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Hydrodynamic Lyapunov modes in coupled map lattices.
1Institute of Physics, Chemnitz University of Technology, D-09107 Chemnitz, Germany. hongliu.yang@physik-tu-chemnitz.de
Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|February 21, 2006
Summary
Hydrodynamic Lyapunov modes (HLMs) exist in coupled map lattices (CMLs), with distinct dispersion relations for standard and circle maps. Their existence depends on system properties like damping and potentials, not solely on Hamiltonian structure or conservation laws.
Area of Science:
- Nonlinear dynamics
- Statistical physics
- Chaos theory
Background:
- Coupled map lattices (CMLs) are crucial models for studying complex spatiotemporal dynamics.
- Hydrodynamic Lyapunov modes (HLMs) are characteristic features of chaotic systems, previously studied in different contexts.
- Understanding the conditions for HLMs in CMLs is essential for characterizing their stability and behavior.
Purpose of the Study:
- To investigate the existence and properties of hydrodynamic Lyapunov modes (HLMs) in coupled map lattices (CMLs).
- To determine the universality classes of HLMs in different types of CMLs.
- To elucidate the influence of system parameters such as Hamiltonian structure, conservation laws, translational invariance, and damping on HLMs.
Main Methods:
- Numerical simulations of coupled map lattices.
- Analytical derivations of dispersion relations.
- Analysis of Lyapunov exponents and Lyapunov vectors.
Main Results:
- HLMs exist in CMLs, exhibiting two universality classes: lambda ~ k for coupled standard maps and lambda ~ k^2 for coupled circle maps.
- Hamiltonian structure is not required for HLMs; conservation laws or translational invariance alone are insufficient.
- Damping does not destroy HLMs in coupled Hamiltonian maps but alters their dispersion relation to lambda ~ k^2; HLMs are destroyed in coupled circle maps under damping.
- On-site potentials eliminate HLMs.
- Zero-value Lyapunov exponents and vectors reveal distinct roles of translational invariance and conservation laws in tangent space dynamics.
Conclusions:
- The existence of HLMs in CMLs is robust and depends on specific system configurations, not just general symmetries.
- Damping and potentials significantly impact HLMs, highlighting the importance of detailed system analysis.
- HLMs are not limited to 1D CMLs, as demonstrated in a 2D system.