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Published on: November 18, 2020
Comparison between covariant and orthogonal Lyapunov vectors
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
|January 15, 2011
Summary
Covariant Lyapunov vectors (CLVs) and orthogonal Lyapunov vectors (OLVs) both detect hydrodynamic Lyapunov modes (HLMs) in chaotic systems. Their significance varies between Hamiltonian and dissipative systems, with CLVs reducing long-wavelength structures in general dissipative systems.
Area of Science:
- Dynamical systems theory
- Nonlinear dynamics
- Statistical mechanics
Background:
- Covariant Lyapunov vectors (CLVs) and orthogonal Lyapunov vectors (OLVs) are key tools for analyzing linear stability in chaotic systems.
- Hydrodynamic Lyapunov modes (HLMs) are important features influencing system dynamics and stability.
Purpose of the Study:
- To compare CLVs and OLVs, focusing on their impact on HLMs in both Hamiltonian and dissipative systems.
- To investigate the universality classes of HLMs when using CLVs.
- To analyze the changing significance of HLMs with the replacement of OLVs by CLVs.
Main Methods:
- Numerical simulations of Hamiltonian and dissipative systems.
- Analysis of Lyapunov vectors (CLVs and OLVs) and their relation to Lyapunov exponents.
- Comparison of HLM detection and significance across different system types and vector sets.
Main Results:
- HLMs detected by OLVs are also found using CLVs in both Hamiltonian and dissipative systems.
- The two universality classes (linear for Hamiltonian, quadratic for dissipative) are preserved for CLVs.
- CLVs reduce the significance of long-wavelength HLMs in general dissipative systems but not in Hamiltonian systems.
- Symmetry relations differ between CLVs and OLVs in Hamiltonian systems, with CLVs being statistically indistinguishable due to microreversibility.
Conclusions:
- CLVs are a viable alternative to OLVs for detecting HLMs across various chaotic systems.
- The choice between CLVs and OLVs impacts the analysis of HLMs, particularly in dissipative systems with nonhyperbolic dynamics.
- Understanding these vector properties is crucial for characterizing the stability and dynamics of complex systems.
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