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

Abstract

A scalar delay-differential equation with diffusion term in one space dimension, where the diffusivity D is a bifurcation parameter, is considered. The center manifold theory and the method of Lyapunov-Schmidt are used to describe two bifurcations from spatially constant solutions as D decreases. By modifying the equation the order of these bifurcations can be reversed. Then the existence of a compact attractor for a class of such equations is shown and the structure of part of the attractor for the modified equation is investigated. It is known that the solutions are globally L2-bounded; bounds on the solution operator from one intermediate space to another are constructed to obtain an attractor in the W e'2 sense.

Normal forms and Hopf bifurcation for partial differential equations with delays

“…For the first PFDE, (H5) holds and the results of Section 4 are applicable; however, this does not happen for the second PFDE, and then the associated FDE does not provide complete information. In both cases, we shall use complex coordinates, i.e., we shall consider C = C([−r, 0]; C), since complex variables allow us to diagonalize the matrix B in (2.9), which implies that the operators M [2], [9], [10], [11], [14]):…”

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].…”

“…Notice that if we introduce a transformation v(x, t) = (a + b)u(x, t), then v satisfies the following equation which is same as (1.4) ignoring a scalar factor. Note that we can rewrite I (n, a, b) = I (n, b) as follows: 27) where K (n, b) is a function of variables b ∈ (0.5, 1) and n = 0, 1, 2, · · ·.…”

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.…”