Generalizations of Laminar Chaos

• Applicant: Professor Dr. Martin Weigel, since 8/2024
• Subject Area: Statistical Physics, Nonlinear Dynamics, Complex Systems, Soft and Fluid Matter, Biological Physics
• Term: from 2021 to 2023
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 456546951

The goal of this project continues to be the exploration of the generalizations of the concept of Laminar Chaos, which we found for systems with periodically varying time-delay. We want to continue our research along the originally planned lines, but also add new, interesting aspects not considered previously. The first line was to consider scalar delay differential equations with increasing complexity in the delayed argument: periodic - quasi-periodic - random - state-dependent. Here it remains to finalize our studies on random delays and to consider the most complicated case of state-dependent delay. The second line was about generalizations to non-scalar systems, which we could not yet consider due to lack of time. They should now receive enhanced attention, because other groups, inspired by our work, started already to consider effects, especially of synchronization, for vectorial systems showing Laminar Chaos in one of its components. The main work in this line, however, is planned to follow our original proposal from the first funding period. A hitherto neglected aspect attracted our attention while publishing our results: all our previous results were obtained for one discrete time-dependent delay. In reality, however, the delay time may be perturbed by small, but fast fluctuations, which can be modelled by a distributed delay with a small, but non-zero width of its distribution. On one hand, one would naively expect that the concept of Laminar Chaos survives such small perturbations, but on the other hand, one of our fundamental tools, the access map, can no longer be applied. For broad delay distributions, discrete or continuous, we do not expect that Laminar Chaos survives in general. But this poses the question in which way time-dependent distributed delays allow or destroy the phenomenon of Laminar Chaos, which constitutes a new and important third line of research. Correspondingly, the planned work can be divided into three parts: I. Generalization of Laminar Chaos to systems with random delays (completion and weak disorder) and systems with state-dependent delays II. Laminar Chaos for scalar systems with distributed delays III. Generalization of the concept of Laminar Chaos to non-scalar systems. The three parts are logically connected. I. and II. both treat scalar systems, but with increasing complexity in the delayed argument: Whereas understanding the occurrence of Laminar Chaos for random delays is a prerequisite for understanding its occurrence in systems with state-dependent delay, these investigations are also necessary for understanding scalar systems with distributed delay in II. All these investigations also aim at a geometrical understanding of the qualitative dynamics in abstract state space leading to Laminar Chaos. This may allow us to define it eventually without reference to specific forms of the delay equations.

• DFG Programme: Research Grants
• Ehemaliger Antragsteller: Professor Dr. Günter Radons, until 7/2024 (†)