Andronov–Hopf Bifurcation in Logistic Delay Equations with Diffusion and Rapidly Oscillating Coefficients

Abstract

A logistic delay equation with diffusion, which is important in applications, is studied. It is assumed that all of its coefficients, as well as the coefficients in the boundary conditions, are rapidly oscillating functions of time. An averaged equation is constructed, and the relation between its solutions and the solutions of the original equation is studied. A result on the stability of the solutions is formulated, and the problem of local dynamics in the critical case is studied. An algorithm for constructing the asymptotics of the solutions and an algorithm for studying their stability are proposed. It is important to note that the corresponding algorithm contains both a regular and a boundary layer component. Meaningful examples are given.

1. STATEMENT OF THE PROBLEM

The Krylov–Bogolyubov averaging principle [1]–[4] is a powerful tool for studying nonlinear systems of differential equations with rapidly oscillating coefficients. This principle was extended to delay equations in [3]–[6]. Some applications of the principle of averaging, including those for bifurcation problems, were studied in detail in [7], [8] for the following logistic delay equation with diffusion, which is important in applications (see, e.g., [9]):

$$\dot u=-ru(t-T)(u+1). $$

(1.1)

The coefficients $r$ and $T$ were assumed positive and periodic functions rapidly oscillating with respect to $t$. In [7] and [8], averaged equations were constructed for various cases and conclusions concerning the relationship between the solutions of the original and averaged equations were formulated. Note specially that the oscillations of the delay coefficient $T$, for example, may lead to the appearance of various classes of nonlinear averaged equations.

For systems of parabolic equations, the principle of averaging was justified in numerous papers (see, e.g., [10]–[14]). Relatively little progress has been made here, because, in general, averaging in the case of a rapidly oscillating diffusion coefficient is not well defined. A class of equations with rapidly oscillating coefficients in which the diffusion matrix is the product of a scalar rapidly oscillating function and a constant matrix was singled out in [15], where problems in which the boundary conditions also rapidly oscillate with respect to time were studied.

In the present paper, results from [7] and [8], together with those from [15], are applied to the study of the dynamics of the following nonlinear logistic delay equation with diffusion:

$$\frac{\partial u}{\partial t}=d(\omega t)\,\frac{\partial^2u}{\partial x^2} -r(\omega t)u(t-T(\omega t),x)[1+u] $$

(1.2)

under the boundary conditions

$$\frac{\partial u}{\partial x}\biggr|_{x=0}=0,\qquad \frac{\partial u}{\partial x}\biggr|_{x=1} =\gamma(\omega t)u|_{x=1}. $$

(1.3)

The positive coefficients $d(s)$, $r(s)$, and $T(s)$, as well as the coefficient $\gamma(s)$, are assumed $2\pi$-periodic. For the space of initial conditions for (1.2), (1.3), it is convenient to choose the space

$$W=C[-\max_sT(s),0]\times W^2_2(t).$$

A number of results on the dynamical properties of the solutions of delay equations with diffusion were given in [16].

We immediately note that, for simplicity, we shall restrict ourselves to studying the case in which the coefficients in (1.2) are independent of the spatial variable $x\in[0,1]$. For the same reason, oscillation is present only at one of the endpoints in the boundary conditions. The averaging problem for a logistic parabolic equation with coefficients rapidly oscillating with respect to the spatial variable was studied in [17].

The paper is structured as follows. In Sec. 2, an averaged equation will be constructed and main statements of a general nature will be presented. It is of interest to note that the average of the product $d(s)\gamma(s)$ plays an important role in the construction of the averaged boundary-value problem. A second important fact setting apart equations with diffusion from ordinary differential equations is the appearance of boundary-layer asymptotics at the boundary points of the closed interval $[0,1]$ in the asymptotic approximations for the solutions of (1.2), (1.3). Here we essentially use the approach of [17]–[19]. Section 3 is devoted to the construction of solutions of a special auxiliary equation that arises in Sec. 2. Section 4 deals with the Andronov–Hopf bifurcation for the boundary-value problem (1.2), (1.3). Using concrete examples, we shall study the role of oscillations in the diffusion coefficient $d(s)$ and the boundary condition coefficient $\gamma(s)$ in problems of bifurcation from the state of equilibrium of the boundary-value problem (1.2), (1.3).

Let us introduce some notation. By $M(*)$ we denote the average over $s=\omega t$ of the expression which is the argument of the operator $M$. By $M_0(v)$ we denote the average over $s$ of the “diffusionless” part:

$$M_0(v)=-M\bigl(r(s)v(t-T(s),x)[1+v(t,x)]\bigr).$$

Explicit expressions for $M_0(v)$ for the most interesting and important functions $r(s)$ and $T(s)$ were given in [7], [8].

2. CONSTRUCTION OF THE AVERAGED EQUATION

Consider the boundary-value problem

$$\frac{\partial v}{\partial t} =d_0\,\frac{\partial^2v}{\partial x^2}+M_0(v),$$

(2.1)

$$\frac{\partial v}{\partial x}\biggr|_{x=0} =0,\qquad \frac{\partial v}{\partial x}\biggr|_{x=1}=\gamma_0v|_{x=1}.$$

(2.2)

By $d_0$ and $\gamma_0$ we denote the expressions

$$d_0=M(d(s)),\qquad \gamma_0=d^{-1}_0M(d(s)\gamma(s)).$$

We shall need an auxiliary boundary-value problem for the function $v(s,y)$ which is $2\pi$-periodic in $s$ and exponentially decreasing with respect to $y$ as $y\to\infty$:

$$\frac{\partial v}{\partial s} =d(s)\,\frac{\partial^2v}{\partial y^2}+f(s,y),\qquad \frac{\partial v}{\partial y}\biggr|_{y=0}=g(s), $$

(2.3)

where $f(s,y)$ and $g(s)$ are $2\pi$-periodic in $s$ and, for some positive constants $f_0$ and $\delta_0$, the following estimate holds:

$$|f(s,y)|\le f_0\exp(-\delta_0y).$$

Lemma 1.

For the existence and uniqueness of a solution $v(s,y)$ of the boundary-value problem (2.3) which is $2\pi$-periodic in $s$ and satisfies the following estimate for some positive constants $v_0$ and $\delta_0$:

$$|v(s,y)|\le v_0\exp(-\delta_0y),$$

it is necessary and sufficient that the following equality hold:

$$M\biggl[d(s)g(s)+\int_0^\infty f(s,y)\,dy\biggr]=0.$$

Thus, it follows from Lemma 1 that the boundary-value problem

$$\frac{\partial w}{\partial s} =d(s)\,\frac{\partial^2w}{\partial y^2}\mspace{2mu},\qquad \frac{\partial w}{\partial y}\biggr|_{y=0}=\gamma(s)-\gamma_0 $$

(2.4)

has a unique $2\pi$-periodic (in $s$) solution $w(s,y)$ such that, for some universal positive constants $c_0$ and $\delta_0$, the following inequality holds:

$$|w(s,y)|\le c_0\exp(-\delta_0y). $$

(2.5)

The justification this lemma is standard. All the necessary constructions are given in Sec. 3.

Let $v(t,x)$ be a solution of the boundary-value problem (2.1), (2.2). By $z(s,t,x,\omega)$ we denote the $2\pi$-periodic (in $s$) function

$${z(s,t,x,\omega) =[1+v(t,x)]\int_0^s \biggl\{(d(s)-d_0)\,\frac{\partial^2v(t,x)}{\partial x^2}+M_0(v) -r(s)v(t-T(s),x)\biggr\}\,ds.} $$

(2.6)

Let $\psi(t,x)$ be an arbitrary function continuous in $t\in[-\max_sT(s)+t_0,t_0]$ and continuously differentiable with respect to $x\in[0,1]$. By $u(t,x,\omega)$ and $v(t,x)$ we denote the solutions of the boundary-value problem (1.2), (1.3) and (2.1), (2.2), respectively, with initial condition $\psi(t,x)$ given for $t=t_0$. Finally, by $\mu>0$ we denote an arbitrary quantity which is sufficiently small and independent of $\omega$.

The following statement establishes a detailed relation between the solutions with the same initial conditions of the original and averaged boundary-value problems.

Theorem 1.

For any fixed $\mu>0$ and $c>0$ , for $t\in[t_0+\mu,t_0+c]$ , the following asymptotic representation holds:

$$u(t,x,\omega)=v(t,x,\omega)+\omega^{-1/2}w(\omega t,y,t,\omega) +\omega^{-1}Z(\omega t,x,t,\omega), $$

(2.7)

in which

$$\begin{aligned} \, y &=(1-x)\omega^{1/2}, \nonumber \\ v(t,x,\omega) &=v(t,x)+\omega^{-1/2}v_1(t,x,\omega)+\dotsb, \end{aligned}$$

(2.8)

$$w(x,y,t,\omega) =w(s,y)v(t,1)+\omega^{-1/2}w_1(s,y,t,\omega)+\dotsb,$$

(2.9)

$$Z(s,x,t,\omega) =z(s,t,x,\omega)+\omega^{-1/2}z_1(s,t,x,\omega)+\dotsb\,.$$

(2.10)

We note that the appearance of the second summand on the right-hand side of (2.9) is due to the variable boundary condition. The principal part of this summand is determined from Lemma 1.

Further, we shall consider the periodic solutions of the boundary-value problem (1.2), (1.3).

Theorem 2.

Let $v_0(t,x)$ be an $h$-periodic solution of the boundary-value problem (2.1), (2.2), and let only one multiplier of the boundary-value problem linearized on $v_0(t,x)$ be equal to $1$ in absolute value. Then there exists a function $\alpha(\omega)$ of the parameter $\omega$ for which $\alpha(\omega)=O(\omega^{-1/2})$ and the boundary-value problem (1.2), (1.3) has an almost periodic solution $u_0(t,x,\omega)$ of the same stability as that of $v_0(t,x)$ and the following asymptotic equality holds:

$${u_0(t,x,\omega)=v_0((1+\alpha(\omega))t,x) +\omega^{-1/2}w(s,(1-x)\omega^{1/2})v_0((1+\alpha(\omega))t,1) +O(\omega^{-1}).} $$

(2.11)

We note that the coefficients of the formal series (2.8), (2.9), and (2.10) in powers of $\omega^{-1/2}$ are bounded functions of the parameter $\omega$. This effect is a consequence of the time delay in (1.2). The justifications of Theorems 1 and 2 are standard (see, e.g., [20]); so we shall not dwell on them.

3. ON THE SOLUTIONS OF THE AUXILIARY BOUNDARY-VALUE PROBLEM (2.3)

3.1. Phase Trajectories

It follows from the equality

$$\int_0^s\,d(s)\,\mathrm ds=d_0(s+a(s)),\qquad (s>0)$$

that the function $a(s)$ is $2\pi$-periodic. The condition $d(s)>0$ implies that

$$a'(s)>-1. $$

(3.1)

In (2.3), we make the change of time

$$\tau=s+a(s). $$

(3.2)

By virtue of (3.1), the value of $s$ can be expressed in terms of $\tau$:

$$s=\tau+b(\tau), $$

(3.3)

where $b(\tau)$ is $2\pi$-periodic. It follows from (3.2) and (3.3) that, to find $b(\tau)$, we can use the functional equation

$$b(\tau)=-a(\tau+b(\tau)). $$

(3.4)

This yields

$$\begin{gathered} \, b(0)=-a(b(0)), \\ \begin{alignedat}{2} a'(s+b(s))&=-b'(1+b'(s))^{-1},&\qquad b'(0)&=-a'(b(0))(1+a'(b(0))), \\ a''(s+b(s))&=-b''(s)(1+b'(s)),&\qquad b''(0)&=-a''(b(0))(1+b'(0)), \end{alignedat} \\ \dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots \dots\dots\dots\dots\dots\dots\dots\dots, \\ a^{(n)}(s+b(s))=P_n(b'(s),\dots,b^{(n)}(s)). \end{gathered}$$

Let us find a $2\pi$-periodic solution of Eq. (3.4). From (3.4) we directly obtain the following statement.

Lemma 2.

Let $a(\tau)$ be a $2\pi$ -periodic solution of the following equation of $n$ th order:

$$\frac{d^na}{d\tau^n}=F(a,a',\dots,a^{(n-1)}). $$

(3.5)

Then the function $b(\tau)$ is a $2\pi$ -periodic solution of the equation

$$\frac{d^nb}{ds^n}=G(b,b',\dots,b^{(n)}), $$

(3.6)

where

$$G(b,b',\dots,b^{(n)}) =F\biggl(-b,-\frac{b'}{1+b'}\mspace{2mu},P_2(b',b''), \dots,P_n(b',\dots,b^{(n)})\biggr),$$

with initial conditions

$$b(0)=a(b(0))=0,\qquad a^{(j)}(b(0))=P_j(b'(0),\dots,b^{(j)}(0)),\quad 1\le j\le n.$$

The expression (3.4) used to define $b(\tau)$ can be regarded as the operator $\Pi$ which, to a $2\pi$-periodic function $a(\tau)$, puts in correspondence the $2\pi$-periodic function $b(\tau)$:

$$b(\tau)=\Pi(a(\tau)). $$

(3.7)

The converse also holds, i.e.,

$$a(\tau)=\Pi(b(\tau)). $$

(3.8)

Consider the following example. Let

$$a(s)=a_0\sin(s),\qquad |a_0|<1. $$

(3.9)

Figure 1.

Phase trajectories for (3.10).

Figure 2.

Phase trajectories for (3.10).

Then

$$a''+a=0, $$

(3.10)

and hence

$$b''+b(1+b')^{-2}=0,\qquad b(0)=0,\qquad b'(0)=-a_0(1+a_0)^{-1}. $$

(3.11)

The phase trajectories for (3.10) and (3.11) are given in Fig. 1 and Fig. 2; Fig. 3 shows the graphs of $b(\tau)$ for different $a_0$.

Figure 3.

The graph of $b(\tau)$.

3.2. Formula for Solving the Boundary-Value Problem (2.3) Satisfying Estimate (2.5)

In (2.4), let us make the change of “time” (3.3). As a result, we obtain the boundary-value problem

$$\frac{\partial v}{\partial\tau} =d_0\,\frac{\partial^2v}{\partial y^2}\mspace{2mu},\qquad \frac{\partial v}{\partial y}\biggr|_{y=0} =\gamma(\tau+b(\tau))-\gamma_0. $$

(3.12)

We expand $\gamma(\tau+b(\tau))$ in the Fourier series

$$\gamma(\tau+b(\tau)) =\sum_{k\ne0,\,k=-\infty}^\infty\gamma_k\exp(ik\tau) $$

(3.13)

and, in (3.12), set

$$v=\sum_{k\ne0,\,k=-\infty}^\infty v_k(y)\exp(ik\tau). $$

(3.14)

Then, for $v_k(y)$, we obtain the second-order equation

$$d_0v_k''-ikv_k=0,\qquad v_k'(0)=\gamma_k. $$

(3.15)

Combining this with the condition $v_k\to 0$ as $y\to\infty$, we obtain

$$v_k(y)=-\gamma_ki(\delta_k)^{-1}\exp(\delta_ky), $$

(3.16)

where

$$\delta_k^2=ikd_0^{-1}, $$

(3.17)

i.e., $\delta_k$ is a root of Eq. (3.17) for which $\operatorname{Re}\delta_k<0$:

$$\delta_k=\begin{cases} \dfrac{(kd_0^{-1})^{1/2}}{2}\sqrt{2}(-1+i) &\text{for}\quad k>0, \\ \dfrac{(-kd_0^{-1})^{1/2}}{2}\sqrt{2}(-1-i) &\text{for}\quad k<0. \end{cases}$$

As a result, we see that the desired solution is of the form

$$v=-\sum_{k\ne 0,\,k=-\infty}^\infty i\delta_k^{-1}\gamma_k\exp(ik\tau+\delta y). $$

(3.18)

Hence it is easy to obtain estimate (2.5). The lemma 1 is proved.

From formulas (3.13) and (3.18) we conclude that

$$\int_0^{2\pi}\gamma(s+\varphi(s))v(s+\varphi(s),0)\,ds=0. $$

(3.19)

Note that, for $v(s,y)$, we could also use formulas (30) and (38) on p. 236 and p. 240, respectively, in [21].

Let us also give two useful formulas for calculating averages. Let $g(\tau)$ be an arbitrary $2\pi$-periodic function. Then

$$\begin{cases} M(g(\tau+b(\tau)))d^{-1}(\tau+b(\tau))=M(g(\tau))d_0^{-1}, \\ M(g(\tau+b(\tau)))=M(g(\tau)d(\tau)). \end{cases} $$

(3.20)

4. BIFURCATION FROM THE STATE OF EQUILIBRIUM

In (1.2) and (1.3), we set

$$d(s)=d_0(1+a\cos(s)),\qquad \gamma(s)=b\cos(s+\phi); $$

(4.1)

the values of $r$ and $T$ are independent of $s$. Without loss of generality, we may assume that $T\equiv 1$.

Using (4.1) as an example, let us study the dependence of the local dynamics of the boundary-value problem (1.2), (1.3) on the coefficients $d(s)$ and $\gamma(s)$. We consider the behavior of all the solutions of (1.2), (1.3) with initial conditions from a sufficiently small (but independent of $\omega$) neighborhood of the zero state of equilibrium.

By the results given above, in the description of the solutions of the boundary-value problem (1.2), (1.3), the defining role in condition (4.1) is played by the averaged equation

$$\frac{\partial v}{\partial t} =d_0\,\frac{\partial^2 v}{\partial x^2}-rv(t-1,x)[1+v],$$

(4.2)

$$\frac{\partial v}{\partial x}\biggr|_{x=0}=0,\qquad \frac{\partial v}{\partial x}\biggr|_{x=1} =\gamma_0v|_{x=1},\qquad \gamma_0=\frac{1}{2}\mspace{2mu}ab\cos(\varphi).$$

(4.3)

In turn, the local dynamics (in a neighborhood of the zero state of equilibrium of this boundary-value problem) depends largely on the properties of the solutions of the linearized (at zero) boundary-value problem

$$\frac{\partial v}{\partial t} =d_0\,\frac{\partial^2v}{\partial x^2}-rv(t-1,x),\qquad \frac{\partial v}{\partial x}\biggr|_{x=0}=0, \qquad \frac{\partial v}{\partial x}\biggr|_{x=1}=\gamma_0v|_{x=1}. $$

(4.4)

4.1. Study of the Properties of the Stability of the Boundary-Value Problem (4.4)

To construct the characteristic equation of this boundary-value problem, in (4.4), we set $v=w(x)\exp(\lambda t)$. As a result, we obtain the problem of finding the eigenvalues of the operator

$$\frac{\partial^2 w}{\partial x^2}=\mu w,\qquad w'(0)=0, \qquad w'(1)=\gamma_0w(1). $$

(4.5)

Here

$$\mu=d_0^{-1}(\lambda+r\exp(-\lambda)). $$

(4.6)

Let us recall that all the eigenvalues of (4.5) are real, and they can be arranged in decreasing order: $\infty>\mu_0>\mu_1>\dotsb$ . The eigenfunction corresponding to the eigenvalue $\mu_j$ is the function $\cosh(x\sqrt{\mu_j}\,)$. The value of $\mu_j$ is a function of the parameter $\gamma_0$: $\mu_j=\mu_j(\gamma_0)$. The graph of the dependence of $\mu_0$ on $\gamma$ is given in Fig. 4.

Figure 4.

The graph of $\mu_0(\gamma)$ for $d=1$.

To find whether the solutions of the boundary-value problem (4.4) are stable, we must study the roots of the characteristic equation

$$\mu_j=d_0^{-1}(\lambda+r\exp(-\lambda)),\qquad j=0,1,\dots, $$

(4.7)

and find the root (4.7) with the greatest real part. Thus, for this root, we obtain the equation

$$\lambda+r\exp(-\lambda)=d_0\mu_0(\gamma_0). $$

(4.8)

Lemma 3.

Let all the roots of (4.8) have negative real parts. Then the solutions of the boundary-value problem (4.4) are exponentially stable.

It follows that, for sufficiently large $\omega$, the zero solution of the boundary-value problem (4.2), (4.3), and hence also the zero solution of the boundary-value problem (1.3), (2.1), is exponentially stable.

Lemma 4.

Let Eq. (4.8) have a root with positive real part. Then the solutions of the boundary-value problem (4.4) are unstable.

We can assert that, under the assumptions of Lemma 4, the zero solutions of the boundary-value problems (4.2), (4.3) and (1.3), (2.1) (for sufficiently large $\omega$) are unstable and, in a sufficiently small (and independent of $\omega$) neighborhood of the zero state of equilibrium, there are no attractors.

For each fixed value of $r>0$, we let $\gamma(r)$ denote a value of the parameter $\gamma_0$ (if it exists) for which Eq. (4.8) has a root with zero real part and has no roots with positive real parts.

We note that, for $\gamma_0>\gamma(r)$, Eq. (4.8) has roots with positive real parts. Let $\sigma_0$ be a root of the equation $\tanh(\sigma)=-4\sigma(d\pi^2)^{-1}$ lying in the interval $\sigma\in(\pi/2,\pi)$. We set $r^*=(\sigma_0^2+(1/16)(d\pi^2)^2)$. For the example in which $d_0=1$, the graph of $\gamma(r)$ is given by Fig. 5.

Figure 5.

The graph of $\gamma(r)$ for $d_0=1$, $r^*\approx3.43$.

Figure 6.

The graph of $\gamma(r)$ for $d_0=1$, $r^*\approx3.43$.

In what follows, we assume that

$$0<r<r^*. $$

(4.9)

Under the condition $0<r<1$, the equation $\lambda+r\exp(-\lambda)=\mu_0(\gamma_0(r))$ has a simple zero root and, for $1<r<r^*$, it has a pair $\lambda_\pm=\pm i\sigma$ of pure imaginary roots. The graph of $\sigma=\sigma(r)$ ($\sigma>0$) is given in Fig. 6.

4.2. Bifurcations in the Boundary-Value Problem (4.4) Under the Condition $0<r<1$

Let

$$0<r<1. $$

(4.10)

We set

$$\gamma_0=\gamma(r)+\varepsilon\gamma_1, $$

(4.11)

where $\gamma_1$ is an arbitrarily fixed number and $\varepsilon$ is a small positive parameter such that

$$0<\varepsilon\ll 1. $$

(4.12)

Then, in the phase space (depending on $\varepsilon$) of the boundary-value problem (4.2), (4.3), there is a stable one-dimensional local invariant integral manifold on which this boundary-value problem can be written as a single ordinary differential equation of first order:

$$\frac{d\xi}{d\tau}=\lambda_1(r)\xi+c(r)\xi^2+O(\varepsilon), $$

(4.13)

where

$$\tau=\varepsilon t,\qquad \lambda_1(r) =\frac{\gamma_1\cosh(\sqrt{\mu_0(\gamma(r))}\,)}{1-r}\mspace{2mu},\qquad c(r)=-\frac{4r(2\sqrt{r}+2)(9sh(\sqrt{r}\,)+sh(3\sqrt{r}\,))} {(1-r)(sh(2\sqrt{r}\,)+2\sqrt{r}\,)}\mspace{2mu}.$$

The graph for $c(r)$ is given in Fig. 7.

Figure 7.

The graph of $c(r)$ for $d=1$.

The solutions $v(t,x,\omega)$ on this manifold are related to the solutions of Eq. (4.13) by the formula

$$v(t,x,\tau)=\varepsilon\xi(\tau)\cosh(\sqrt{\mu_0(\gamma(r))}\,) +o(\varepsilon).$$

The following assertion on the structure of the solutions in a small neighborhood of the zero state of equilibrium holds.

Theorem 3.

Let conditions (4.10), (4.11) hold, and let $\gamma_1>0$$(<0)$. Then the boundary-value problem (4.2), (4.3) has an unstable (respectively, stable) zero state of equilibrium and a stable (respectively, unstable) nonzero state of equilibrium $v_0(x,\varepsilon)$, and

$$v_0(x,\varepsilon)=-\varepsilon\lambda_1(r)(c(r))^{-1} \cosh(\sqrt{\mu_0(\gamma(r))}\,)+O(\varepsilon^2).$$

4.3. Andronov–Hopf Bifurcation

Let

$$1<r<r^*, $$

(4.14)

and let equality (4.11) hold. Then, in a neighborhood of the zero solution of the boundary-value problem (4.2), (4.3), there exists a two-dimensional stable local invariant manifold on which this boundary-value problem can be written as the following single complex ordinary differential equation of first order:

$$\frac{d\xi}{d\tau} =\lambda_+\xi+\beta(r)\xi|\xi|^2+O(\varepsilon), $$

(4.15)

where

$$\begin{aligned} \, \beta(r) &=-r\biggl\{(1+e^{-i\sigma}) \biggl[\frac{A_1}{2\sqrt{{r}/{d}}\,(r/d-4\mu_0d^{-1})} \\&\hphantom{=-r\biggl\{(1+e^{-i\sigma})\biggl[}\qquad{} \times \biggl(\sinh\biggl(\sqrt{\frac{r}{d}}\,\biggr) \biggl(\frac{r}{d}\cosh(2\sqrt{\mu_0(\gamma(r))d^{-1}}\,) +\frac{r}{d}-4\mu_0d^{-1}\biggr) \\&\hphantom{=-r\biggl\{(1+e^{-i\sigma})\biggl[{}\times \biggl(}\qquad\qquad{} -2\sqrt{\mu_0\frac{r}{d}} \cosh\biggl(\sqrt{\frac{r}{d}}\biggr) \sinh(2\sqrt{\mu_0(\gamma(r))d^{-1}}\,)\biggr) \\&\hphantom{=-r\biggl\{(1+e^{-i\sigma})\biggl[}\quad{} +\frac{A_2}{16} \biggl(\frac{4\sinh(2\sqrt{\mu_0(\gamma(r))d^{-1}}\,) +\sinh(4\sqrt{\mu_0(\gamma(r))d^{-1}}\,)} {\sqrt{\mu_0(\gamma(r))d^{-1}}}+4\biggr) \\&\hphantom{=-r\biggl\{(1+e^{-i\sigma})\biggl[}\quad{} +\frac{A_3}{4} \biggl(\frac{\sinh(2\sqrt{\mu_0(\gamma(r))d^{-1}}\,)} {\sqrt{\mu_0(\gamma(r))d^{-1}}}+2\biggr)\biggr] \\&\hphantom{=-r\biggl\{} +(e^{i\sigma}+e^{-2i\sigma}) \biggl[\frac{B_1}{2\sqrt{\rho}(\rho-4\mu_0d^{-1})} \bigl(\sinh(\sqrt{\rho}\,) (\rho\cosh(2\sqrt{\mu_0(\gamma(r))d^{-1}}\,)+\rho-4\mu_0d^{-1}) \\&\quad\hphantom{=-r\biggl\{}\quad{}\hphantom{+(e^{i\sigma}+e^{-2i\sigma}) \biggl[\frac{B_1}{2\sqrt{\rho}(\rho-4\mu_0d^{-1})}\bigl(} -2\sqrt{\mu_0d^{-1}\rho}\cosh(\sqrt{\rho}\,) \sinh(2\sqrt{\mu_0(\gamma(r))d^{-1}}\,)\bigr) \\&\hphantom{=-r\biggl\{+(e^{i\sigma}+e^{-2i\sigma}) \biggl[}+\frac{B_2}{16} \biggl(\frac{4\sinh(2\sqrt{\mu_0(\gamma(r))d^{-1}}\,) +\sinh(4\sqrt{\mu_0(\gamma(r))d^{-1}}\,)} {\sqrt{\mu_0(\gamma(r))d^{-1}}}+4\biggr) \\&\hphantom{=-r\biggl\{+(e^{i\sigma}+e^{-2i\sigma}) \biggl[}+\frac{B_3}{4} \biggl(\frac{\sinh(2\sqrt{\mu_0(\gamma(r))d^{-1}}\,)} {\sqrt{\mu_0(\gamma(r))d^{-1}}}+2\biggr)\biggr]\biggr\} \biggl(\frac{\sinh(2\sqrt{\mu_0(\gamma(r))d^{-1}}\,)} {4\sqrt{\mu_0(\gamma(r))d^{-1}}} +\frac{1}{2}\biggr)^{-1} \end{aligned}$$

and

$$\begin{gathered} \, A_1=\frac{\gamma(A_2\cosh(2\sqrt{\mu_0(\gamma(r))d^{-1}}\,)+A_3) -2A_2\sqrt{\mu_0(\gamma(r))d^{-1}} \sinh(2\sqrt{\mu_0(\gamma(r))d^{-1}}\,)} {\sqrt{r/d}\sinh(\sqrt{r/d}\,)-\gamma\cosh(\sqrt{r/d}\,)}\mspace{2mu}, \\ A_2=\frac{\cos(\sigma)}{4\mu_0d-r}\mspace{2mu},\qquad A_3=-\cos(\sigma),\qquad \rho=\frac{r\exp(-2i\sigma)+2i\sigma}{d}\mspace{2mu}, \\ B_1 =\frac{\gamma(B_2\cosh(2\sqrt{\mu_0(\gamma(r))d^{-1}}\,)+B_3) -2B_2\sqrt{\mu_0(\gamma(r))d^{-1}}\sinh(2\sqrt{\mu_0(\gamma(r))d^{-1}}\,)} {\sqrt{\rho}\sinh(\sqrt{\rho}\,)-\gamma\cosh(\sqrt{\rho}\,)}\mspace{2mu}, \\ B_2=\frac{r\exp(-i\sigma)}{2d(4\mu_0-\rho)}\mspace{2mu},\qquad B_3=-\frac{r\exp(-i\sigma)}{2d\rho}\mspace{2mu}. \end{gathered}$$

The graph of $\operatorname{Re}\beta(r)$ is given in Fig. 8. We note that

$$\operatorname{Re}\beta(r)<0. $$

(4.16)

Figure 8.

The graph of $\operatorname{Re}\beta(r)$ for $d=1$.

Let us state the final result.

Theorem 4.

Let conditions (4.14) and (4.9) hold. Then, for $\gamma_1<0$ and for sufficiently small $\varepsilon$, all the solutions of (4.2), (4.3) from a sufficiently small neighborhood of zero which is independent of $\varepsilon$ tend to zero as $t\to\infty$. But if $\gamma_1>0$, then the zero state of equilibrium is unstable, while the cycle $v_0(t,x,\varepsilon)$ for which

$$\begin{gathered} \, v_0(t,x,\varepsilon) =\varepsilon^{1/2}[\xi_0\exp(i\sigma t) +\overline\xi_0\exp(-i\sigma t)] \cosh(\sqrt{\mu_0(\gamma(r))}\,)+O(\varepsilon), \\ \xi_0 =-\operatorname{Re}\lambda_+(r)(\operatorname{Re}\beta(r))^{-1},\qquad \varphi_0=\operatorname{Im}\lambda_+(r) +\xi_0^2\operatorname{Im}\beta(r). \end{gathered}$$

is stable in (4.2), (4.3).

We note that, for $r=1$ and $\gamma_0=\gamma(1)$, we have the critical case of a double zero root with one group of solutions.

A number of results from this section were given in the authors’ paper [22].

4.4. Bifurcations from the State of Equilibrium in the Boundary-Value Problem (1.2), (1.3) Under the Condition $\gamma_0=\gamma(r)$

Let

$$\gamma_0=\gamma(r). $$

(4.17)

For $0<r<1$, just as in the boundary-value problem (4.2), (4.3), all solutions from a small neighborhood of the zero state of equilibrium of the boundary-value problem (1.2), (1.3) (for sufficiently large values of $\omega$) tend to a one-dimensional integral manifold on which this boundary-value problem can be expressed in the form

$$\frac{d\xi}{d\tau}=p(\omega)\xi +(c(r)+O(\omega^{-1/2}))\xi^2 +O(\omega^{-1/2}),\qquad \tau=\omega^{-1/2}t, $$

(4.18)

which is “close” to Eq. (4.13).

But if $1<r<r^*$, then the corresponding integral manifold is two-dimensional and, on it, the boundary-value problem (1.2), (1.3) has the form

$$\frac{d\xi}{d\tau}=p(\omega)\xi +(\beta(r)+O(\omega^{-1/2}))\xi|\xi|^2,\qquad \tau=\omega^{-1/2}t, $$

(4.19)

which is “close” to Eq. (4.15).

To determine the behavior of the solutions (1.2), (1.3) from a small neighborhood of the zero state of equilibrium, it remains to write out the asymptotics for $p(\omega)$.

Thus, we need to study the stability of the linear boundary-value problem

$$\frac{\partial u}{\partial t} =d(s)\,\frac{\partial^2u}{\partial x^2}-ru(t-1,x),\qquad \frac{\partial u}{\partial x}\biggr|_{x=0}=0,\quad \frac{\partial u}{\partial x}\biggr|_{x=1} =\gamma(s)u|_{x=1},\qquad s=\omega t. $$

(4.20)

4.5. Algorithm for Calculating the Quantity $p(\omega)$

This algorithm is based on results from [7], [8], and [15]. Under condition (4.17), the averaged boundary-value problem (4.20)

$$\frac{\partial u}{\partial t} =d_0\,\frac{\partial^2u}{\partial x^2}-ru(t-1,x),\qquad \frac{\partial u}{\partial x}\biggr|_{x=0}=0,\qquad \frac{\partial u}{\partial x}\biggr|_{x=1}=\gamma_0(r)u|_{x=1} $$

(4.21)

has the periodic solution

$$u_0(t,x)=\cosh\Bigl(x\sqrt{\mu_0(r)d_0^{-1}}\,\Bigr)\exp(i\delta t)\qquad (\delta\equiv 0\quad \text{for}\quad 0<r<1),$$

which is constant with respect to $t$ for $0<r<1$.

Consider the formal series

$$\begin{aligned} \, u(t,x,\omega) &=[u_0(t,x)+U(t,x,\omega)+V_0(s,z,\omega) +V_1(s,y,\omega)+W(s,x,\omega)] \nonumber \\&\qquad\qquad{}\times \exp[(i\delta+\omega^{-1/2}p_1 +\omega^{-1}p_2+\dotsb)t], \end{aligned}$$

(4.22)

where

$$U(t,x,\omega) =\omega^{-1/2}u_1(t,x,\omega)+\omega^{-1}u_2(t,x,\omega)+\dotsb,$$

(4.23)

$$V_0(s,z,\omega) =\omega^{-1/2}v_{01}(s,z,\omega) +\omega^{-1}v_{02}(s,z,x,\omega)+\dotsb,\qquad s=\omega t,\quad z=\omega^{1/2}x,$$

(4.24)

$$V_1(s,y,\omega) =\omega^{-1/2}v_{11}(s,y,\omega) +\omega^{-1}v_{12}(s,y,x,\omega)+\dotsb,\qquad y=\omega^{-1/2}(1-x),$$

(4.25)

$$W(s,x,\omega) =\omega^{-1}w_2(s,x,\omega) +\omega^{-3/2}w_3(s,x,\omega)+\dotsb\,.$$

(4.26)

Here all the elements of the series are bounded functions of $\omega$ as $\omega\to\infty$. Once again we note that the dependence of the elements of these series on $\omega$ is related to the delay in (4.21).

Let us substitute the formal series (4.22)–(4.26) into (4.20) and collect the coefficients of equal powers of $\omega$.

At the first step, we obtain the boundary-value problems for the functions $v_{01}(s,z)$ and $v_{11}(s,y)$:

$$\frac{\partial v_{01}}{\partial s} =d(s)\,\frac{\partial^2v_{01}}{\partial z^2}\mspace{2mu},\qquad \frac{\partial v_{01}}{\partial z}\biggr|_{z=0}=0,$$

(4.27)

$$\frac{\partial v_{11}}{\partial s} =d(s)\,\frac{\partial^2v_{11}}{\partial y^2}\mspace{2mu},\qquad \frac{\partial v_{11}}{\partial y}\biggr|_{y=0} =(\gamma(s)-\gamma_0(r))\cosh(\sqrt{\mu_0(r)}\,).$$

(4.28)

According to Lemma 5, this determines unique solutions $v_{01}(s,z)$ and $v_{11}(s,y)$ that are functions $2\pi$-periodic in $s$ and exponentially decreasing with respect to $y$ and $z$ as $y\to\infty$ and $z\to\infty$; moreover, $v_{01}\equiv 0$.

At the second step, taking into account Lemma 4, we obtain the equations

$$\frac{\partial w_2}{\partial s} =(d(s)-d_0)\,\frac{d}{dx^2}(\cosh(x\sqrt{\mu_0(r)}\,)),$$

(4.29)

$$\frac{\partial u_1}{\partial t} =d_0\,\frac{\partial^2u_1}{\partial x^2} -\mu_0(r)u_1(t-1,x)-p_1(1-\mu_0(r)+i\delta) \cosh(x\sqrt{\mu_0(r)}\,),$$

(4.30)

$$\frac{\partial u_1}{\partial x}\biggr|_{x=0}=0,\qquad \frac{\partial u_1}{\partial x}\biggr|_{x=1} =\gamma_0(r)u_1|_{x=1}+\Delta_0,$$

(4.31)

$$\Delta_0=d_0^{-1}M(d(s)\gamma(s)v_{11}(s,0,\omega)),$$

(4.32)

$$\frac{\partial v_{02}}{\partial s} =d(s)\,\frac{\partial^2v_{20}}{\partial z^2}\mspace{2mu},\qquad \frac{\partial v_{02}}{\partial z}\biggr|_{z=0}=0,$$

(4.33)

$$\frac{\partial v_{12}}{\partial s} =d(s)\,\frac{\partial^2v_{12}}{\partial y^2}\mspace{2mu},\qquad \frac{\partial v_{12}}{\partial y}\biggr|_{y=0} =(\gamma(s)-\gamma_0(r)u_1(t,1)+\gamma(s)v_{11}(s,0,\omega)).$$

(4.34)

From (4.29), (4.33), and (4.34), we obtain the functions $w_2(s,x,\omega)$, $v_{02}(s,z,\omega)\equiv 0$, and $v_{12}(s,y,\omega)$. We note that (4.28) yields the relation

$$M(d(s)[\gamma(s)-\gamma_0(r)]v_{11}(s,0))\cosh\sqrt{\mu_0(r)} =M\biggl(d(s)\int_0^\infty \biggl(\frac{\partial v}{\partial y}\biggr)^2\,dy\biggr)>0,$$

which can be used to determine the quantity $\Delta_0$.

To find $w_3(t,x)$, we use the following simple statement.

Lemma 5.

For all the solutions of the boundary-value problem

$$\begin{aligned} \, \frac{\partial u}{\partial t} &=d_0\,\frac{\partial^2u}{\partial x^2}-\mu_0(r)u(t-1,x)+p(x), \\ \frac{\partial u}{\partial x}\biggr|_{x=0} &=q_0,\qquad \frac{\partial u}{\partial x}\biggr|_{x=1} =\gamma_0(r)u|_{x=1}+q_1 \end{aligned}$$

to be bounded as $t\to\infty$ for $x\in[0,1]$ , it is necessary and sufficient that the following equality hold:

$$\int_0^1p(x)\cosh(x\sqrt{\mu_0}\,)\,dx -d_0q_0+d_0q_1\cosh(\sqrt{\mu_0}\,)=0. $$

(4.35)

It follows from this lemma that, for the solution of the boundary-value problem (4.30), (4.31) to be bounded in $t$, it is necessary and sufficient that

$$p_1=(1-d_0\mu_0+i\delta)^{-1}c_0\Delta_0d_0,\qquad \text{where}\quad c_0=\biggl(\int_0^1\cosh^2(x\sqrt{\mu_0}\,)\,dx\biggr)^{-1} \cosh(\sqrt{\mu_0(r)}\,). $$

(4.36)

After that, we obtain the corresponding relations at the third step and find the functions $w_3(s,x,\omega)$, $u_2(t,x,\omega)$, $v_{03}(s,z,\omega)$, and $v_{13}(s,y,\omega)$ and, which is most important, the quantity $p_2(\omega)$, etc.

Let us state the final result. Let

$$\operatorname{Re}(p_1(\omega)) \equiv\operatorname{Re}(p_2(\omega)) \equiv\dotsb\equiv\operatorname{Re}(p_{m-1}(\omega))\equiv 0,\qquad \operatorname{Re}(p_m(\omega))\ne 0, $$

(4.37)

and let, for some sequence $\omega_n\to\infty$ and some positive constant $c>0$, the inequality $|p_m(\omega_n)|\ge\varepsilon^{m/2}c$ hold. Then, for sufficiently large $n$, the stability of the solutions of the boundary-value problem (4.20) is determined by the sign of the quantity $\operatorname{Re}(p_m(\omega_n))$.

By the constructions given above, for such $\omega_n$ and for $p(\omega_n)=p_m(\omega_n)+o(\omega^{-m/2})$, the dynamics of the logistic boundary-value problem with delay (4.2), (4.3) is described by Eq. (4.18) for $0<r<1$ and by Eq. (4.19) for $1<r<r^*$.

Example.

Consider the boundary-value problem

$$\frac{\partial u}{\partial t} =\frac{\partial^2u}{\partial x^2} -\frac{\pi}{2}\mspace{2mu}u(t-1,x)[1+u],$$

(4.38)

$$\frac{\partial u}{\partial x}\biggr|_{x=0} =0,\qquad \frac{\partial u}{\partial x}\biggr|_{x=1} =b\cos(s)u|_{x=1},\qquad s=\omega t.$$

(4.39)

By averaging, we obtain the boundary-value problem

$$\frac{\partial u}{\partial t} =\frac{\partial^2u}{\partial x^2} -\frac{\pi}{2}\mspace{2mu}u(t-1,x)[1+u],\qquad \frac{\partial u}{\partial x}\biggr|_{x=0} =\frac{\partial u}{\partial x}\biggr|_{x=1}=0. $$

(4.40)

In it, $\gamma_0=\mu_0=0$, $\delta=\pi/2$. Thus, the Andronov–Hopf bifurcation conditions hold: the characteristic equation for the linearized the boundary-value problem (4.39)

$$\lambda+\frac{\pi}{2}\exp(-\lambda)=-\pi^2k^2,\qquad k=0,1,\dots,$$

has a pair of pure imaginary roots $\lambda_\pm=\pm i\delta$, while its other roots have negative real parts. In Eq. (4.19), for the case under consideration, the following representation is valid:

$$\beta\biggl(\frac{\pi}{2}\biggr) =-\frac{\pi}{2}[3\pi-2+i(\pi+6)] \biggl(10\biggl(1+\frac{4}{\pi^2}\biggr)\biggr)^{-1},\qquad \operatorname{Re}\beta\biggl(\frac{\pi}{2}\biggr)<0.$$

Let us determine the value of $p_1(\omega)$. By (4.23)–(4.36), we have

$$u_0(t,x)\equiv 1,\qquad w_2(s,x,0)\equiv 0,\qquad v_{01}(s,z,0)\equiv 0,$$

and the function $v_{11}(s,y)$ is a solution of the boundary-value problem

$$\frac{\partial v_{11}}{\partial s} =\frac{\partial^2v_{11}}{\partial y^2}\mspace{2mu},\qquad \frac{\partial v_{11}}{\partial y}\biggr|_{y=0}=b\cos(s). $$

(4.41)

Hence we see that $v_{11}(s,y)=A(y)\cos(s)+B(y)\sin(s)$, where

$$\begin{aligned} \, A(y) &=2^{-1/2}b\exp\biggl(-\frac{\sqrt{2}}{2}\mspace{2mu}y\biggr) \biggl[\sin\biggl(\frac{\sqrt{2}}{2}\mspace{2mu}y\biggr) -\cos\biggl(\frac{\sqrt{2}}{2}\mspace{2mu}y\biggr)\biggr], \\ B(y) &=-2^{-1/2}b\exp\biggl(-\frac{\sqrt{2}}{2}\mspace{2mu}y\biggr) \biggl[\sin\biggl(\frac{\sqrt{2}}{2}\mspace{2mu}y\biggr) +\cos\biggl(\frac{\sqrt{2}}{2}\mspace{2mu}y\biggr)\biggr]. \end{aligned}$$

Here $\mu_0(r)=0$, $c_0=1$, and $\delta=\pi/2$. For $\Delta_0$, we then obtain the equality $\Delta_0=-b^2(2\sqrt{2}\,)^{-1};$ therefore,

$$p_1=-\biggl(2^{3/2}\biggl(1+\frac{\pi^2}{4}\biggr)\biggr)^{-1} \biggl(1-i\frac{\pi}{2}\biggr)b^2.$$

Thus, we see that, for $b\ne 0$, the inequality $\operatorname{Re}p_1<0$ holds. Hence we can draw the interesting conclusion that rapid oscillations of the boundary conditions in the logistic boundary-value problem (4.21) stabilize the state of equilibrium. Here we note the analogy with the well-known problem of the stability of the upper state of equilibrium of a pendulum with vibrating point of suspension (see, e.g., [23]).

5. CONCLUSIONS

An averaged equation for the logistic delay equation with diffusion and variable boundary conditions is constructed. It should be noted that the averaging in the boundary conditions depends largely on the oscillations of the diffusion coefficient. Results on the relation between the solutions of the original and the averaged equations are formulated. The local dynamics in a neighborhood of the state of equilibrium of the averaged equation in critical cases is considered. An algorithm for finding the asymptotic expansions of the coefficients of the normal form that determine the dynamics of the original problem is developed. In particular, it is shown that the corresponding asymptotic expansions contain both regular components typical of averaging theory in the case of ordinary differential equations and boundary-layer asymptotics playing an important role in the theory of singular perturbations. It is shown that oscillations in the boundary conditions possess a strong stabilizing effect.