Rare Events and Scale--Invariant Dynamics of Perturbations in Delayed Dynamical Systems

TL;DR

This paper reveals that perturbation dynamics in delayed dynamical systems exhibit scale-invariant, rare event-driven behavior similar to surface growth models, leading to non-Gaussian fluctuations and multiscaling.

Contribution

It establishes a novel mapping between Lyapunov vector dynamics in delayed systems and the Zhang surface growth model, highlighting the role of rare events.

Findings

Lyapunov vector dynamics map to the Zhang surface growth model.

Perturbations show non-Gaussian fluctuations and multiscaling.

Rare events dominate the dynamics, affecting stability analysis.

Abstract

We study the dynamics of perturbations in time delayed dynamical systems. Using a suitable space-time coordinate transformation, we find that the time evolution of the linearized perturbations (Lyapunov vector) can be mapped to the linear Zhang surface growth model [Y.-C. Zhang, J. Phys. France {\bf 51}, 2129 (1990)], which is known to describe surface roughening driven by power-law distributed noise. As a consequence, Lyapunov vector dynamics is dominated by rare random events that lead to non-Gaussian fluctuations and multiscaling properties.