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This summary is machine-generated.

This study explores a random dynamical system using a doubling map. It reveals a critical transition from chaotic to regular dynamics, marked by anomalous behavior and power-law correlations.

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Area of Science:

  • Dynamical Systems
  • Chaos Theory
  • Statistical Physics

Background:

  • The doubling map is a fundamental model for chaotic dynamics.
  • Random dynamical systems combine deterministic maps with probabilistic sampling.
  • Understanding transitions in dynamical systems is crucial for various scientific fields.

Purpose of the Study:

  • To analyze the invariant density and autocorrelation function of a random dynamical system based on the doubling map.
  • To investigate the transition from chaotic to regular dynamics as a function of sampling probability.
  • To characterize the anomalous dynamics observed at a critical sampling probability.

Main Methods:

  • Analytical calculations of invariant density and autocorrelation function.
  • Numerical simulations to verify analytical findings.
  • Variation of the sampling probability (p) for the expanding and contracting maps.

Main Results:

  • The random dynamical system exhibits a transition at a critical probability (pc) where the Lyapunov exponent is zero.
  • Anomalous long-time dynamics are observed at pc, characterized by an infinite invariant density.
  • Weak ergodicity breaking and power-law decay in correlations are identified as hallmarks of this anomalous regime.

Conclusions:

  • The studied random dynamical system displays a non-trivial transition between chaotic and regular behaviors.
  • A critical sampling probability leads to anomalous dynamics with unique statistical properties.
  • This work provides insights into the complex dynamics of random systems and their transitions.