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On the Contraction Analysis of Nonlinear System with Multiple Equilibrium Points

Riddhi Mohan Bora, Bhabani Shankar Dey, Indra Narayan Kar

TL;DR

This paper tackles global stability for planar nonlinear systems with multiple isolated equilibria by fusing 2-contraction theory with Poincaré index theory to identify global regions of attraction without relying on local linearization. It develops a concrete procedure to construct a 2-contraction region $oldsymbol{\\Omega}$ and a bounded energy sublevel set $oldsymbol{\\mathcal{U}}=ig\\{E(f x)<rig frac{ig frac{dE}{dt}igleq 0}$, yielding a common BOA $oldsymbol{\\mathcal{D}_0}=oldsymbol{\\Omega igcap oldsymbol{\\mathcal{U}}}$ that contains all stable equilibria. The method integrates computation of $oldsymbol{\\Omega}$, equilibrium classification via $I_C(f f)$, and an energy-based bound to preclude nontrivial periodic orbits in certain regions, with validation on benchmark planar systems and extensions to networked opinion-dynamics models. This framework provides a scalable approach for obtaining global-like stability guarantees in multi-equilibria settings, complementing traditional Lyapunov and 1-contraction analyses and offering practical guidance for multi-agent network stability analysis.

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.

On the Contraction Analysis of Nonlinear System with Multiple Equilibrium Points

TL;DR

This paper tackles global stability for planar nonlinear systems with multiple isolated equilibria by fusing 2-contraction theory with Poincaré index theory to identify global regions of attraction without relying on local linearization. It develops a concrete procedure to construct a 2-contraction region and a bounded energy sublevel set , yielding a common BOA that contains all stable equilibria. The method integrates computation of , equilibrium classification via , and an energy-based bound to preclude nontrivial periodic orbits in certain regions, with validation on benchmark planar systems and extensions to networked opinion-dynamics models. This framework provides a scalable approach for obtaining global-like stability guarantees in multi-equilibria settings, complementing traditional Lyapunov and 1-contraction analyses and offering practical guidance for multi-agent network stability analysis.

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.

Paper Structure

This paper contains 11 sections, 6 theorems, 27 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Contraction analysis
  4. Standard contraction or 1-contraction
  5. 2-contraction
  6. Poincar$\acute{\textbf{e}}$ Index theory meiss2007differentialstrogatz2001nonlinear
  7. Finding a common BOA for a nonlinear system with multiple equilibria
  8. Computation of 2-contraction region
  9. Nature of equilibrium points
  10. Definition of common BOA for a planar system
  11. General steps to compute common BOA

Key Result

Lemma 1

WU2022110048 If there exists a scalar $\eta > 0$ and a vector norm $\parallel . \parallel$, with induced matrix measure $\mu: \Re^{n \times n} \to \Re$, such that

Figures (5)

  • Figure 1: Evolution of virtual dynamics $\delta \mathbf{x}(t)$ with respect to time.
  • Figure 2: Convergence of two trajectories in a contraction region.
  • Figure 3: The phase trajectories originating from 10 different initial conditions for system \ref{['2nd order nonlinear system example 1']}.
  • Figure 4: Evolution of area $z(t)$ with respect to time.
  • Figure 5: Definition of the Poincar$\acute{e}$ index.

Theorems & Definitions (11)

  • Definition 1
  • Definition 2
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Proposition 1
  • proof
  • Remark 3
  • Definition 4
  • ...and 1 more