The transition to phase and amplitude turbulent states in the one-dimensional complex Ginzburg-Landau equation is investigated by a numerical simulation. In order to describe existing attractors quantitatively, the largest Lyapunov exponent is estimated. The largest Lyapunov exponent is positive but small for the phase turbulence and it is much larger for the amplitude turbulence. Moreover, it seems independent on the system size.
The driven damped nonlinear Schrödinger equation is solved numerically and different attractors - periodic and chaotic ones - are obtained for different ranges of the forcing field amplitude. It is shown that these attractors are in both cases spatially nonuniform in time. Spatial inhomogeneity of the chaotic attractor is investigated by the estimation of the correlation dimension at different space points.
Three spatially extended one-dimensional dynamical systems are examined by numerical simulations and the time-delay technique is applied to find their dynamics at different positions in space. On the basis of this technique the correlation dimension is calculated from time series obtained at different positions. It is found that the value of the correlation dimension may vary from one position to another, reflecting the spatial inhomogeneity of the system.
JavaScript is turned off in your web browser. Turn it on to take full advantage of this site, then refresh the page.