Laminar chaos in systems with random and chaotically time-varying delay

A type of chaos called laminar chaos was found in singularly perturbed dynamical systems with periodically [D. Müller et al., Phys. Rev. Lett. 120, 084102 (2018)10.1103/PhysRevLett.120.084102] and quasiperiodically [D. Müller-Bender and G. Radons, Phys. Rev. E 107, 014205 (2023)10.1103/PhysRevE.107.014205] time-varying delay. Compared to high-dimensional turbulent chaos that is typically found in such systems with large constant delay, laminar chaos is a very low-dimensional phenomenon. It is characterized by a time series with nearly constant laminar phases that are interrupted by irregular bursts, where the intensity level of the laminar phases varies chaotically from phase to phase. In this paper, we demonstrate that laminar chaos, and its generalizations, can also be observed in systems with random and chaotically time-varying delay. Moreover, while for periodic and quasiperiodic delays the appearance of (generalized) laminar chaos and turbulent chaos depends in a fractal manner on the delay parameters, it turns out that short-time correlated random and chaotic delays lead to (generalized) laminar chaos in almost the whole delay parameter space, where the properties of circle maps with quenched disorder play a crucial role. It follows that introducing such a delay variation typically leads to a drastic reduction of the dimension of the chaotic attractor of the considered systems. We investigate the dynamical properties and generalize the known methods for detecting laminar chaos in experimental time series to random and chaotically time-varying delay.