On the Contraction Analysis of Nonlinear System with Multiple Equilibrium Points

TL;DR

This paper introduces a global stability analysis framework for nonlinear systems with multiple equilibria using 2-contraction theory and geometric tools, enabling characterization of equilibrium nature, regions of periodic solutions, and basins of attraction.

Contribution

It develops a novel methodology combining 2-contraction and index theory to analyze stability, identify regions of periodic solutions, and approximate basins of attraction in nonlinear systems with multiple equilibria.

Findings

The framework effectively characterizes equilibrium points and their stability.

It provides a way to approximate basins of attraction for multiple stable equilibria.

Numerical simulations validate the theoretical approach.

Abstract

In this work, we leverage the 2-contraction theory, which extends the capabilities of classical contraction theory, to develop a global stability framework. Coupled with powerful geometric tools such as the Poincare index theory, the 2-contraction theory enables us to analyze the stability of planar nonlinear systems without relying on local equilibrium analysis. By utilizing index theory and 2-contraction results, we efficiently characterize the nature of equilibrium points and delineate regions in 2-dimensional state space where periodic solutions, closed orbits, or stable dynamics may exist. A key focus of this work is the identification of regions in the state space where periodic solutions may occur, as well as 2-contraction regions that guarantee the nonexistence of such solutions. Additionally, we address a critical problem in engineering the determination of the basin of attraction (BOA) for stable equilibrium points. For systems with multiple equilibria identifying candidate BOAs becomes highly nontrivial. We propose a novel methodology leveraging the 2-contraction theory to approximate a common BOA for a class of nonlinear systems with multiple stable equilibria. Theoretical findings are substantiated through benchmark examples and numerical simulations, demonstrating the practical utility of the proposed approach. Furthermore, we extend our framework to analyze networked systems, showcasing their efficacy in an opinion dynamics problem.