Abstract
This paper investigates the questions about the local dynamics in the neighborhood of the equilibrium state for the spatially distributed delay logistic equation with diffusion. The critical cases in the stability problem are singled out. The equations for their invariant manifolds that determine the structure of the solutions in the equilibrium state neighborhood are constructed. The dominant bulk of this paper is devoted to the consideration of the most interesting and important cases of either the translation (advection) coefficient is large enough or the diffusion coefficient is small enough. Both of this cases convert the original problem to a singularly perturbed one. It is shown that under these conditions the critical cases are infinite–dimensional in the problems of the equilibrium state stability for the singularly perturbed problems. This means that infinitely many roots of the characteristic equations of the corresponding linearized boundary value problems tend to the imaginary axis as the small parameter tends to zero. Thus, we are talking about infinite–dimensional bifurcations. Standard approaches to the study of the local dynamics based on the application of the invariant integral manifolds methods and normal forms methods are not applicable. Therefore, special methods of infinite–dimensional normalization have been developed which allow one to construct special nonlinear boundary value problems called quasinormal forms. Their nonlocal dynamics determine the behavior of the initial boundary value problem solutions in the neighborhood of the equilibrium state. The bifurcation features arising in the case of different boundary conditions are illustrated.
Highlights
We consider the spatially distributed delay logistic equation ∂u ∂2 u= d 2 + b + r [1 − u(t − T, x )]u ∂t ∂xPublisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.with the periodic boundary conditions u(t, x + 2π ) ≡ u(t, x ).Copyright: Licensee MDPI, Basel, Switzerland. (1) (2)
We study the local dynamics of the boundary value problem (1), (2) in a neighborhood of a positive equilibrium state, that is, the behavior of the (1), (2) solutions with initial conditions from some sufficiently small neighborhood of the equilibrium state u0 ≡ 1
Special nonlinear equations that do not contain small parameters are constructed as the main results. Their nonlocal dynamics determine the behavior of the boundary value problem (1), (2) solutions in the neighborhood of the equilibrium state of u0
Summary
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Special nonlinear equations that do not contain small parameters are constructed as the main results Their nonlocal dynamics determine the behavior of the boundary value problem (1), (2) solutions in the neighborhood of the equilibrium state of u0. These equations are classical normal forms on invariant manifolds in finite-dimensional critical cases. There are no invariant manifolds in infinite-dimensional critical cases, but the formal method of normal forms allows us to construct special boundary value problems of the parabolic type, the so-called quasinormal forms, which play the role of normal forms.
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