Contraction After Small Transients

TL;DR

This paper extends contraction theory to include systems that exhibit contraction after small transients, providing conditions for such generalized contractive systems and demonstrating their relevance in biological models.

Contribution

It introduces generalized contraction with respect to a norm, offers checkable conditions for these systems, and shows they preserve key asymptotic properties despite small transients.

Findings

Existence of simple conditions for generalized contraction

Small transients do not negate convergence or entrainment

Parameter changes can induce generalized contraction before losing contractivity

Abstract

Contraction theory is a powerful tool for proving asymptotic properties of nonlinear dynamical systems including convergence to an attractor and entrainment to a periodic excitation. We consider three generalizations of contraction with respect to a norm that allow contraction to take place after small transients in time and/or amplitude. These generalized contractive systems (GCSs) are useful for several reasons. First, we show that there exist simple and checkable conditions guaranteeing that a system is a GCS, and demonstrate their usefulness using several models from systems biology. Second, allowing small transients does not destroy the important asymptotic properties of contractive systems like convergence to a unique equilibrium point, if it exists, and entrainment to a periodic excitation. Third, in some cases as we change the parameters in a contractive system it becomes a GCS just before it looses contractivity with respect to a norm. In this respect, generalized contractivity is the analogue of marginal stability in Lyapunov stability theory.