Characterization of Faraday patterns and spatiotemporal chaos in parametrically driven dissipative systems

In this work, we have studied numerically the dynamics of the parametrically driven damped nonlinear Schr & ouml;dinger equation (PDDNLS). The PDDNLS is a universal model to describe parametrically driven systems. In particular, we have characterized stationary Faraday patterns, periodic, quasi-periodic, and spatiotemporal chaos as a function of the amplitude and the frequency of the parametric driving force. We have computed the Lyapunov spectra, the Fourier spectra, the amplitude norm, and the Kaplan-Yorke dimensions as valuable indicators for the identification of several dynamical regimes. We show that in the Faraday regime, close to the bifurcation of the trivial state, the pattern amplitude scales with power one-fourth (1/4) of the bifurcation parameter. Furthermore, we have found that the pattern wavelength decreases when the detuning parameter increases. In the case of the high dimensional spatiotemporal chaotic states, we have found that the Kaplan- Yorke dimension increases linearly with the length of the system, showing its extensive character in this dynamical regime. We have also found a transition from low to high dimensional chaos when the forcing amplitude is increased.