Rate-induced tipping from periodic attractors: partial tipping and connecting orbits

TL;DR

This paper investigates how rapid parameter changes can cause partial or complete tipping in systems with periodic attractors, revealing new phenomena and thresholds using advanced mathematical methods.

Contribution

It introduces the concept of partial tipping in systems with periodic attractors and characterizes tipping thresholds using periodic-to-periodic and periodic-to-equilibrium connections.

Findings

Identification of partial tipping phenomena.

Thresholds for rate-induced tipping determined.

Use of Lin's method for connection analysis.

Abstract

We consider how breakdown of the quasistatic approximation for attractors can lead to rate-induced tipping, where a qualitative change in tracking/tipping behaviour of trajectories can be characterised in terms of a critical rate. Associated with rate-induced tipping (where tracking of a branch of quasistatic attractors breaks down) we find a new phenomenon for attractors that are not simply equilibria: partial tipping of the pullback attractor where certain phases of the periodic attractor tip and others track the quasistatic attractor. For a specific model system with a parameter shift between two asymptotically autonomous systems with periodic attractors we characterise thresholds of rate-induced tipping to partial and total tipping. We show these thresholds can be found in terms of certain periodic-to-periodic (PtoP) and periodic-to-equilibrium (PtoE) connections that we determine using Lin's method for an augmented system.