Main Topics:
This subject concerns the study of chaotic behaviours arising in spatially extended systems. Examples of spatio-temporal chaos are given by Rayleigh-Benard convection, chemical reaction-diffusion systems, Josephson junction arrays, plasma turbulence, nonlinear optics, etc. The complete understanding of such phenomena within a comprehensive "scenario" would represent a primary scientific achievement.
The pricipal aim of these studies has been to characterize spatially extended systems exhibiting high dimensional chaos. In particular, we have examined one dimensional chains of diffusively couple map lattices (CML) and the complex Ginzburg-Landau equation (CGLE), as prototypes of spatio-temporal chaotic systems.
In our first works we have shown limits and possibilities of application to spatially extended systems of the usual tools developed to characterize low dimensional chaos (i.e. Lyapunov exponents, fractal dimensions and metric entropies). Moreover, new and effective algorithms have been developed to estimate both Lyapunov spectra and fractal dimension densities for extended systems (see Refs. [1,2,3,4,5] of the publication list ). In particular, we have identified the existing relationship between the Lyapunov spectrum for 1-d diffusively coupled maps and the spectrum of the discrete Schroedinger operator [2]. This analogy has allowed us to estimate analitically the full Lyapunov spectrum in some particular case.
For the same class of models (i.e. CML), fractal dimensions have been carefully estimated for short sub-chains of different lenghts M (spatial embedding). The value of the dimensions at finite resolution turn out to be smaller than $M$ and show a converge towards the corresponding emdedding dimension M in the limit of very high resolution. A similar behaviour has been observed considering short time series obtained at a certain spatial location along the chain (temporal embedding). The rate of convergence to the asymptotic limit (namely, to the considered embedding dimension) decreases drastically increasing the length of the sub-chain or of the time series. In the last few years several new works have been published on this subject, but our results still remain valid and fundamental in the field [1,5].
In succesive papers a technique to extract spatiotemporal periodic orbits for extended chains of 1-d and 2-d CML's has been developed. Moreover, an analytic procedure has been also developed to determine the Lyapunov spectra of periodic orbits of short spatial or temporal period embedded in an infinite chains. An improved zeta-function formalism was applied, for the first time, to evaluate multifractal spectra for extended systems. Finally, the extraction of periodic orbits both in space and in time has allowed us to analyze the spatially chaotic structure of the invariant measure of such systems (see Refs. [7,9]).
The mechanisms of propagation of perturbations have been also studied, revealing that in some model infinitesimal and finite disturbances spread along the chain with two different propagation velocities (see Ref. [17]). The two velocities are related to two different mechanisms of information production: one connected to the linear evolution of the system and the other to the nonlinear evolution. A theoretical scheme has been proposed in order to explain the origin of such linear and nonlinear mechanism and a strong analogy with propagation into unstable steady states (usually described by the Kolmogorov-Petrovsky-Piskunov equation) has been also found (see Ref. [21,22]). Preliminary results concerning propagation of disturbances in the 1-d CGLE are reported in Ref. [28]. However, for the CGLE only linear propagation mechanisms have been detected in the examined parameter range. Recently, the spreading of perturbations has been analyzed also in systems of CML with long-range power law coupling [32].
Two complementary approaches have been introduced, which allow to measure either the temporal or the spatial growth rate of given space- and time-periodic perturbations. The first method is the usual tecnique to determine the spectrum of Lyapunov exponents introduced for low dimensional chaos, once an initial perturbation with a spatial exponential profile is considered. The second one leads to the definition of spatial Lyapunov exponents, which are useful to determine the localization properties of Lyapunov vectors. These different indicators are related to each other and they can account for every possible (linear) instability arising in extended systems. An unified approach to describe the stability of spatio-temporal orbits, which treats space and time simmetrically, has been derived. In particular, it has been found that all the stability properties (spatial and temporal) of spatially extended 1-d systems can be obtained from a single function, that has been termed "Entropy Potential", since it gives directly the Kolmogorov-Sinai entropy density. Our results represent the first indication of such a common origin for both spatial and temporal Lyapunov exponents [23,29,34].
New possibilities of integration schemes for Partial Differential Equations, substituting the usual pseudo-spectral codes, have been determined. The new integration algorithms, avoiding the estimation of Fourier transforms, turn out to be faster than the pseudo-spectral ones and to maintain the same precision (in particular, algorithms to integrate the 1-d complex Ginzburg-Landau equation have been developed [28,31]). More recently such approach have been extended to 1-d and 2-d reaction diffusion models [43].
The maximal conserved phase gradient has been identified as a new order parameter to characterize the transition from phase- to defect-turbulence in the one-dimensional complex Ginzburg-Landau equation (CGLE). It has been also observed that the degree of "chaoticity" of the solutions of the CGLE decreases for increasing values of the phase gradient and finally leads to stable travelling wave solutions for the CGLE. These solutions represent a new class of non-chaotic solutions of the CGLE. The main features characterizing these stable waves can be reproduced via a modified Kuramoto-Sivashinsky equation for the phase dynamics [27,31].