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Finding the Effective Dynamics to Make Rare Events Typical in Chaotic Maps.

Ricardo Gutiérrez1, Adrián Canella-Ortiz2, Carlos Pérez-Espigares2,3

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Area of Science:

  • Complex Systems
  • Nonlinear Dynamics
  • Statistical Physics

Background:

  • Chaotic maps exhibit sensitive dependence on initial conditions, making rare events and atypical trajectories difficult to predict.
  • Identifying specific initial conditions that lead to stability islands or unusual behavior in chaotic systems is a significant challenge.

Purpose of the Study:

  • To propose a novel framework for finding atypical trajectories in chaotic maps.
  • To develop a method for identifying effective topologically conjugate maps that mirror atypical dynamics.

Main Methods:

  • Constructing a topologically conjugate map whose typical trajectories correspond to the original map's atypical ones.
  • Applying the generalized Doob transform, analogous to stochastic dynamics in Markov chains and quantum systems.

Main Results:

  • Demonstrated the framework's effectiveness in counterbalancing fixed point and periodic orbit instability.
  • Characterized a dynamical phase transition using the finite-time Lyapunov exponent.
  • Successfully controlled rare fluctuations in dynamical observables.

Conclusions:

  • The proposed framework provides a method to characterize and control rare fluctuations in chaotic maps.
  • This work extends the application of large-deviation formalism to chaotic systems, aligning with stochastic processes.