## 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.