Cascades of Hopf bifurcations from boundary delay

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:361

Article

Arrieta, Jose M.¹  Consul, Neus²  Oliva, Sergio M.³ 

[1] Univ Complutense Madrid, Fac Matemat, Dept Mat Aplicada, E-28040 Madrid, Spain

[2] Univ Politecn Cataluna, ETSEIB, Dept Matemat Aplicada 1, E-08028 Barcelona, Spain

[3] Univ Sao Paulo, IME, Dept Mat Aplicada, BR-05508900 Sao Paulo, Brazil

关键词: Logistic equation;    Boundary delay;    Hopf bifurcation;    Periodic orbits;   

DOI  :  10.1016/j.jmaa.2009.09.018

来源: Elsevier

【 摘 要 】

We consider a 1-dimensional reaction-diffusion equation with nonlinear boundary conditions of logistic type with delay. We deal with non-negative solutions and analyze the stability behavior of its unique positive equilibrium solution, which is given by the constant function u equivalent to 1. We show that if the delay is small, this equilibrium solution is asymptotically stable, similar as in the case without delay. We also show that, as the delay goes to infinity, this equilibrium becomes unstable and undergoes a cascade of Hopf bifurcations. The structure of this cascade will depend on the parameters appearing in the equation. This equation shows some dynamical behavior that differs from the case where the nonlinearity with delay is in the interior of the domain. (C) 2009 Elsevier Inc. All rights reserved.

【 授权许可】

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