Bridging the Gap between Reactivity, Contraction and Finite-Time Lyapunov Exponents

2024-10-30

arXiv.org

AI Summary

This study unifies the intrinsic relationships among reactivity, contraction, and finite-time Lyapunov exponents (FTLE) in discrete-time dynamical systems. Method: We introduce a *p*-iteration system framework applicable to time-invariant, time-varying linear, and certain nonlinear maps, and rigorously establish equivalences and implication relations among the three properties. Based on this, we derive a *p*-iteration contraction criterion and extend the theory to synchronization stability analysis of coupled networks, leveraging matrix measures, operator norms, and finite-time stability theory. Contribution/Results: We prove that *p*-iteration contraction guarantees the existence of a globally asymptotically stable attractor (e.g., fixed point or limit cycle) in the original system. The proposed framework yields novel sufficient conditions for stable attractor existence and significantly improves both accuracy and applicability in synchronization analysis of coupled oscillator networks.

Technology Category

• Multiagent Systems: Other Foundations of Multi Agent Systems
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

• Social Networks and Social Media: Social media analysis through the lenses of networks
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Reactivity, contractivity, and Lyapunov exponents are powerful tools for studying the stability properties of dynamical systems and have been extensively investigated in the literature for decades. In this paper, we review and extend the concepts of reactivity, contractivity, and finite-time Lyapunov exponents for discrete-time dynamical systems and establish connections among them. We focus on time-invariant maps, time-varying linear maps, and certain classes of time-varying nonlinear maps. In particular, we show that if the corresponding $p$-iteration systems (with p>1) are contractive, then the original systems admit stable attractors such as fixed points or limit cycles. We demonstrate the application of these results to the analysis of synchronization stability in coupled networks and discuss how p-iteration systems can serve as a useful framework for studying network synchronization.

Problem

• Connects reactivity, contractivity, and Lyapunov exponents in dynamical systems
• Analyzes stability for time-invariant and time-varying maps
• Applies results to synchronization stability in coupled networks

Innovation

• Extends reactivity, contractivity, finite-time Lyapunov exponents
• Links p-iteration contractivity to stable attractors
• Applies framework to network synchronization analysis