstable manifolds of the hyperbolic point. It is known that
where f k and f –k are the degrees of the diffeomorphisms f and f –1.
Let point w be such that w ≠ 0, w ∈ W s(0) ∩ W u(0); this point is a homoclinic point. Suppose that Ws(0) and Wu(0) are tangent to each other at point w; then, w is a non-transversal homoclinic point. It is clear that
Assume that f has the following form in some bounded neighborhood V0 of the origin
for (x, y) ∈ V0.
Let w1 = (0, y0) and w2 = (x0, 0) be two points of the orbit of homoclinic point w such that w1 ∈ V0 and w2 ∈ V0. Suppose that, for certain \(\bar {\lambda }\) and \(\bar {\mu }\) such that λ < \(\bar {\lambda }\) <1 and 1 < \(\bar {\mu }\) < μ, we have inclusion
Assume that
It follows from the definition of a homoclinic point that there is natural number ω > 1 such that f ω(w1) = w2. We assume that f k(w1) ∉ V, k = 1, 2, …, ω − 1.
Let U be a neighborhood of point w1 such that U ⊂ V, f ω(U) ⊂ V, f k(U) ∩ V = \(\not {0}\), k = 1, 2, …, ω − 1, and sets U, f(U), …, f ω(U) do not intersect pairwise.
We call
the extended neighborhood of the homoclinic point.
Periodic point u ∈ U is called an n-pass periodic point if its trajectory is located in U0 and the intersection of its orbit with U consists of n different points.
We denote restriction f ω|U by L. It is clear that L is a mapping of class C1. We write mapping L in coordinates
where F1(x, y − y0) and F2(x, y − y0) are C1-functions defined in U such that F1(0, 0) = F2(0, 0) = 0.
Point w2 is called a homoclinic point of finite-order tangency if there exists natural number l > 1 such that
It was assumed in [2–6] that a homoclinic point is a point with a finite order of tangency. In this paper, as in [7, 8], a different way of touching stable and unstable manifolds is studied.
3 FORMULATION OF THEOREMS
Let f be a C1-self-diffeomorphism of the plane with a hyperbolic fixed point at the origin and a non-transversal point homoclinic to it. Suppose that conditions (2) are satisfied in some bounded neighborhood of the origin. Let w1 = (0, y0) and w2 = (x0, 0) be two points of the orbit of homoclinic point w such that conditions (3) and (4) hold and extended neighborhood U0 is defined. Mapping L = f ω|U is defined in neighborhood U of point w1.
Assume that the coordinate functions of mapping L has form
where a and b are real numbers such that
and functions g, φ1, and φ2 are such that φ1(0, 0) = φ2(0, 0) = 0 and g(0) = \(\frac{{dg(0)}}{{dy}}\) = 0. It is clear that φ1, φ2, and their first-order derivatives are bounded in U, while function g and its derivative are bounded in a neighborhood of zero.
The nature of tangency of stable and unstable manifolds at point w2 is specified by the properties of function g. Let us describe the properties of this function using sequences. Let σ = σ(k) and ε = ε(k) be positive vanishing sequences such that, for any k,
Let i0 = i0(k) be an increasing sequence of natural numbers for which there exists natural number s ≥ n such that
for any k.
Suppose that, for any k,
Let function g be such that there exist sequences with the above properties such that, for any k,
where Δ = Δ(k) is a vanishing sequence of real numbers.
Suppose that there exists real number α > 1 such that, for any k, inequality
holds for t ∈ (σ(k) − ε(k), σ(k) + ε(k)).
It follows from (11) and (12) that point w2 is not a point of tangency of finite order. Obviously, conditions (5) are not satisfied in this case.
Let i1 = i1(k), i2 = i2(k), …, in – 1 = in – 1(k) be increasing sequences of natural numbers such that
where m = 1, 2, …, n − 1, l = 1, 2, …, n − 1, and m ≠ l.
Suppose that function g be such that there exists set of sequences i1 = i1(k), i2 = i2(k), …, in − 1 = in − 1(k) satisfying conditions (13) such that, for any k, the following system of equations has a solution:
Let ξ2 = ξ2(k), ξ3 = ξ3(k), …, ξn − 1 = ξn − 1(k) be a solution of system (14). Sequence ξ1 = ξ1(k) is defined as follows
Theorem 1. Let f be a self-diffeomorphism of the plane with a fixed hyperbolic point at the origin and a non-transversal point homoclinic to it. Let conditions (1)–(4), (6)–(12) hold. Suppose that that there exists a set of increasing sequences of natural numbers i1 = i1(k), i2 = i2(k), …, in − 1 = in − 1(k) satisfying conditions (13) such that system of equations (14) is solvable. Assume that the following inequalities hold for any k:
Then, any neighborhood of homoclinic point w1 contains a countable set of n-pass stable periodic points the characteristic exponents of which are bounded away from zero.
Note that in order to meet conditions (15), it is necessary that ξ1(k) ∉ (σ(k) − ε(k), σ(k) + ε(k)). Otherwise, inequalities (15) contradict (11) and (12).
Since g(t) is a C1-function of one variable defined in a neighborhood of zero that satisfies (11) and (12), there exist sequences τ1 = τ1(k) and τ2 = τ2(k) such that (τ1(k), τ2(k)) ⊂ (σ(k) + ε(k), σ(k − 1) − ε(k − 1)), and
for t ∈ (τ1(k), τ2(k)).
Theorem 2. Let g(t) be a C1-function of one variable defined in a neighborhood of zero and satisfying (8)–(12). Suppose that, for any k, inclusions
hold, where sequences τ1(k) and τ2(k) are defined by conditions (16). Then, for any set of increasing natural sequences i1 = i1(k), i2 = i2(k), …, in − 1 = in − 1(k) satisfying conditions (13), system of equations (14) is solvable.
4 PROOF OF THEOREM 1
Let k be a sufficiently large natural number. Let σ, ε, Δ, i0, i1, …, in − 1, ξ1, ξ2, …, ξn − 1 be elements of the corresponding sequences with sufficiently large k. In the proof of the theorem, index k of the sequences is omitted.
For any k, we define x0 = \({{\lambda }^{{{{i}_{{n - 1}}}}}}({{x}^{0}}\) + aξn − 1), x1 = \({{\lambda }^{{{{i}_{0}}}}}({{x}^{0}}\) + aσ), and xm = \({{\lambda }^{{{{i}_{{m - 1}}}}}}({{x}^{0}}\) + aξm − 1), where m = 2, 3, …, n − 1. Let us define sets
where m = 1, 2, …, n − 1. We consider that Um ⊂ U for m = 0, 1, …, n − 1.
We show that inclusions \({{f}^{{{{i}_{m}}}}}L({{U}_{m}})\) ⊂ Um + 1 (where m = 0, 1, …, n − 2) and \({{f}^{{{{i}_{{n - 1}}}}}}L({{U}_{{n - 1}}})\) ⊂ U0 hold.
Let (x, y) ∈ U0. Clearly, x = x0 + u0, y = y0 + σ + \({{{v}}_{0}}\), where |u0| ≤ \({{\lambda }^{{{{i}_{0}}}}}\)(|a| + 1)ε and \(\left| {{{{v}}_{0}}} \right|\) ≤ ε. Define
From conditions (2) and (6), we have
whence, in view of conditions (11), we obtain
From conditions (12), we have
from which, taking into account the properties of functions φ1 and φ2 and conditions (10), we get
Inclusion \({{f}^{{{{i}_{0}}}}}L({{U}_{0}})\) ⊂ U1 is proven.
Let (x, y) ∈ U1. Clearly, x = x1 + u1, y = y0 + ξ1 + \({{{v}}_{1}}\), where |u1| ≤ \({{\lambda }^{{{{i}_{0}}}}}\)(|a| + 1)ε and \(\left| {{{{v}}_{1}}} \right|\) ≤ \(\varepsilon {{\mu }^{{ - ({{i}_{1}} + {{i}_{2}} + \ldots + {{i}_{{n - 1}}})}}}\). Define
From conditions (2) and (6), we obtain
From conditions (15), we have
It follows from the properties of functions g that
hence, we obtain
Inclusion \({{f}^{{{{i}_{1}}}}}L({{U}_{1}})\) ⊂ U2 is proven.
Inclusions \({{f}^{{{{i}_{m}}}}}L({{U}_{m}})\) ⊂ Um + 1, where m = 2, 3,…, n − 1, and \({{f}^{{{{i}_{{n - 1}}}}}}L({{U}_{{n - 1}}})\) ⊂ U0 can be proven in a similar way taking conditions (14) into account; consequently, we obtain \({{f}^{{{{i}_{{n - 1}}}}}}L \ldots {{f}^{{{{i}_{0}}}}}L({{U}_{0}})\) ⊂ U0.
It follows from the previous reasoning that there exists point z0 ∈ U0, z0 = (\(x_{0}^{*}\), \(y_{0}^{*}\)) such that \({{f}^{{{{i}_{{n - 1}}}}}}L \ldots {{f}^{{{{i}_{0}}}}}L({{z}_{0}})\) = z0 and U0 contains a periodic point of the original diffeomorphism. Let zm = (\(x_{m}^{*}\), \(y_{m}^{*}\)), zm ∈ Um be points from the orbit of the periodic point such that zm = \({{f}^{{{{i}_{{m - 1}}}}}}L \ldots {{f}^{{{{i}_{0}}}}}L({{z}_{0}})\), m = 1, 2, …, n − 1.
Define
In order to prove the stability of points z0, we estimate the eigenvalues of this matrix. It is clear that
where m = 0, 1, 2, …, n − 1.
It is easy to see that
where quantity A depends on k but is bounded for all k.
We introduce notation
where m = 0, 1, 2, …, n − 1. The following equalities define matrices Ψ and Φm for m = 0, 1, 2, …, n − 1:
Elements of the matrices Ψ and Φm, where m = 0, 1, 2, …, n − 1, depend on k but is bounded for all k.
It is easy to show by induction on l for l = 1, 2, …, n − 1 that
where β11(l), β12(l), β21(l), β22(l), l = 1, 2, …, n − 1, depend on k but is bounded for all k.
From (12) we have |g0| < \({{\mu }^{{ - \alpha n{{i}_{0}}}}}\); consequently, there exist a quantity B independent of k such that
where TrΞ is the trace of the matrix Ξ and γ = min[θ − n, n(α − 1)].
Let ρ1 and ρ2 be the eigenvalues of the matrix Ξ. It is known that
Let us show that there exist C > 0 and k0 such that, for any k > k0 and m = 1, 2, we have the inequalities
Suppose that (19) do not hold, then, for any C > B, there exists a sequence of indices k such that
therefore,
hence,
The last inequality contradicts equalities (18). Inequalities (19) are proven.
Characteristic exponents νm, m = 1, 2, of periodic points z0 ∈ U0 are defined as follows:
where m = 1, 2.
It follows from conditions (13), (19) that, for sufficiently large k,
where m = 1, 2.
The last inequalities prove Theorem 1.
5 PROOF OF THEOREM 2
Let k be a sufficiently large natural number. In the proof of the theorem, the index k of the sequences σ, ε, Δ, i0, i1, …, in − 1, τ1, and τ2 is omitted. It is assumed that ξ2, ξ3, …, ξn − 1 are real variables.
Let P = {(ξ2, ξ3, …, ξn − 1), ξm ∈ (τ1, τ2), m = 2, 3, …, n − 1}. On these sets, for any k, we define mapping G : P → \({{\mathbb{R}}^{{n - 2}}}\), or in coordinate form
where
Let Hn − 2 be the determinant of order n − 2 of form
It is clear that
Using the properties of the determinant, we can easily show that
\(\operatorname{Det} DG({{\xi }_{2}},{{\xi }_{3}}, \ldots ,{{\xi }_{{n - 1}}}) = {{H}_{{n - 2}}} + {{( - 1)}^{{n - 1}}}{{(ab)}^{{n - 1}}}{{\lambda }^{{{{i}_{1}} + {{i}_{2}} + \ldots + {{i}_{{n - 1}}}}}}{{\mu }^{{{{i}_{0}}}}}.\)
It follows from the last formulas and conditions (16) and (17) that
for any (ξ2, ξ3, …, ξn − 1) ∈ P. It is easy to see that the mapping G is one-to-one on P.
From conditions (13) for sufficiently large k and any m = 2, 3, …, n − 1, we have
It follows from these inequalities and conditions (17) that the following inclusions hold for sufficiently large k:
The last inclusions imply the existence of a solution to system (14).
Theorem 2 is proven.
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Funding
This study was supported by the Russian Foundation for Basic Research (grant no. 19-01-00388).
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Translated by I. Tselishcheva
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Vasil’eva, E.V. Multi-Pass Stable Periodic Points of Diffeomorphism of a Plane with a Homoclinic Orbit. Vestnik St.Petersb. Univ.Math. 54, 227–235 (2021). https://doi.org/10.1134/S1063454121030092
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DOI: https://doi.org/10.1134/S1063454121030092