Lyapunov Spectra in Spatially Extended Systems

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