The ergodic theory of chaotic feedback systems
Abstract
The notion of an invariant density for a dynamical system is defined, and it is shown that by the Birkhoff ergodic theorem such densities may be used to compute various interesting statistics for the trajectories. It is also shown how, for one class of systems, such densities satisfy a certain functional equation. Existence and uniqueness results for these functional equations are presented. It is shown that for some chaotic systems no continuous invariant densities exist whereas for a related class of systems there exist unique invariant uniform densities. The principal difference between these two classes of systems is that for the former there do not exist positively invariant sets with nonzero content.
 Publication:

In: Conference on Decision and Control
 Pub Date:
 1980
 Bibcode:
 1980deco....1...80B
 Keywords:

 Control Theory;
 Ergodic Process;
 Feedback Control;
 Stochastic Processes;
 Trajectory Analysis;
 Invariance;
 Mathematical Models;
 Numerical Analysis;
 Scalars;
 Electronics and Electrical Engineering