|
Recent progress in characterizing the dynamics of spatially extended dynamical systems by covariant Lyapunov vectors is reviewed. A splitting of tangent space into at least two distinct subspaces, where in one a partial domination of the Oseledec splitting is prevailing, and another, where Lyapunov vectors are dynamically entangled, is found in Hamiltonian and dissipative systems with translational symmetry. In Hamiltonian systems this leads to well observable Hydrodynamic Lyapunov Modes [1], and in dissipative systems the entangled modes define a physically relevant finite-dimensional subspace [2], which is supposed to characterize inertial manifolds of such systems.
These findings are exemplified and discussed for paradigmatic systems such as Coupled Map Lattices and standard partial differential equations exhibiting spatio-temporal chaos.
[1] H.-l. Yang and G. Radons; When can one observe good hydrodynamic Lyapunov modes? Phys. Rev. Lett. 100, 024101 (2008) [2] H.-l. Yang, K. A. Takeuchi, F. Ginelli, H. Chaté, and G. Radons; Hyperbolicity and the Effective Dimension of Spatially Extended Dissipative Systems, Phys. Rev. Lett. 102, 074102 (2009). |
|