3,982 research outputs found
Dynamics of kinks in the Ginzburg-Landau equation: Approach to a metastable shape and collapse of embedded pairs of kinks
We consider initial data for the real Ginzburg-Landau equation having two
widely separated zeros. We require these initial conditions to be locally close
to a stationary solution (the ``kink'' solution) except for a perturbation
supported in a small interval between the two kinks. We show that such a
perturbation vanishes on a time scale much shorter than the time scale for the
motion of the kinks. The consequences of this bound, in the context of earlier
studies of the dynamics of kinks in the Ginzburg-Landau equation, [ER], are as
follows: we consider initial conditions whose restriction to a bounded
interval have several zeros, not too regularly spaced, and other zeros of
are very far from . We show that all these zeros eventually disappear
by colliding with each other. This relaxation process is very slow: it takes a
time of order exponential of the length of
The Definition and Measurement of the Topological Entropy per Unit Volume in Parabolic PDE's
We define the topological entropy per unit volume in parabolic PDE's such as
the complex Ginzburg-Landau equation, and show that it exists, and is bounded
by the upper Hausdorff dimension times the maximal expansion rate. We then give
a constructive implementation of a bound on the inertial range of such
equations. Using this bound, we are able to propose a finite sampling algorithm
which allows (in principle) to measure this entropy from experimental data.Comment: 26 pages, 1 small figur
Complexity for extended dynamical systems
We consider dynamical systems for which the spatial extension plays an
important role. For these systems, the notions of attractor, epsilon-entropy
and topological entropy per unit time and volume have been introduced
previously. In this paper we use the notion of Kolmogorov complexity to
introduce, for extended dynamical systems, a notion of complexity per unit time
and volume which plays the same role as the metric entropy for classical
dynamical systems. We introduce this notion as an almost sure limit on orbits
of the system. Moreover we prove a kind of variational principle for this
complexity.Comment: 29 page
A concentration inequality for interval maps with an indifferent fixed point
For a map of the unit interval with an indifferent fixed point, we prove an
upper bound for the variance of all observables of variables
which are componentwise Lipschitz. The proof is based on
coupling and decay of correlation properties of the map. We then give various
applications of this inequality to the almost-sure central limit theorem, the
kernel density estimation, the empirical measure and the periodogram.Comment: 26 pages, submitte
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