Cycling chaotic attractors in two models for dynamics with invariant subspaces

Area of Science:

• Dynamical Systems and Chaos Theory
• Nonlinear Dynamics
• Mathematical Physics

Background:

• Nonergodic attractors, characterized by cycles between chaotic saddles, are robust in symmetric systems.
• These cycling attractors exhibit complex dynamics, including "cycling chaos," due to connections within invariant subspaces.
• Understanding the structure and behavior of these attractors is crucial for analyzing complex systems.

Purpose of the Study:

• To investigate the effects of internal symmetries and phase-resetting on cycling attractors in symmetric systems.
• To analyze two specific models: cyclically coupled Lorenz equations and a magnetoconvection return map.
• To explore the robustness and complexity of connections within chaotic saddles and the resulting attractors.

Main Methods:

• Detailed examination of two established models exhibiting cycling attractors.
• Analysis of internal symmetries within chaotic saddles and their impact on trajectory connections.
• Investigation of phase-resetting phenomena and its implications for attractor dynamics, including skew product structures and resonance effects.

Main Results:

• A "false phase-resetting" effect was identified in cyclically coupled Lorenz equations due to skew product dynamics.
• Internal symmetries in chaotic saddles prevent clean connections, though anomalous connections are rare.
• Genuine phase-resetting and stable periodic orbits of long periods were found near resonance in the magnetoconvection model, with complex, pseudo-riddled basins of attraction.

Conclusions:

• Symmetric systems can exhibit robust cycling chaos, with dynamics influenced by internal symmetries and phase-resetting.
• The structure of connections within chaotic saddles and the resulting basins of attraction can be highly complex.
• These findings provide deeper insights into the behavior of nonergodic attractors in various scientific models.