Convective Lyapunov Spectra

We generalize the concept of the convective (or velocity-dependent) Lyapunov exponent from the maximum rate Lambda (v) to an entire spectrum Lambda (v, n). Our results are derived by following two distinct computational protocols: (i) Legendre transform within the chronotopic approach (Lepri et al 1996 J. Stat. Phys. 82 1429); (ii) by letting evolve an ensemble of initially localized perturbations. The two approaches turn out to be mutually consistent. Moreover, we find the existence of a phase transition: above a critical value n = n(c) of the integrated density of exponents, the zero-velocity convective exponent is strictly smaller than the corresponding Lyapunov exponent. This phenomenon is traced back to a change of concavity of the so-called temporal Lyapunov spectrum for n > n(c), which, therefore, turns out to be a dynamically invariant quantity.