Characterization of the transition from defect- to phase-turbulence

For the complex Ginzburg-Landau equation on a large periodic interval, we show that the transition from defect- to phase-turbulence is more accurately described as a smooth crossover rather than as a sharp continuous transition. We obtain this conclusion by using a powerful parallel computer to calculate various order parameters, especially the density of space-time defects, the Lyapunov dimension density, and the correlation lengths of the field phase and amplitude. Remarkably, the correlation length of the field amplitude is, within a constant factor, equal to the length scale defined by the dimension density. This suggests that a correlation measurement may suffice to estimate the fractal dimension of some large homogeneous chaotic systems.