Hydrodynamic Lyapunov modes and effective degrees of freedom of extended systems

This work reviews the authors' contributions to the Lyapunov analysis of extended dynamical systems. Hydrodynamic Lyapunov modes (HLMs), the special Lyapunov vectors (LVs) associated with near-zero Lyapunov exponents exhibiting long wavelength structures and slow oscillations, have recently been observed in many extended systems with continuous symmetry, such as hard sphere systems, dynamic XY models, Lennard-Jones fluids, coupled map lattices and partial differential equations. They are of potential importance for the connection between nonlinear dynamics and its coarse-grained description. In the first part of this paper, we review our recent results on Lyapunov modes in an extended system, which includes the universality of HLMs in Hamiltonian and dissipative systems, the condition for the appearance of significant or 'vague' modes and the appearance of branch splitting in the Lyapunov spectra of diatomic Hamiltonian systems. In particular, most results on HLMs obtained via orthogonal LVs have been checked by using covariant LVs. An emerging picture from these studies is that HLMs in Hamiltonian systems can be viewed as the generalization of normal modes of harmonic systems to nonlinear, chaotic systems. The second part is devoted to the hyperbolicity and the effective degrees of freedom of dissipative partial differential equations. By using covariant LVs, a hyperbolic separation between two sets of LVs was found in these infinite-dimensional systems. The finite set of mutually entangled LVs, named 'physical modes', was conjectured to span a local linear approximation of the inertial manifold, a finite-dimensional smooth manifold containing the physically relevant dynamics of the original PDE system. Going beyond the Lyapunov analysis, a projection method was proposed to probe the geometric structures of the inertial manifold, which enables us to provide a direct relation between the physical modes and the inertial manifold of dissipative PDEs.