Delay differential equations: On periodic solutions, Floquet multipliers, and global attractors
H. O. Walther
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
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.
For any remark concerning the website.
Last modified: Mon Sep 09 00:02:09 Pacific Daylight Time 2002