Numerical Study of a Lyapunov Functional for the Complex Ginzburg-Landau Equation

TL;DR

This study numerically investigates the validity of a Lyapunov functional for the one-dimensional Complex Ginzburg-Landau equation, confirming its decreasing behavior in non-chaotic regimes and identifying limitations in turbulent and chaotic regimes.

Contribution

It provides the first numerical validation of Graham's Lyapunov functional for the complex Ginzburg-Landau equation across different dynamical regimes.

Findings

Functional decreases monotonically in non-chaotic regions.

Potential relaxes to a characteristic value in phase turbulence.

Functional is ill-defined in highly chaotic regimes with phase singularities.

Abstract

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.