Finding positively invariant sets and proving exponential stability of limit cycles using Sum-of-Squares decompositions

TL;DR

This paper introduces a computational method using sum-of-squares decompositions and semidefinite programming to find positively invariant sets and prove exponential stability of limit cycles in polynomial dynamical systems of any dimension.

Contribution

It extends SOS-based stability analysis by enabling the computation of invariant sets and exponential stability proofs for limit cycles in high-dimensional polynomial systems.

Findings

Successfully computed positively invariant sets for classical systems

Proved exponential stability of limit cycles in systems like van der Pol oscillator

Applicable to systems of any dimension, surpassing classical 2D methods

Abstract

The dynamics of many systems from physics, economics, chemistry, and biology can be modelled through polynomial functions. In this paper, we provide a computational means to find positively invariant sets of polynomial dynamical systems by using semidefinite programming to solve sum-of-squares (SOS) programmes. With the emergence of SOS programmes, it is possible to efficiently search for Lyapunov functions that guarantee stability of polynomial systems. Yet, SOS computations often fail to find functions, such that the conditions hold in the entire state space. We show here that restricting the SOS optimisation to specific domains enables us to obtain positively invariant sets, thus facilitating the analysis of the dynamics by considering separately each positively invariant set. In addition, we go beyond classical Lyapunov stability analysis and use SOS decompositions to computationally implement sufficient positivity conditions that guarantee existence, uniqueness, and exponential stability of a limit cycle. Importantly, this approach is applicable to systems of any dimension and, thus, goes beyond classical methods that are restricted to two-dimensional phase space. We illustrate our different results with applications to classical systems, such as the van der Pol oscillator, the Fitzhugh-Nagumo neuronal equation, and the Lorenz system.