Destabilization of periodic solutions arising in delay-diffusion systems in several space dimensions

Abstract

We considera diffusion equation with time delay having a stable spatially homogeneous periodic solution bifurcating from a steady state. We show that under certain circumstances the bifurcating periodic solution loses its stability very near the bifurcation point if the diffusion coefficients are sufficiently small. Such a destabilization phenomenon also occurs when in place of the diffusion coefficients, the shape of the domain is varied instead. Sufficient conditions for the occurrence of such phenomena, along with some specific examples, will be presented.

Hopf Bifurcation in a Diffusive Logistic Equation with Mixed Delayed and Instantaneous Density Dependence

“…The local stability and Hopf bifurcations from the constant steady state solution were studied in [27,29,47], and global stability for this case has been proved in [12,20,21,31]. Similar analysis for a constant steady state solution in a Dirichlet boundary value problem has also been investigated [41].…”

Global Dynamics for a Reaction–Diffusion Equation with Time Delay

“…To mention a few examples, in 1982 Yoshida [28] studied the Hopf bifurcation and stability of spatially homogeneous solutions for a single delay diffusion equation, a modified plant-eating population model, under Neumann boundary condition by using a local center manifold approach developed by Chow and Mallet-Paret [3]. Yoshida's work was extended later on by Morita [21] and in particular, by Memory [19] who proved the existence of a second Hopf bifurcation as well as the existence of a compact attractor. At the same time Green and Stech [11] studied the local stability of a positive equilibrium for the same type of equation with Dirichlet boundary condition and showed numerically that an increase of delay would destabilize the positive equilibrium and lead to periodic oscillation.…”

Bifurcation and Asymptotic Behavior of Solutions of a Delay-Differential Equation with Diffusion

“…Morita considers this question for a class of problems in [12]. He treats (1) (for f c Rn) as an example and shows that for any/z > 0 there is a D D(/z) for which the spatially constant periodic orbit is unstable.…”

“…Given a solution of (2) and its stability properties, what are its stability properties as a solution of (1)? In 2-4, we will build on the work of Yoshida [15] and Morita [12] to deal with this question for the zero solution and the periodic solution arising in the Hopf bifurcation. Morita shows that, for fixed /x > 0, this periodic solution is unstable for D less than a certain Do.…”