Local Dynamics of Chains of Van der Pol Coupled Systems
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Local Dynamics of Chains of Van der Pol Coupled Systems S. A. Kashchenko1* 1
Demidov Yaroslavl State University, Yaroslavl, 150000 Russia
Received July 30, 2020; in final form, July 30, 2020; accepted September 9, 2020
DOI: 10.1134/S0001434620110322 Keywords: logistic equation, stability, van der Pol equation.
This note considers the local dynamics of circular chains of coupled van der Pol systems. It is assumed that the number of elements in the chain is sufficiently large. The passage to a nonlinear boundary-value problem with continuous spatial variable is performed. For relations of diffusion type, critical cases in the problem of the stability of an equilibrium state are distinguished. It is shown that all of them have infinite dimension. The main result is the construction of special nonlinear parabolic boundary-value problems, which play the role of the first approximation. Their nonlocal dynamics determines the local behavior of the solutions to the initial problem. 1. PROBLEM STATEMENT The van der Pol equation f (u, u) ˙ = −uu ˙ 2
u ¨ + au˙ + u = f (u, u), ˙
(1)
is the simplest and one of the most popular types of oscillators. Let us write this equation as the system u˙ 1 = αu1 + u2 ,
u˙ 2 = (αa − 1 − α2 )u1 − (αu1 + u2 )u22 − (a + α)u2
and consider a coupled chain of N such systems: N αij vi − vj , v˙ j = A(α)vj + F (vj ) + D A(α) =
α −αa − 1 − α2
i=1, i=j
1 , a−α
D=
d1 0
0 , d2
j = 1, . . . , N,
(2)
(3)
F (v) = (0, (αv1 + v2 )v12 ).
Systems of the form (3) have been studied by many authors (see, e.g., [1]–[9]). Let us make several assumptions. We assume that N
αij = 1,
j = 1, . . . , N.
i=1, i=j
This implies that system (3) has a homogeneous solution vj (t) ≡ v(t), where v(t) is a solution of (2). We also assume that system (3) is circular, i.e., vj ≡ vj±N , and homogeneous, i.e., αij = αi−j . It is natural to relate the values vj (t) with the values of a vector function v(t, xj ) of two variables, where xj = 2πjN −1 is an angular variable with values uniformly distributed on a circle. In addition, we assume that N is sufficiently large, so that the parameter ε = 2πN −1 is sufficiently small: 0 < ε 1. *
E-mail: [email protected]
901
(4)
902
KASHCHENKO
This assumption allows us to consider a continuous variable x ∈ [0, 2π]; then system (3) takes the form ˆ ∞ ∂v = A(x)v + F (v) + D F (s, ε)v(t, x + s) ds − v (5) ∂t −∞ with periodic boundary conditions v(t, x + 2π) ≡ v(t, x).
(6)
In what follows, we assume that 1 √ exp(−(εσ)−2 (s + ε)2 ). 2 2σ ε π Note that, as σ → 0 with v(t, x) fixed, the last summand in (5) takes the form 1 D[v(t, x + ε) − 2v(t, x) + v(t, x − ε)]. 2 Let us study the behavior as t → ∞ of all solutions of the boundary-value problem (5), (6) with small ε and initial conditions small enough in the norm of C[0,2π] (R2 ). The characteristic equation for system (5), (6) linearized at zero is F (s, ε) = Fε (s) + F−ε (s),
Fε (s) =
λ2 + λ[a − (d1 + d2 )g] + [α(d2 − d1 ) − ad1 ]g + 1 + d1 d2 g2 = 0,
(7)
in which g = g(z) = cos z ·
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