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Area of Science:

  • Dynamical systems theory
  • Nonlinear dynamics
  • Statistical physics

Background:

  • Rate-induced tipping occurs when dynamical systems cannot track equilibrium states during rapid parameter changes.
  • Early-warning indicators (EWIs) like increased variance and autocorrelation are commonly used to predict tipping.
  • The timing and reliability of these EWIs under various conditions remain an active research area.

Purpose of the Study:

  • To investigate the timing of commonly used early-warning indicators (EWIs) in a prototypical model of rate-induced tipping.
  • To explain the observed delay in EWI signals relative to the rate of parameter change.
  • To systematically analyze the most likely tipping time under the influence of noise and varying proximity to the tipping threshold.

Main Methods:

  • Studied a saddle-node normal form model subjected to time-varying equilibrium drift and noise.
  • Employed numerical continuation techniques to find solutions to the variational problem defining the most likely tipping path.
  • Analyzed the relationship between parameter change rate, noise intensity, and the timing of EWIs.

Main Results:

  • Found that early-warning indicators (variance and autocorrelation) signal tipping with a delay, not concurrently with the fastest parameter drift.
  • Demonstrated that this delay arises because the most probable tipping trajectory itself crosses the critical threshold with a delay.
  • Systematically mapped the most likely tipping time across a parameter space defined by proximity to the tipping threshold and noise intensity.

Conclusions:

  • The commonly used early-warning indicators for rate-induced tipping do not provide immediate signals but exhibit a characteristic delay.
  • Understanding this delay is crucial for accurate prediction and mitigation strategies in systems susceptible to critical transitions.
  • The study provides a framework for analyzing tipping times in stochastic dynamical systems with time-varying parameters.