Mitigating long transient time in deterministic systems by resetting

TL;DR

This paper demonstrates that applying resetting strategies to deterministic dynamical systems can significantly reduce transient times and fluctuations, improving system stability and response speed.

Contribution

It introduces the novel application of resetting to deterministic systems to control transient durations and fluctuations, supported by numerical studies on complex oscillators.

Findings

Resetting dramatically decreases mean transient time.

Resetting reduces fluctuations around the mean transient time.

Effective resetting strategies depend on system dynamics.

Abstract

How long does a trajectory take to reach a stable equilibrium point in the basin of attraction of a dynamical system? This is a question of quite general interest, and has stimulated a lot of activities in dynamical and stochastic systems where the metric of this estimation is often known as the transient or first passage time. In nonlinear systems, one often experiences long transients due to their underlying dynamics. We apply resetting or restart, an emerging concept in statistical physics and stochastic process, to mitigate the detrimental effects of prolonged transients in deterministic dynamical systems. We show that stopping an ongoing process at intermittent time only to restart all over from a spatial control line, can dramatically expedite its completion, resulting in a huge decrease in mean transient time. Moreover, our study unfolds a net reduction in fluctuations around the mean. Our claim is established with detailed numerical studies on the Stuart-Landau limit cycle oscillator and chaotic Lorenz system under different resetting strategies. Our analysis opens up a door to control the mean and fluctuations in transient time by unifying the original dynamics with an external stochastic or periodic timer, and poses open questions on the optimal way to harness transients in dynamical systems.