Cycle Stability for Hopf Bifurcation Generated by Sublinear Terms

✍ Scribed by Alexander Krasnosel'skii; Reinhard Mennicken; Dmitrii Rachinskii

• Publisher: John Wiley and Sons
• Year: 2002
• Tongue: English
• Weight: 346 KB
• Volume: 233-234
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/1522-2616(200201)233:1<171::aid-mana171>3.0.co;2-6

✦ Synopsis

The paper is concerned with the study of small stable cycles of autonomous quasilinear systems depending on a parameter. Sufficient conditions are presented for the existence of such cycles for control theory equations with scalar nonlinearities. The principal distinction of the case considered from usual results on Hopf bifurcations is that the linear part of the problem is degenerate for all the parameter values (not only at a bifurcation point). Small sublinear nonlinearities play the main role in our results. The proofs are based on the theory of monotone operators.

1. Problem statement

Consider the system x = ϕ(x, λ), x ∈ IR n with the scalar parameter λ ∈ [0, 1]. Suppose that zero is an equilibrium of this system: ϕ(0, λ) ≡ 0. We study the socalled Hopf bifurcation phenomenon: the value λ 0 of the parameter is called 1) a Hopf bifurcation point if there exist arbitrarily small (in some appropriate sense) nonzero periodic solutions of the system with λ arbitrarily close to λ 0 . In other words, small nonzero cycles arise from the equilibrium in the neighborhood of a Hopf bifurcation point.

Usual approach to study the problem is as follows.

The function ϕ(x, λ) is supposed to be differentiable at zero, i. e. ϕ(x, λ) ≡ A(λ)x + ψ(x, λ) where the n × n Jacobian matrix A(λ) is continuous in λ and the continuous nonlinearity ψ(x, λ) : IR n × [0, 1] → IR n is sublinear, i. e. lim |x|→0 ψ(x, λ) |x| -1 = 0