A converse sum of squares lyapunov function for outer approximation of minimal attractor sets of nonlinear systems

Morgan Jones; Matthew M. Peet

doi:10.3934/jcd.2022019

Article Contents

2023, Volume 10, Issue 1: 48-74. Doi: 10.3934/jcd.2022019

This issue Previous Article Review on contraction analysis and computation of contraction metrics Next Article A novel moving orthonormal coordinate-based approach for region of attraction analysis of limit cycles

A converse sum of squares lyapunov function for outer approximation of minimal attractor sets of nonlinear systems

Morgan Jones^1, , and
Matthew M. Peet^2,

1.
The Department of Automatic Control and Systems Engineering, The University of Sheffield, Sheffield, S1 3JD UK
2.
The School for the Engineering of Matter, Transport and Energy Tempe, Arizona State University, AZ, 85298 USA

^* Corresponding author: Morgan Jones
^* Corresponding author: Morgan Jones

Received: August 2021

Revised: April 2022

Early access: August 2022

Published: December 2022

The authors of this manuscript were supported by NSF grant NSD CMMI-1931270.
We would like to thank the reviewers of this paper for their valuable comments and suggestions

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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.

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Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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Figure 1. Graph showing an estimation of the Lorenz attractor (Example 1) given by the red transparent surface. This surface is the $ 1 $-sublevel set of a solution to the SOS Problem (68). The grayed shaded surfaces represent the projection of the our Lorenz attractor estimation on the xy, xz, and yz axes. The black line is an approximation of the attractor found by simulating a Lorenz trajectory using Matlab's $ \texttt{ODE45} $ function

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Figure 2. Graph showing an estimation of the attractor (given by the red area) of the ODE (70) in Example 2. This red area is the $ 1 $-sublevel set of a solution to the SOS Problem (68). The two black lines are simulated solution maps of the ODE (71) using Matlab's $ \texttt{ODE45} $ function initialized outside of the limit cycle

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Figure 3. Graph showing an estimation of the attractor (given by the red area) of the ODE (71) in Example 3. This red area is the $ 1 $-sublevel set of a solution to the SOS Problem (68). The four black lines are simulated solution maps initialized at $ [ \pm 1, \pm 1]^T $ of the ODE (71) using Matlab's $ \texttt{ODE45} $ function

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