Peter Takac
Dynamics on the Attractor for the Complex Ginzburg-Landau Equation
The paper is published: Rostocker Mathematische Kolloquium, Rostock. Math. Kolloq. 49, 163-184 (1995)

MSC:

76E30 Nonlinear effects

58F12 Structure of attractors (and repellors)

Abstract: We present a numerical study of
the large-time asymptotic behavior of solutions to
the one-dimensional complex {\sc Ginzburg-Landau} equation with
periodic boundary conditions.
Our parameters belong to the {\sc Benjamin-Feir} unstable region.
Our solutions start near a pure-mode rotating wave
that is stable under sideband perturbations for
the {\sc Reynolds} number $R$
ranging over an interval $(R_{sub}, R_{sup})$.
We find sub- and super-critical bifurcations from
this stable rotating wave to a stable 2-torus as
the parameter $R$ is decreased or increased past
the critical value $R_{sub}$ or $R_{sup}$.
As $R > R_{sup}$ further increases,
we observe a variety of dynamical phenomena, such as
a local attractor consisting of
three unstable manifolds of periodic orbits or 2-tori
cyclically connected by manifolds of connection orbits.
We compare our numerical simulations to both
rigorous mathematical results and
experimental observations for binary fluid mixtures.

Keywords: Periodic orbit, 2- and 3-tori, stability, local attractor, psedo-spectral method, Fourier modes