Symmetries, Stability, and Control in Nonlinear Systems and Networks

TL;DR

This paper explores how symmetries and stability concepts can be integrated to analyze and control nonlinear systems and networks, using contraction theory and equivariance to understand invariance and synchrony.

Contribution

It introduces a novel framework combining symmetry analysis with contraction theory for nonlinear systems and networks, enhancing stability and control strategies.

Findings

Symmetries can be used to analyze invariance in nonlinear networks.

Contraction theory provides a convergence-based approach to stability.

Structural symmetries facilitate pattern control in complex networks.

Abstract

This paper discusses the interplay of symmetries and stability in the analysis and control of nonlinear dynamical systems and networks. Specifically, it combines standard results on symmetries and equivariance with recent convergence analysis tools based on nonlinear contraction theory and virtual dynamical systems. This synergy between structural properties (symmetries) and convergence properties (contraction) is illustrated in the contexts of network motifs arising e.g. in genetic networks, of invariance to environmental symmetries, and of imposing different patterns of synchrony in a network.