Weak Chaos, Anomalous Diffusion, and Weak Ergodicity Breaking in Systems with Delay

TL;DR

This paper investigates how delay systems with specific nonlinearities exhibit weak chaos, subdiffusive behavior, and ergodicity breaking, with effects influenced by delay modulation and fixed points in function space.

Contribution

It reveals the conditions under which weak chaos and anomalous diffusion occur in delay systems, including the impact of delay modulation and non-hyperbolic fixed points.

Findings

Delay systems can show weak chaos and subdiffusive behavior.

Delay modulation reduces effective system dimension and induces anomalous diffusion.

Anomalous behavior stems from non-hyperbolic fixed points in function space.

Abstract

We demonstrate that standard delay systems with a linear instantaneous and a delayed nonlinear term show weak chaos, asymptotically subdiffusive behavior, and weak ergodicity breaking if the nonlinearity is chosen from a specific class of functions. In the limit of large constant delay times, anomalous behavior may not be observable due to exponentially large crossover times. A periodic modulation of the delay causes a strong reduction of the effective dimension of the chaotic phases, leads to hitherto unknown types of solutions, and the occurrence of anomalous diffusion already at short times. The observed anomalous behavior is caused by non-hyperbolic fixed points in function space.