Colloquium: Prof. Jack W. Macki, Mathematics, University of Alberta

Periodic Solutions to Systems of Ordinary Differential Equations with Delay and/or Hysteresis

We consider systems of the form $x'(t) - A x(t) = F[x(.)]$ with $x(t)$ in $Rn$ and $F$ a nonlinear operator with memory. We are principally interested in delay problems, and in problems with hysteresis. We apply a truncated Galerkin method to identify a centre for a ball in the space of two-pi periodic functions to which we can apply Schauder's Theorem. Since the actual period of the unknown periodic solutions is unknown, we normalize the unknown period to two-pi, thus placing the period in the equation as a parameter. In a delay problem, this leads to a two parameter problem. For hysteresis problems, we use multivalued maps with a parameter. We are able to predict periodic solutions which seem to be not detectable by other means (e.g., hopf bifurcation). The class of problems we can treat is limited but interesting.

The research is joint with Professors Pietro Zecca and Paolo Nistri of the University of Florence, Italy.

*Please note Dr. Halperin's talk precedes this colloquium and will be held in Geography 200. Refreshments are for both colloquiums and are compliments of John Wiley \& Sons Canada Ltd. from 1:30-3:30 p.m. in Math Annex Lounge , Room 1115. As well a book exhibit and software demo will take place.