Using SOS for Optimal Semialgebraic Representation of Sets: Finding Minimal Representations of Limit Cycles, Chaotic Attractors and Unions

TL;DR

This paper demonstrates how Sum-of-Squares optimization can be used to find minimal volume semialgebraic representations of sets, including attractors and unions, with applications to chaotic and nonlinear systems.

Contribution

It introduces a heuristic SOS-based method for optimal set representation focusing on volume minimization under various constraints.

Findings

Successfully applied to Lorenz attractor

Effectively finds minimal representations of limit cycles

Handles implicit and explicit set definitions

Abstract

In this paper we show that Sum-of-Squares optimization can be used to find optimal semialgebraic representations of sets. These sets may be explicitly defined, as in the case of discrete points or unions of sets; or implicitly defined, as in the case of attractors of nonlinear systems. We define optimality in the sense of minimum volume, while satisfying constraints that can include set containment, convexity, or Lyapunov stability conditions. Our admittedly heuristic approach to volume minimization is based on the use of a determinantlike objective function. We provide numerical examples for the Lorenz attractor and the Van der Pol limit cycle.