Recently, several different dynamical phases occuring in various parameter ranges for the 1-d CGLE have been identified. In particular, two distinct chaotic regimes have been seen: the phase-turbulent (PT) regime and the defect-turbulent (DT) phase. Moreover, a non chaotic phase and an intermittent one have been also observed. We plan to perform extensive studies in order to characterize more carefully these regimes. The analysis performed by Chate' et al. has been limited to a phenomenological observation of these phases, instead we plan to use indicators appropriate for extended chaotic systems, like Lyapunov exponents or propagation velocities of disturbances, in order to characterize more quantitatively such phases.
We have already taken in exam the PT regime, and we have found a new class of stable solutions for the CGLE [27,31]. In the next future, we plan to study the stability properties of these solutions and to investigate if solutions of the same class exist also for the two dimensional CGLE. The relevance of such solutions is due to the fact that only few other exact stable solutions are known for the CGLE. Moreover, we expect that their stability analysis should clarify also the origin of the transition from the PT phase to the DT regime.
We are presently carring out a systematic study of traveling waves with modulated amplitude (MAW) in the 1D-CGLE by means of the continuation software AUTO97 and a numerical stability code. It is found that modulated amplitude waves emerge from the well known plane waves with constant amplitude via Hopf bifurcation in the appropriate traveling wave equations (3 ordinary differential equations). To make the continuation problem treatable by AUTO94, we have to put additional constraint namely the system length L, the winding number \nu and the number of periods n of the modulated amplitude equations. It is found that all branches end in a turning point if one increases the parameter of the CGLE towards the defect turbulent regime. The location of the turning point or saddle-node bifurcation (SN) depends on the values of \nu, L and n. In addition, we study the stability of these MAWs and found that can be destabilized both by period doublings, Hopf bifurcation as well as a characteristic short wavelength instability for small enough \nu. Our results explain many numerical observations in the regime of wound up phase turbulence and give hints concerning the mechanism responsible for the appearance of defect turbulence in the 1D CGLE [46,49].
Recently. MAW's have been observed in various experiments on hydrothermal waves: namely, in annular convection channels (Janiaud et al. (1992) and Mukolobwiez et al. (1998)) and in linear convection cells (Garnier et al. (2001)). Moreover, recent experimental observation of MAW's have been reported for a Taylor-Dean system (Bot et al. (2000)) and also in a 2d oscillatory variant of the Belousov-Zhabotinsky reaction in connection with the occurence of the so-called ``super-spirals'' (Ouyang et al. (2000)). In collaboration with the Ouyang's group we are presently investigating the origin of the super-spirals breakup, leading to defect turbulence, in a 2d variant of the CGLE. Due to the relevance of the MAWS's for the experimentalists, we are planning to extend our investigation on the stability and existence of MAW's also to more realistic systems, like the Oregonator or the FitzHugh-Nagumo equation,
The cited references refer to the publication list