Speaker:
Title:
Abstract: Alternative to the popular OGY and targeting techniques of controlling
chaos is the statistical based inverse Frobenius-Perron Problem (IFPP).
In our approach,
we reduce the question of stabilizing an arbitrary invariant measure, to
the question of whether or not a corresponding hyperplane intersects a
unit hyperbox. Based on this simple observation, we introduce a
straight forward matrix-based control algorithm, and we assert several
easily tested controllability theorems. A major requirement of our
transition matrix approach, is our ability to approximate the dynamical
system by a stochastic matrix using the Ulam conjecture, and then our
ability to pass from the modified stochastic matrix back to an
appropriate and nearby dynamical system, which we call the inverse Ulam
problem (IUP). We present here a new class of piecewise affine
transformations, generalizing the Baker's maps for an arbitrary
grammatical symbol dynamics, designed to solve the IUP.
In this talk, we will give background information, including discussion
of action of a dynamical system on a distribution ensemble of initial
condition, whose long-term distribution defines ``observed statistics."
This action on distributions is defined by an associated
Frobenius-Perron operator on L^1 distributions, and the Koopman
operator, the dual operator on measures. Several examples will be given
in the more easily accessible one-dimensional case. We will also
discuss the so-called Ulam conjecture, and its relationship to our
problem.