Review on contraction analysis and computation of contraction metrics

TL;DR

This review discusses contraction analysis for dynamical systems, highlighting its theoretical foundations, extensions to various system types, and recent algorithms for computing contraction metrics to assess system stability and behavior.

Contribution

It provides a comprehensive overview linking mathematical and engineering approaches, and details recent developments in algorithms for contraction metric computation.

Findings

Contraction analysis applies to diverse system types including discrete and delay systems.

Recent algorithms enable practical computation of contraction metrics.

Contraction analysis helps estimate attractor dimensions and entropy.

Abstract

Contraction analysis considers the distance between two adjacent trajectories. If this distance is contracting, then trajectories have the same long-term behavior. The main advantage of this analysis is that it is independent of the solutions under consideration. Using an appropriate metric, with respect to which the distance is contracting, one can show convergence to a unique equilibrium or, if attraction only occurs in certain directions, to a periodic orbit. Contraction analysis was originally considered for ordinary differential equations, but has been extended to discrete-time systems, control systems, delay equations and many other types of systems. Moreover, similar techniques can be applied for the estimation of the dimension of attractors and for the estimation of different notions of entropy (including topological entropy). This review attempts to link the references in both the mathematical and the engineering literature and, furthermore, point out the recent developments and algorithms in the computation of contraction metrics.