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Updated: Jul 8, 2025
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Mapping Cortical Dynamics Using Simultaneous MEG/EEG and Anatomically-constrained Minimum-norm Estimates: an Auditory Attention Example
Published on: October 24, 2012
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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
1Grupo Interdisciplinar de Sistemas Complejos (GISC), Departamento de Matemáticas, Universidad Carlos III de Madrid, 28911 Leganés, Madrid, Spain.
Physical Review Letters
|December 15, 2023
Summary
Researchers developed a new method to find rare, atypical trajectories in chaotic maps by creating a conjugate map. This approach helps control system behavior and understand dynamical phase transitions.
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.
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