Kantorovich-Rubinstein-Wasserstein distance between overlapping attractor and repeller

Area of Science:

• Dynamical Systems and Chaos Theory
• Nonlinear Dynamics
• Statistical Mechanics

Background:

• Overlapping attractors and repellers are key features in complex dynamical systems.
• Understanding their behavior is crucial for predicting system evolution.
• Controllable dissipation offers a novel approach to study these phenomena.

Purpose of the Study:

• To investigate the emergence and characteristics of overlapping attractors and repellers.
• To analyze the influence of controllable dissipation on these dynamical structures.
• To quantify the distinction between attractors and repellers using a robust metric.

Main Methods:

• Introduction of controllable dissipation into prototypic chaotic models (Anosov cat map, Chirikov standard map, ABC-flow).
• Analysis of ergodic non-chaotic behavior using the skew-shift map.
• Application of the Kantorovich-Rubinstein-Wasserstein distance for characterization.

Main Results:

• Demonstration of overlapping attractor-repeller pairs in various dynamical systems.
• Quantification of the difference between attractors and repellers as a function of dissipation.
• Systematic analysis across chaotic and non-chaotic models.

Conclusions:

• Controllable dissipation effectively generates overlapping attractor-repeller scenarios.
• The Kantorovich-Rubinstein-Wasserstein distance provides a sensitive measure of attractor-repeller separation.
• Findings offer insights into the fundamental dynamics of complex systems.