Topological entropy of switched nonlinear and interconnected systems

TL;DR

This paper develops bounds for the topological entropy of switched nonlinear and interconnected systems, providing tools to estimate system complexity based on Jacobian measures and network structure.

Contribution

It introduces new upper and lower bounds for topological entropy of switched systems, including network-level bounds and bounds requiring less switching information.

Findings

Upper bounds depend on matrix measures of Jacobians.

Lower bounds are based on traces of Jacobians.

Numerical examples demonstrate the bounds on ecosystem models.

Abstract

A general upper bound for topological entropy of switched nonlinear systems is constructed, using an asymptotic average of upper limits of the matrix measures of Jacobian matrices of strongly persistent individual modes, weighted by their active rates. A general lower bound is constructed as well, using a similar weighted average of lower limits of the traces of these Jacobian matrices. In a case of interconnected structure, the general upper bound is readily applied to derive upper bounds for entropy that depend only on "network-level" information. In a case of block-diagonal structure, less conservative upper and lower bounds for entropy are constructed. In each case, upper bounds for entropy that require less information about the switching signal are also derived. The upper bounds for entropy and their relations are illustrated by numerical examples of a switched Lotka-Volterra ecosystem model.