Anomalous Behavior in Systems with Delay

Abstract

In a recent letter [Phys. Rev. E 112, L042201 (2025)], we demonstrated the occurrence of anomalous diffusion, weak chaos, and weak ergodicity breaking in a certain class of time-delayed feedback systems with linear instantaneous and nonlinear delayed term. This anomalous behavior is caused by the trapping of chaotic solutions close to nonhyperbolic fixed-point solutions in function space. In this paper, we extend our previous study, present more detailed insights, and show surprising new results. We first investigate the case of a constant delay, where the dynamical behavior during such trapping events can be explained by center manifold theory. In the large-delay limit, the probability for the trapping events becomes small because of a high effective dimension of the chaotic phases. As a consequence, anomalous behavior is only observable on very large time scales. By introducing a periodic modulation of the delay time, trapping events become more probable with increasing modulation amplitude due to a reduction of the effective dimension and the occurrence of laminar chaos. For large values of the modulation amplitude, the anomalous behavior is more complex and the most relevant aspects can be reproduced by a two-dimensional iterated map and a stochastic model.