From Partial and Horizontal Contraction to $k$-Contraction

TL;DR

This paper explores the relationships between three generalized contraction frameworks in dynamical systems, providing new conditions under which one form implies another, and illustrating these results with the Andronov-Hopf oscillator.

Contribution

It establishes new sufficient conditions linking partial, horizontal, and $k$-contraction, clarifying their differences and implications in system analysis.

Findings

Partial contraction implies horizontal contraction under certain conditions.

Horizontal contraction implies $k$-contraction with specific criteria.

Theoretical results are demonstrated using the Andronov-Hopf oscillator.

Abstract

A geometric generalization of contraction theory called~$k$-contraction was recently developed using $k$-compound matrices. In this note, we focus on the relations between $k$-contraction and two other generalized contraction frameworks: partial contraction (also known as virtual contraction) and horizontal contraction. We show that in general these three notions of contraction are different. We here provide new sufficient conditions guaranteeing that partial contraction implies horizontal contraction, and that horizontal contraction implies $k$-contraction. We use the Andronov-Hopf oscillator to demonstrate some of the theoretical results.