Trapping Phenomenon Attenuates Tipping Points for Limit Cycles

TL;DR

This paper investigates how trapping phenomena in periodically forced nonlinear systems can smooth out the transition at tipping points of limit cycles, complicating detection of critical transitions.

Contribution

It introduces the concept of a transient channel that preserves limit cycle characteristics, revealing a new mechanism that moderates abrupt tipping in oscillatory systems.

Findings

Channels act as ghosts of destroyed limit cycles, leading to smooth transitions.

Traditional tipping indicators may fail due to the channel's effects.

Limit cycle tipping differs fundamentally from equilibrium tipping.

Abstract

Nonlinear dynamical systems may be exposed to tipping points, critical thresholds at which small changes in the external inputs or in the systems parameters abruptly shift the system to an alternative state with a contrasting dynamical behavior. While tipping in a fold bifurcation of an equilibrium is well understood, much less is known about tipping of oscillations (limit cycles) though this dynamics are the typical response of many natural systems to a periodic external forcing, like e.g. seasonal forcing in ecology and climate sciences. We provide a detailed analysis of tipping phenomena in periodically forced systems and show that, when limit cycles are considered, a transient structure, so-called channel, plays a fundamental role in the transition. Specifically, we demonstrate that trajectories crossing such channel conserve, for a characteristic time, the twisting behavior of the stable limit cycle destroyed in the fold bifurcation of cycles. As a consequence, this channel acts like a ghost of the limit cycle destroyed in the critical transition and instead of the expected abrupt transition we find a smooth one. This smoothness is also the reason that it is difficult to precisely determine the transition point employing the usual indicators of tipping points, like critical slowing down and flickering.