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Exploring the role of diffusive coupling in spatiotemporal chaos.
1Department of Mechanical Engineering, Virginia Tech, Blacksburg, Virginia 24061, USA.
Chaos (Woodbury, N.Y.)
|October 7, 2024
Summary
We studied chaotic dynamics in coupled maps using covariant Lyapunov vectors (CLVs). Increasing diffusion strength initially leads to periodic behavior before returning to chaos, with CLVs revealing spatial structure evolution.
Area of Science:
- Nonlinear dynamics
- Statistical physics
- Complex systems
Background:
- Chaotic dynamics in extended systems are fundamental to understanding complex phenomena.
- Coupled map lattices (CMLs) provide a simplified yet powerful model for studying spat-temporal chaos.
- Covariant Lyapunov vectors (CLVs) offer insights into the geometric and dynamical properties of chaotic systems.
Purpose of the Study:
- To investigate the impact of diffusive coupling strength on the chaotic dynamics of a one-dimensional lattice of coupled maps.
- To quantify the evolution of spatial structures and Lyapunov spectra as a function of diffusion.
- To analyze the spatial features of CLVs and their relation to the coupling operator.
Main Methods:
- Utilized a one-dimensional lattice of diffusively coupled quadratic maps.
- Employed covariant Lyapunov vectors (CLVs) to analyze chaotic dynamics.
- Calculated Lyapunov exponents and fractal dimensions analytically using eigenvalues of the coupling operator.
- Quantified spatial features of CLVs and compared them with eigenvectors of the coupling operator.
Main Results:
- A rapid decrease in the leading Lyapunov exponent from positive to zero with increasing diffusion strength, creating a window of periodic dynamics.
- Chaotic dynamics beyond the periodic window showed minimal change in the leading Lyapunov exponent, except at the highest diffusion strengths.
- Analytical descriptions of the Lyapunov spectrum and fractal dimension were derived as functions of diffusion strength.
- CLVs were found to be composed of physical modes, with the leading CLV exhibiting decreasing localization as coupling strength increased.
- Violation of Oseledets splitting dominance indicated increased entanglement between neighboring CLVs with stronger diffusion.
Conclusions:
- Diffusive coupling significantly alters the chaotic dynamics of CMLs, introducing periodic windows and influencing spatial structure formation.
- CLVs provide a robust tool for characterizing the spat-temporal behavior and mode localization in coupled chaotic systems.
- The observed entanglement of CLVs suggests a breakdown of independent error growth in strongly coupled chaotic lattices.
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