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Characterizing dynamics with covariant Lyapunov vectors.
1Service de Physique de l'Etat Condensé, CEA-Saclay, 91191 Gif-sur-Yvette, France.
Physical Review Letters
|October 13, 2007
Summary
A new method determines covariant Lyapunov vectors for dynamical systems, quantifying system hyperbolicity. These vectors reveal unique spatial properties in extended systems, differing from standard Lyapunov exponent calculations.
Area of Science:
- Dynamical Systems Theory
- Nonlinear Dynamics
- Chaos Theory
Background:
- Lyapunov exponents quantify the rate of separation of infinitesimally close trajectories in dynamical systems.
- Standard methods often rely on orthonormalized bases, which may obscure intrinsic system properties.
- Understanding system hyperbolicity is crucial for characterizing chaotic behavior.
Purpose of the Study:
- Introduce a general method for calculating covariant Lyapunov vectors.
- Quantify the hyperbolicity of discrete- and continuous-time dynamical systems.
- Investigate the properties of covariant Lyapunov vectors in spatially extended systems.
Main Methods:
- Development of a general algorithm to compute covariant Lyapunov vectors.
- Analysis of the transversality of these vectors to quantify hyperbolicity.
- Examination of localization properties and spatial Fourier spectra of covariant Lyapunov vectors.
Main Results:
- The introduced method successfully determines covariant Lyapunov vectors for various dynamical systems.
- Quantification of hyperbolicity is achieved through the transversality of these intrinsic vectors.
- Covariant Lyapunov vectors exhibit distinct localization and spectral properties in spatially extended systems compared to standard methods.
Conclusions:
- The covariant Lyapunov vector approach offers a more comprehensive understanding of dynamical system properties.
- This method provides deeper insights into hyperbolicity and the behavior of extended systems.
- The findings suggest a potential refinement of existing analyses in nonlinear dynamics and chaos theory.
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