Cellular Automata (CA) are a class of discrete dynamical systems that have been widely used to model complex systems in which the dynamics is specified at local cell-scale. Classically, CA are run on a regular lattice and with perfect synchronicity. However, these two assumptions have little chance to truthfully represent what happens at the microscopic scale for physical, biological or social systems.One may thus wonder whether CA do keep their behavior when submitted to small perturbations of synchronicity.This work focuses on the study of one-dimensional (1D) asynchronous CA with two states and nearest-neighbors. We define what we mean by ``the behavior of CA is robust to asynchronism'' using a statistical approach with macroscopic parameters.and we present an experimental protocol aimed at finding which are the robust 1D elementary CA. To conclude, we examine how the results exposed can be used as a guideline for the research of suitable models according to robustness criteria.

An Experimental Study of Robustness to Asynchronism for Elementary Cellular Automata

This paper introduces a new notion of chaotic algorithms. These algorithms are iterative and are based on so-called chaotic iterations. Contrary to all existing studies on chaotic iterations, we are not interested in stable states of such iterations but in their possible unpredictable behaviors. By establishing a link between chaotic iterations and the notion of Devaney's topological chaos, we give conditions ensuring that these kind of algorithms produce topological chaos. This leads to algorithms that are highly unpredictable. After presenting the theoretical foundations of our approach, we are interested in its practical aspects. We show how the theoretical algorithms give rise to computer programs that produce true topological chaos, then we propose applications in the area of information security.

Topological chaos and chaotic iterations application to hash functions

https://pageperso.lif.univ-mrs.fr/~sylvain.sene/files/publi_pres/djs08.pdf

Boundary conditions and phase transitions in neural networks. Theoretical results

Dans cette these, nous nous interessons a l'influence des conditions de bords dans les reseaux d'automates booleens a seuil, qui sont des objets mathematiques discrets classiquement utilises pour modeliser les systemes de regulation biologiques L'objectif est de mettre en evidence que les elements determinant les bords de ces reseaux, que l'on peut rapprocher dans le contexte biologique de potentiels electriques, d'hormones ou encore de micro-ARN, sont des composants d'un systeme dont l'effet peut etre primordial sur le comportement de ce dernier Cet objectif est atteint en suivant deux axes distincts Le premier consiste a montrer les liens entre l'influence des conditions de bord et les transitions de phase emergeant du comportement asymptotique de reseaux theoriques, a savoir des reseaux d'automates cellulaires Le deuxieme axe se focalise sur les systemes biologiques reels et developpe l'idee selon laquelle les conditions de bord dans ces systemes ont une influence particuliere sur les bassins d'attraction des systemes dynamiques par lesquels ces systemes sont modelises

Influence des conditions de bord dans les réseaux d'automates booléens à seuil et application à la biologie

This paper gives a formulation of the basins of fixed point states of fully asynchronous discrete-time discrete-state dynamic networks. That formulation provides two advantages. The first one is to point out the different behaviors between synchronous and asynchronous modes and the second one is to allow us to easily deduce an algorithm which determines the behavior of a network for a given initialization. In the context of this study, we consider networks of a large number of neurons (or units, processors, etc.), whose dynamic is fully asynchronous with overlapping updates . We suppose that the neurons take a finite number of discrete states and that the updating scheme is discrete in time. We make no hypothesis on the activation functions of the nodes, so that the dynamic of the network may have multiple cycles and/or basins. Our results are illustrated on a simple example of a fully asynchronous Hopfield neural network.

Basins of attraction in fully asynchronous discrete-time discrete-state dynamic networks

We consider networks of a large number of neurons (or units, processors, ...), whose dynamics are fully asynchronous with overlapping updating. We suppose that the neurons take a finite number of states (discrete states), and that the updating scheme is discrete in time. We make no hypotheses on the activation function of the neurons; the networks may have multiple cycles and basins. We derive conditions on the initialization of the networks, which ensures convergence to fixed points only. Application to a fully asynchronous Hopfield neural network allows us to validate our study.

S. Contassot-Vivier

Papers

Stability of fully asynchronous discrete-time discrete-state dynamic networks