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

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 $v_0$ whose restriction to a bounded interval $I$ have several zeros, not too regularly spaced, and other zeros of $v_0$ are very far from $I$. 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 $I$

A Model of Heat Conduction

We define a deterministic ``scattering'' model for heat conduction which is continuous in space, and which has a Boltzmann type flavor, obtained by a closure based on memory loss between collisions. We prove that this model has, for stochastic driving forces at the boundary, close to Maxwellians, a unique non-equilibrium steady state

Dynamics of Triangulations

We study a few problems related to Markov processes of flipping triangulations of the sphere. We show that these processes are ergodic and mixing, but find a natural example which does not satisfy detailed balance. In this example, the expected distribution of the degrees of the nodes seems to follow the power law d −

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