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How to test for partially predictable chaos.

Hendrik Wernecke1, Bulcsú Sándor1, Claudius Gros2

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This study introduces a new 0-1 chaos indicator to reliably detect chaotic systems, even those with partial predictability. The method analyzes trajectory distances to distinguish chaos from laminar flow.

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Area of Science:

  • Nonlinear dynamics
  • Chaos theory
  • Complex systems analysis

Background:

  • Chaotic systems exhibit decorrelation of trajectories, leading to unpredictability.
  • Partially predictable chaos, characterized by fractal attractors, poses challenges for standard chaos detection methods.
  • Inter-orbital correlations are commonly used but yield ambiguous results for partially predictable chaos.

Purpose of the Study:

  • To develop a robust method for discriminating chaotic from laminar flow.
  • To accurately identify partially predictable chaos, which is difficult with existing techniques.
  • To introduce a novel 0-1 indicator for chaos detection based on trajectory scaling.

Main Methods:

  • Introducing a novel 0-1 indicator for chaos based on the cross-distance scaling of initially close trajectories.
  • Analyzing the finite time cross-correlation of trajectory pairs.
  • Utilizing properties of trajectory pairs to identify different flow regimes.

Main Results:

  • The novel 0-1 indicator robustly discriminates chaos, including partially predictable chaos, from laminar flow.
  • The method successfully identifies laminar flow, strong chaos, and partially predictable chaos.
  • The analysis relies solely on the properties of pairs of trajectories.

Conclusions:

  • The new 0-1 chaos indicator provides a reliable tool for identifying various types of chaotic behavior.
  • This method overcomes limitations of standard inter-orbital correlation tests, especially for partially predictable chaos.
  • The approach offers a clear distinction between chaotic dynamics and laminar flow based on trajectory behavior.