Transient chaos in time-delayed systems subjected to parameter drift

TL;DR

This paper investigates how slow parameter drift in time-delayed systems induces transient chaos, revealing probabilistic tipping behavior, lifetime distributions, and scaling laws that differ from non-delayed systems.

Contribution

It introduces a novel analysis of transient chaos caused by parameter drift in infinite-dimensional time-delayed systems, including lifetime distributions and scaling laws.

Findings

Transient chaos occurs due to parameter drift crossing bifurcations.

Transient lifetimes follow a gamma distribution, not a normal distribution.

The parameter change rate influences tipping probability and transient lifetime, following a derived scaling law.

Abstract

External and internal factors may cause a system's parameter to vary with time before it stabilizes. This drift induces a regime shift when the parameter crosses a bifurcation. Here, we study the case of an infinite dimensional system: a time-delayed oscillator whose time delay varies at a small but non-negligible rate. Our research shows that due to this parameter drift, trajectories from a chaotic attractor tip to other states with a certain probability. This causes the appearance of the phenomenon of transient chaos. By using an ensemble approach, we find a gamma distribution of transient lifetimes, unlike in other non-delayed systems where normal distributions have been found to govern the process. Furthermore, we analyze how the parameter change rate influences the tipping probability, and we derive a scaling law relating the parameter value for which the tipping takes place and the lifetime of the transient chaos with the parameter change rate.