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Updated: Jun 21, 2026
04:35
Preparation of Free-Surface Hyperbolic Water Vortices
Published on: July 28, 2023
Violation of hyperbolicity in a diffusive medium with local hyperbolic attractor.
Pavel V Kuptsov1, Sergey P Kuznetsov
1Department of Informatics, Saratov State Law Academy, Volskaya 1, Saratov 410056, Russia. p.kupstov@rambler.ru
Summary
This study explores how spatial coupling affects chaotic dynamics in a one-dimensional medium. Hyperbolicity is lost as the medium length increases, leading to extensive spatiotemporal chaos and altered Lyapunov exponents.
Area of Science:
- Nonlinear Dynamics
- Complex Systems
- Chaos Theory
Background:
- Hyperbolic chaotic dynamics are typically observed in low-dimensional systems.
- Understanding the transition to spatiotemporal chaos in extended systems is crucial for complex system analysis.
Purpose of the Study:
- To investigate the impact of spatial coupling on hyperbolic chaotic dynamics.
- To analyze the loss of hyperbolicity and emergence of spatiotemporal chaos in a one-dimensional extended system.
Main Methods:
- Construction of a one-dimensional medium from coupled nonautonomous amplitude equations.
- Analysis of Lyapunov exponents and Kaplan-Yorke dimension as a function of system length.
- Identification of bifurcations leading to the violation of hyperbolicity.
Main Results:
- Synchronous oscillations and hyperbolicity are observed for small system lengths, characterized by a single positive Lyapunov exponent.
- Hyperbolicity is maintained with spatial inhomogeneity until the third Lyapunov exponent becomes positive, indicating a loss of hyperbolicity.
- Further increases in length lead to extensive spatiotemporal chaos, with linear growth in Kaplan-Yorke dimension and the number of positive Lyapunov exponents.
Conclusions:
- Spatial coupling can lead to the loss of hyperbolicity and the emergence of extensive spatiotemporal chaos in extended systems.
- The transition is marked by changes in the Lyapunov spectrum and the geometry of tangent subspaces.
- The findings provide insights into the behavior of complex chaotic systems with increasing spatial extent.
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