Characterizing dynamics with covariant Lyapunov vectors

We introduce a general, innovative approach to determine, in both continuous- and discrete-time dynamical systems, an intrinsic set of directions at each point of phase space that are covariant with the dynamics as well as invariant under time reversal. The knowledge of these covariant Lyapunov vectors (CLV) allows for an accurate characterization of the underlying dynamics. In particular, it is possible to quantify the degree of hyperbolicity or to compute the complete spectrum of Lyapunov exponents via ensemble averages over the attractor invariant measure rather than through time averages.
Moreover, regarding the search for "collective modes", we show that the statistical properties of CLV differ from those of the vectors obtained from the standard orthogonalization procedure introduced by Benettin et al. In particular, we show evidence in both dissipative and Hamiltonian spatially extended systems that CLV are strictly localized almost everywhere in the asymptotic spectrum, thus providing a meaningful hierarchical decomposition of spatiotemporal chaos.