Noise-driven topological changes in chaotic dynamics

Area of Science:

• Nonlinear dynamics and chaos theory
• Topological data analysis
• Stochastic processes

Background:

• Deterministic chaotic systems, like the Lorenz attractor, possess a fixed topological structure.
• Stochastic perturbations can qualitatively alter the behavior of chaotic systems.
• Understanding the evolution of chaotic attractors under noise is crucial for nonlinear dynamics.

Purpose of the Study:

• To compare the topological structure of the deterministic Lorenz attractor with its stochastically perturbed version.
• To investigate the temporal evolution of a noise-driven chaotic attractor.
• To apply and extend branched manifold analysis to nonlinear noise-driven systems.

Main Methods:

• Utilizing branched manifold analysis through homologies to examine topological structures.
• Analyzing snapshots of the Lorenz model's random attractor (LORA) at different time instants.
• Extending a technique for deterministic chaotic flows to nonlinear noise-driven systems.

Main Results:

• The stochastically perturbed Lorenz attractor (LORA) evolves in time, unlike its deterministic counterpart.
• Branched manifold analysis reveals sharp transitions in LORA's evolution.
• These transitions manifest as topological tipping points in the system's behavior.

Conclusions:

• Noise-driven chaotic systems can exhibit dynamic topological changes.
• Topological tipping points represent significant qualitative shifts in system behavior.
• Branched manifold analysis is a powerful tool for characterizing evolving chaotic dynamics under noise.