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Parametric evolution of unstable dimension variability in coupled piecewise-linear chaotic maps
R F Pereira1, R L Viana, S R Lopes
1Departamento de Física, Universidade Estadual de Ponta Grossa, 84030-900 Ponta Grossa, Paraná, Brazil.
Summary
Analytical solutions for chaotic systems with unstable dimension variability are crucial. This study provides exact results for the onset and intensity of this phenomenon in coupled bungalow maps during synchronization.
Area of Science:
- Dynamical Systems and Chaos Theory
- Nonlinear Dynamics
- Mathematical Physics
Background:
- Numerical solutions for chaotic systems are limited by unstable dimension variability.
- Analytical results are needed for long-term predictions in chaotic systems.
- Bungalow maps offer intervals of linearity suitable for analytical treatment.
Purpose of the Study:
- To investigate the parametric evolution of unstable dimension variability in coupled bungalow maps.
- To derive analytical results for the onset and intensity of unstable dimension variability.
- To explore the role of synchronization and Markov partitions in this phenomenon.
Main Methods:
- Analysis of two coupled bungalow maps.
- Utilizing intervals of linearity to define Markov partitions.
- Investigating synchronization conditions to recover partitions in the coupled system.
- Calculating exact results for the onset of unstable dimension variability and contrast measure.
Main Results:
- Markov partitions are recovered for the coupled bungalow map system under synchronization.
- Exact results for the onset of unstable dimension variability were obtained.
- The contrast measure, quantifying phenomenon intensity, was derived.
- Results link phenomenon intensity to the stability of periodic orbits in the synchronization subspace.
Conclusions:
- Analytical solutions for unstable dimension variability in coupled chaotic systems are achievable.
- Synchronization plays a key role in simplifying the dynamics of coupled bungalow maps.
- The study provides a framework for understanding and quantifying chaotic behavior analytically.
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