On the bifurcation of species.

We propose and analyze a model of evolution of species based upon a general description of phenotypes in terms of a single quantifiable characteristic. In the model, species spontaneously arise as solitary waves whose members almost never mate with those in other species, according to the rules laid down. The solitary waves in the model bifurcate and we interpret such events as speciation. Our aim in this work is to determine whether a generic mathematical mechanism may be identified with this process of speciation. Indeed, there is such a process in our model: it is the Andronov homoclinic bifurcation. It is robust and is at the heart of the formation of new solitary waves, and thus (in our model) new species.

How much information can one store in a nonequilibrium medium?

It has recently been emphasized again that the very existence of stationary stable localized structures with short-range interactions might allow one to store information in nonequilibrium media, opening new perspectives on information storage. We show how to use generalized topological entropies to measure aspects of the quantities of storable and nonstorable information. This leads us to introduce a measure of the long-term stably storable information. As a first example to illustrate these concepts, we revisit a mechanism for the appearance of stationary stable localized structures that is related to the stabilization of fronts between structured and unstructured states (or between differently structured states).

A new approach to data storage using localized structures.

In this paper we describe how to use the bifurcation structure of static localized solutions in one dimension to store information on a medium in such a way that no extrinsic grid is needed to locate the information. We demonstrate that these principles, deduced from the mathematics adapted to describe one-dimensional media, also allow one to store information on two-dimensional media.

Chaotic self-trapping of a weakly irreversible double Bose condensate.

We analyze the dynamics of a weakly open Bose-Einstein condensate trapped in a double-well potential. Close to the self-trapping bifurcation, numerical simulations of the weakly irreversible one-dimensional Gross Pitaevskii equation reveal chaotic behaviors. A two-mode model is used to derive amplitude equations describing the complex dynamic of the condensate.

Self-parametric instability in spatially extended systems.

We study the stability of almost homoclinic homogeneous limit cycles with respect to spatiotemporal perturbations. It is shown that they are generically unstable. The instability is either the phase instability or a finite wavelength period doubling instability.

Stable static localized structures in one dimension

We study the existence, the stability properties, and the bifurcation structure of static localized solutions in one dimension, near the robust existence of stable fronts between homogeneous solutions and periodic patterns.

Head-on collisions of waves in an excitable FitzHugh-Nagumo system: a transition from wave annihilation to classical wave behavior.

For the particular case of an excitable FitzHugh-Nagumo system with diffusion, we investigate the transition from annihilation to crossing of the waves in the head-on collision. The analysis exploits the similarity between the local and the global phase portraits of the system. We find that the transition has features typical of the nucleation theory of first-order phase transitions, and may be understood through purely geometrical arguments. In the case of periodic boundary conditions, the transition is an infinite-dimensional analog of the creation and the vanishing of limit cycles via a homoclinic Andronov bifurcation. Both before and after the transition, the behavior of a single cell continues to be typical for excitable systems: a stable equilibrium state, and a threshold above which an excitation pulse can be induced. The generality and qualitative character of our argument shows that the phenomenon described can be observed in excitable systems well beyond the particular case presented here.

Chaotic alternation of waves in ring lasers.

We intend to give an analytic description of the mechanisms involved in the periodic and chaotic wave alternation frequently observed in ring lasers. A set of amplitude equations is derived from the Maxwell-Bloch equations. These equations are studied analytically and numerically.

From oscillations to excitability: A case study in spatially extended systems.

This volume is devoted to the presentation of the main contributions to the workshop "From oscillations to excitability: A case study in spatially extended systems," organized by the authors in Nice in June 1993. It gives an overview of the current research on spatiotemporal patterns in a wide range of systems that display self-oscillatory or excitable behavior. It tries to give a better understanding of the transition from the oscillatory to the excitable regime and of its effect on the properties of spiral waves, and to fill the gap between the theories and concepts used to describe both regimes in the so-called "active media."

Excitability in liquid crystal.

The spiral waves observed in a liquid crystal submitted to a vertical electric field and a horizontal rotating magnetic field are explained in the framework of a purely mechanical description of the liquid crystal. The originality of the experiment described in this paper is the presence of the vertical electric field which allows us to analyze the spiral waves in the framework of a weakly nonlinear theory.