Cascades of Hopf bifurcations from boundary delay |
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Authors: | José M. Arrieta,Neus Có nsul |
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Affiliation: | a Dept. Mat. Aplicada, Facultad de Matemáticas, Universidad Complutense de Madrid, 28040 Madrid, Spain b Dept. Matemática Aplicada I, ETSEIB, Universitat Politècnica de Catalunya, 08028 Barcelona, Spain c Dept. Mat. Aplicada, IME, USP, Rua do Matão, 1.010, CEP 05508-900, São Paulo, SP, Brazil |
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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. |
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Keywords: | Logistic equation Boundary delay Hopf bifurcation Periodic orbits |
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