A CONVERSE SUM OF SQUARES LYAPUNOV FUNCTION FOR OUTER APPROXIMATION OF MINIMAL ATTRACTOR SETS OF NONLINEAR SYSTEMS

Morgan Jones, Matthew M.Peet

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

Abstract

Many dynamical systems described by nonlinear ODEs are unstable. Their associated solutions do not converge towards an equilibrium point, but rather converge towards some invariant subset of the state space called an attractor set. For a given ODE, in general, the existence, shape and structure of the attractor sets of the ODE are unknown. Fortunately, the sublevel sets of Lyapunov functions can provide bounds on the attractor sets of ODEs. In this paper we propose a new Lyapunov characterization of attractor sets that is well suited to the problem of finding the minimal attractor set. We show our Lyapunov characterization is non-conservative even when restricted to Sum-of-Squares (SOS) Lyapunov functions. Given these results, we propose a SOS programming problem based on determinant maximization that yields an SOS Lyapunov function whose 1-sublevel set has minimal volume, is an attractor set itself, and provides an optimal outer approximation of the minimal attractor set of the ODE. Several numerical examples are presented including the Lorenz attractor and Van-der-Pol oscillator.

Original languageEnglish (US)
Pages (from-to)48-74
Number of pages27
JournalJournal of Computational Dynamics
Volume10
Issue number1
DOIs
StatePublished - Jan 2023

Keywords

  • attractor sets
  • chaos theory
  • Lyapunov theory
  • Nonlinear systems
  • sum-of-squares programming

ASJC Scopus subject areas

  • Computational Mechanics
  • Computational Mathematics

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