Marshall University Math Colloquium

October 8, 2004

“From Chaos to Stability: Dynamic Equations Parameterized by Time Scales”

Ralph Oberste-Vorth

Marshall University

Abstract

Consider the logistic initial value
problems

For this differential equation, *x*(*t*) = 3/4
is a stable equilibrium. In
forwards time (i.e., as
*t*→+∞), for
*x*_{0}∈(0,1),
all trajectories tend towards 3/4.
Finding a solution of the difference
equation is equivalent to iterating the function

.

Orbits in [0,1]
are chaotic except for countably
many periodic and pre-periodic orbits. Our long-term goal is to try to
understand the differences in behavior between solutions of differential and
difference equations as "limits" and "bifurcations" over
the underlying domains of the solutions. We use the theory of time scales,
developed by S. Hilger in 1988, to do this. The set
of closed subsets of
= [0,∞),
,
is a parameter space for the corresponding dynamic equations

.

The time scales for the forward solutions of the differential equation and the
difference equation are
and
,
respectively. This is joint work with E. R. Duke, K. J. Hall,
and B. A. Lawrence.