Transient dynamics of the Lorenz system with a parameter drift

TL;DR

This paper investigates how slowly changing parameters affect the transient dynamics of the Lorenz system, revealing a scaling law for transient duration and exploring recovery methods by reversing parameter changes.

Contribution

It uncovers a new scaling law linking transient duration to parameter change rate and analyzes transient recovery via parameter reversal in non-autonomous Lorenz systems.

Findings

Discovered a scaling law relating transient duration to parameter change rate.

Analyzed the effectiveness of reversing parameter changes to recover transient dynamics.

Provided insights into transient behavior in systems with chaotic attractors.

Abstract

Non-autonomous dynamical systems help us to understand the implications of real systems which are in contact with their environment as it actually occurs in nature. Here, we focus on systems where a parameter changes with time at small but non-negligible rates before settling at a stable value, by using the Lorenz system for illustration. This kind of systems commonly show a long-term transient dynamics previous to a sudden transition to a steady state. This can be explained by the crossing of a bifurcation in the associated frozen-in system. We surprisingly uncover a scaling law relating the duration of the transient to the rate of change of the parameter for a case where a chaotic attractor is involved. Additionally, we analyze the viability of recovering the transient dynamics by reversing the parameter to its original value, as an alternative to the control theory for systems with parameter drifts. We obtain the relationship between the paramater change rate and the number of trajectories that tip back to the initial attractor corresponding to the transient state.