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Gisela D Charó1, Mickaël D Chekroun2, Denisse Sciamarella3

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Noise-driven chaotic systems exhibit evolving topological structures. This study uses branched manifold analysis to reveal topological tipping points in the Lorenz model

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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.