Dynamics of disordered regimes in spatially extended systems: The complex Ginzburg-Landau equation

Montagne, Raul (Directors E. Hernandez-Garcia and M. San Miguel)
PhD Thesis (1996)

This thesis is devoted to study different aspects of Spatio{temporal complex dynamics.
Spatio-temporal structures are universally present in nature. These structures commonly
referred as patterns, can be formed via bifurcations, often from a uniform reference state.
An interesting aspect of patterns is that many of them have a universal character and can
evolve, under subsequent instabilities, to Spatio{temporal chaos (STC) (M. Cross and P.
Hohenberg, Science 263, 1569 (1994),J. Gollub, Nature 367, 318 (1994)). The universal
character of this phenomenum allows a description through general model equations. A
paradigmatic model is the Complex Ginzburg{Landau Equation (CGLE). The CGLE is
the amplitude equation describing universal features of the dynamics of extended systems
near a Hopf bifurcation (M. Cross and P. Hohenberg, Rev. Mod. Phys. 65, 851 (1993),
W. van Saarloos and P. Hohenberg, Physica D 56, 303 (1992)). The CGLE displays a rich
variety of complex spatio-temporal dynamical regimes that have been recently classified,
for the one (and also two) dimensional case, in a phase diagram in the parameter space
(B. Shraiman et al., Physica D 57, 241 (1992), H. Chate, Nonlinearity 7, 185 (1994)).
In this general context this thesis contributes to current studies of the d = 1 CGLE.
The main issues addressed include the feasibility of a description of states of the CGLE
in terms of a nonequilibrium potential, the characterization of a phase transition between
different states of STC, the control and stabilization of ordered states within a STC phase
and the synchronization of STC in extended systems.
First, in this thesis we study numerically in the one-dimensional case the validity of
the functional calculated by Graham and coworkers (R. Graham and T. Tel, Phys. Rev.
A 42, 4661 (1990), O. Descalzi and R. Graham, Z. Phys. B 93, 509 (1994)) 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
Second, this thesis addresses a statistical characterization of states with nonzero wind-
ing number in the Phase Turbulence (PT) regime of the one-dimensional Complex Ginzburg-
Landau equation. We find that states with winding number larger than a critical one are
unstable, in the sense that they decay to states with smaller winding number. The tran-
sition from Phase to Defect Turbulence is interpreted as an ergodicity breaking transition
which occurs when the range of stable winding numbers vanishes. Asymptotically stable
states (wound states) which are not spatio-temporally chaotic are described within the
PT regime of nonzero winding number. Besides the complete numerical characterization
of these wound states some analytical insight is brought to such states by explaining them
in terms of solutions of a phase equation.
We have also considered the role of nonlinear complex diffusion terms in the stability
of periodic solutions in the regime of STC. This is discussed in the context of control of
STC. The stabilization of unstable plane waves in the Complex Ginzburg Landau equation
in weakly chaotic regimes such as phase turbulence and spatio-temporal intermittency or
in strongly chaotic ones like defect turbulence is demonstrated.
STC behaviour has been also considered for two coupled CGLE. It is shown that the
synchronization of STC extended systems is possible in the context of Coupled Complex
Ginzburg-Landau equations (CCGLE). A regime of coupled spatiotemporal intermittency
is identified and described in terms of distribution functions and information measures.
Additional properties of coupled CGLE are also described as the disappearance of STI
when crossing from weak to strong coupling.

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