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<p>This work leverages the 2-contraction theory, which extends classical contraction theory, to develop a systematic stability framework for systems with multiple equilibria. Coupled with powerful geometric tools, such as Poincaré index theory, the 2-contraction theory enables the stability analysis of the planar nonlinear systems instead of undertaking it locally around equilibrium points. Using index theory and 2-contraction, we characterize the nature of equilibrium points and delineate regions in <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathbb {R}^2\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> where periodic solutions, closed orbits, or stable solutions may exist. A key focus of this work is the identification of regions in the state space where periodic solutions may exist and the regions that guarantee the non-existence of such solutions. Additionally, we address a crucial problem to determine the basin of attraction (BOA) for stable equilibrium points. For systems with multiple equilibria, identifying candidate BOAs is highly nontrivial. We propose a novel methodology adopting the 2-contraction theory to approximate a common BOA for a class of nonlinear systems with multiple equilibria. Theoretical findings are substantiated through benchmark examples and numerical simulations, demonstrating the efficacy of the proposed approach. Furthermore, we extend our framework to analyze networked systems, showcasing their efficacy in an opinion dynamics problem.</p>

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2-Contraction and Poincaré-Index Theory Based Framework for Stability Analysis of Nonlinear Dynamical Systems with Multiple Equilibria

  • 期刊论文

  • 发表日期：

    2026年7月20日

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  • Riddhi Mohan Bora,
  • Bhabani Shankar Dey,
  • Indra Narayan Kar

摘要

This work leverages the 2-contraction theory, which extends classical contraction theory, to develop a systematic stability framework for systems with multiple equilibria. Coupled with powerful geometric tools, such as Poincaré index theory, the 2-contraction theory enables the stability analysis of the planar nonlinear systems instead of undertaking it locally around equilibrium points. Using index theory and 2-contraction, we characterize the nature of equilibrium points and delineate regions in \(\mathbb {R}^2\) R 2 where periodic solutions, closed orbits, or stable solutions may exist. A key focus of this work is the identification of regions in the state space where periodic solutions may exist and the regions that guarantee the non-existence of such solutions. Additionally, we address a crucial problem to determine the basin of attraction (BOA) for stable equilibrium points. For systems with multiple equilibria, identifying candidate BOAs is highly nontrivial. We propose a novel methodology adopting the 2-contraction theory to approximate a common BOA for a class of nonlinear systems with multiple equilibria. Theoretical findings are substantiated through benchmark examples and numerical simulations, demonstrating the efficacy of the proposed approach. Furthermore, we extend our framework to analyze networked systems, showcasing their efficacy in an opinion dynamics problem.