Conservative coupled oscillator systems with nonlinear interaction force are prototypes of complex chaotic dynamics. Few rigorous results on the dynamics are available when the number N of oscillators is large (N > 2), then one must resort to approximation methods and to numerical experiments. Recently, systems with both on-site and nearest-neighbour nonlinear force have been studied which are susceptible of applications to physical and biological systems (nonlinear effects in solids at small temperature and relaxation to equipartition, diffusion of adatoms on surfaces, denaturation of DNA strains, high conductivity in hydrogen bonded systems,...). Our analysis has been limited to the so-called Fermi-Pasta-Ulam beta-model.
It has been shown that the phase-space of the Fermi-Pasta-Ulam nonlinear lattice has invariant sub-manifolds, which are found by composing linear Fourier modes (Poggi and Ruffo). Some one and two-dimensional manifolds contain exact periodic and quasi-periodic solutions, whose explicit analytical form can be derived. Linear stability analysis of the highest zone-boundary (N/2) mode has been fully completed: this solution is stable below a critical energy which vanishes in the thermodynamic (N to infinity) limit, and becomes unstable through a modulational Benjamin-Feir type instability. However, the exponential instability rate also vanishes in this limit, and moreover the modulational instability saturates to an exponential Fourier spectrum (localization in Fourier space) [33,37]. The resulting solution deserves to be studied, expecially in connection with localized mode solutions. A lower dimensional study restricted to the (N/4,N/2) subspace has in fact shown that the instability leads to the formation of a bounded chaotic layer; thus one might be here in the presence of a localized "chaotic" mode. A complete study of the Lyapunov spectrum associated to this "new" localized mode is planned in order to clarify the existence of these states and to characterize their main features.
Preliminary results confirm the existence of such a chaotic localized mode and a detailed Lyapunov analysis of such state is actually in progress. Moreover, this self-emerging localized state appears to be similar in several respect to exact stable non-linear oscillating solutions of the beta-FPU model: the so called "breathers". But breathers are non-chaotic solutions, instead the localized mode we observed is claerly chaotic [36]. Finally, we will investigate the role played by these "chaotic breathers" for the thermalization of the $\beta$-FPU chain.
The cited references refer to the publication list