We are studying, from several years, spatio-temporal instabilities occurring in spatially extended systems, in particular we have usually considered models of coupled map lattices. In order to characterize such instabilities, spatial and temporal Lyapunov exponents have been introduced and relationships between these new classes of chaotic indicators have been identified. In order to generalize such results to systems with continuous time and/or continuous space, we are actually trying to extend our results to coupled Roessler oscillators and to the 1-d complex Ginzburg-Landau equation. Moreover, we have preliminary results on the possibility to introduce a unique Lyapunov indicator to characterize instabilities spreading along any world line within a generic spatio-temporal pattern. Such indicator can be obtained from the knowledge of the spatial and temporal Lyapunov spectra and it is related also to the comoving Lyapunov exponents introduced by Deissler and Kaneko [34].
Within this framework, we have also found first indications that in spatially extended systems the number of nodes of the "spatial" Lyapunov vectors can be related to the Lyapunov "temporal" spectra. This allows us to introduce a new method to estimate Lyapunov spectra in systems with spatial degree of freedom. We have already validated our approach for systems of coupled map lattices [38]. Further studies for systems with continuous time are planned.
More recently we have reconsidered the problem of finite amplitude instabilities in the framework of the Finite Size Lyapunov Exponents, introduced by Aurell, Boffetta, Crisanti, Paladin, Vulpiani in 1996. We have extended their definition to comoving reference frames and shown that with their employment a marginal stability criterion able to capture linear as well as nonlinear propagation phenomena can be introduced. Error propagation in spatio-temporal chaotic systems is then strictly related to front propagation in reaction diffusion systems [48].
The cited references refer to the publication list