On Three Generalizations of Contraction

TL;DR

This paper introduces three new forms of generalized contraction that accommodate initial transients in time and amplitude, broadening the applicability of contraction-based analysis in dynamical systems.

Contribution

The paper proposes three novel generalizations of contraction, providing conditions and examples where these apply to systems not contractive under any norm.

Findings

GC can handle systems with initial transients

Examples show GC applies where traditional contraction fails

Sufficient conditions for GC are established

Abstract

We introduce three forms of generalized contraction (GC). Roughly speaking, these are motivated by allowing contraction to take place after small transients in time and/or amplitude. Indeed, contraction is usually used to prove asymptotic properties, like convergence to an attractor or entrainment to a periodic excitation, and allowing initial transients does not affect this asymptotic behavior. We provide sufficient conditions for GC, and demonstrate their usefulness using examples of systems that are not contractive, with respect to any norm, yet are GC.