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
MATH
For this differential equation, x(t) = 3/4 is a stable equilibrium. In forwards time (i.e., as t→+∞), for x0∈(0,1), all trajectories tend towards 3/4. Finding a solution of the difference equation is equivalent to iterating the function
MATH.
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 $\QTR{Bbb}{R_{+}}$ = [0,∞), MATH, is a parameter space for the corresponding dynamic equations
MATH.
The time scales for the forward solutions of the differential equation and the difference equation are $\QTR{Bbb}{R_{+}}$ and $\QTR{Bbb}{Z_{+}}$, respectively. This is joint work with E. R. Duke, K. J. Hall, and B. A. Lawrence.