Critical speed-up vs critical slow-down: a new kind of relaxation oscillation with application to stick-slip phenomena

TL;DR

This paper introduces a new type of relaxation oscillation observed in stick-slip phenomena, characterized by a speed-up before transition, differing from traditional critical slowing-down, supported by asymptotic analysis and numerical simulations.

Contribution

It identifies and explains a novel relaxation oscillation mechanism with a finite-time singularity, contrasting standard models with an attracting slow manifold.

Findings

Fast and slow steps in dynamics for large parameters

Slow manifold is always attracting, with a finite-time singularity causing transition

System response shows speed-up before transition, unlike critical slowing-down

Abstract

The equations for the sliding of a single block driven by an elastic force show numerically a fast and a slow step in their dynamics when a dimensionless parameter is very large, a limit pertinent for many applications. An asymptotic analysis of the solutions explains well the two sharply different steps of the stick-slip dynamics. The stick (slow) part takes place along a slow manifold in the phase space. But, in contrast with standard relaxation dynamics (of van der Pol type), the slow manifold is always formally attracting and the transition from slow to fast dynamics occurs because the slow dynamics has a finite time singularity breaking the assumption of slowness. This makes a new kind of relaxation oscillation. We show that the response of the stick-slip system to an external noise displays a progressive speed-up before the transition, in contrast with the well known critical slowing-down observed in the standard case.