Numerical Study of a Lyapunov Functional for the Complex Ginzburg-Landau Equation
Montagne, R.; Hernández-García, E.; San Miguel, M.
Physica D 96, 47-65 (1996)
We numerically study in the one-dimensional case the validity of the functional
calculated by Graham and coworkers as a Lyapunov potential for the Complex
Ginzburg-Landau equation. In non-chaotic regions of parameter space the
functional decreases monotonically in time towards the plane wave attractors, as
expected for a Lyapunov functional, provided that no phase singularities are
encountered. In the phase turbulence region the potential relaxes towards a value
characteristic of the phase turbulent attractor, and the dynamics there
approximately preserves a constant value. There are however very small but
systematic deviations from the theoretical predictions, that increase when going
deeper in the phase turbulence region. In more disordered chaotic regimes
characterized by the presence of phase singularities the functional is ill-defined
and then not a correct Lyapunov potential.