Marrakech 2002

Delay differential equations: On periodic solutions, Floquet multipliers, and global attractors

H. O. Walther

U. Giessen

The lectures deal with autonomous delay differential equations, with constant and with state-dependent time lags. In the first part, results by T. Krisztin, J. Wu, and the author on the structure of a global attractor are explained. The underlying delay differential equation is related to neural network theory. The attractor is 3-dimensional, looks like a solid spindle, contains one periodic orbit and three stationary points, and is smooth except at the tips of the spindle which are singularities.

The second part presents joint work with A.L. Skubachevsky on Floquet multipliers of periodic solutions. In cases where the period of the solution and the delay in the differential equation are commensurable, the Floquet multipliers are the solutions of a characteristic equation. The characteristic equation can be analyzed and permits to obtain stability and hyperbolicity of periodic orbits in some cases. Involved are new existence results for periodic solutions.

The third part is devoted to a new, rather elementary method to find attracting periodic orbits. It uses only Lipschitz continuity and the form of the nonlinearities in the delay differential equations, and works also for systems with state dependent delay.

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Last modified: Mon Sep 09 00:02:09 Pacific Daylight Time 2002