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Cascades of Hopf bifurcations from boundary delay

Arrieta Algarra, José María and Cónsul, Neus and Oliva, Sergio M. (2010) Cascades of Hopf bifurcations from boundary delay. Journal of Mathematical Analysis and Applications , 361 (1). pp. 19-37. ISSN 0022-247X

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Abstract

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≡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.

Item Type:Article
Uncontrolled Keywords:Logistic equation; Periodic orbits; One space dimension; Nonlinear boundary conditions
Subjects:Sciences > Mathematics > Differential equations
ID Code:13913
Deposited On:18 Nov 2011 08:32
Last Modified:06 Feb 2014 09:55

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