Lyapunov Spectra in Spatially Extended Systems

Abstract

This series of lectures reviews some aspects of the theory of Lyapunov characteristic exponents (LCE) and its application to spatially extended systems, namely chains of coupled oscillators and coupled map lattices (CML). After introducing the main definitions and theorems and presenting the most widely used computational algorithm (due to Benettin et al.) in Section 1, I give an account, in Section 2, of the existence of a Lyapunov spectral density in the thermodynamic limit (first conjectured by Ruelle for the Navier-Stokes equation and then numerically evidentiated for the Fermi-Pasta-Ulam oscillator chain by Politi et al.). Although not intrinsically defined, Lyapunov eigenvectors are an important tool for studying spatial development of chaos: their localization property (in the sense of Anderson theory) is presented in Section 3, together with a recent generalization known as chronotopic analysis, due to Lepri et al.. The random matrix approximation allows to obtain reasonable estimates of the LCE when chaos is well developed and correlations are weak: some examples of analytical calculation of scaling laws in the perturbation parameter for random Verlet matrices are discussed in Section 4, along the path opened by Parisi et al.. Finally, in Section 5 I discuss a recently discovered phenomenon known as coupling sensitivity (Daido), which is the sharp increase (as 1/ln(e)) of the maximal LCE when the coupling diffusive parameter e is switched on in CML models; the analytical treatment is quite appealing, making reference to an interesting probabilistic model, the random energy model of Derrida.