Résumés des cours

Arezki Kessi, USTHB, Algérie : Systèmes dynamiques en dimension 1.

Le cours est constitué de trois parties : 1) Notions générales sur les systèmes dynamiques.  2) Partition du cercle.  3) Homéomorphismes du cercle

 La première partie est consacrée aux notions générales sur les systèmes dynamiques discrets. On donne des définitions et des exemples de systèmes dynamiques ainsi que certains résultats, sans s’attarder sur les démonstrations. L’exemple qui sera étudié en détail dans suite est le cercle.

Dans la deuxième partie on donne une partition du cercle déterminée par une rotation. On utilise les fractions continues, ce qui permet de représenter géométriquement cette dernière notion. Un petit rappel est nécessaire sur les fractions continues.

Enfin dans la troisième partie on définit le nombre de rotation. On montre son existence ainsi que certaines de ses propriétés.  

Références :

Luis Barreira, Claudia Valls, Dynamical Systems, An introduction, Springer, 2012

Ya. G. Topics in ergodic theory, Princeton university press, 1994.

I. Cornfeld, S. Fomin, Ya. G. Sinai, Ergodic theory, Springer, 1982

Philippe Thieullen, Université de Bordeaux, France : Optimisation ergodique

Ergodic optimization of additive cocyle studies the trajectories of a dynamical system that maximize the time average of an observable. An operator cocycle is a sequence of products of linear operators along a trajectory. Ergodic optimization of operator cocycles studies the vectors of optimal growth under the action of the cocycle. This non-commutative theory of the classical ergodic optimization theory  is new and has many applications in stability, discrete non-autonomous dynamical systems, linear switching systems, or smooth ergodic theory. 

The course will review recent theoretical results where the base dynamical system is simple as in the minimal  case (quasi-periodic in time as in Kessi's course) or in the hyperbolic case (existence of a transversally hyperbolic invariant manifold). The course will also be more applied and oriented toward the study of the stability of constraint linear systems in a neural network (as in Rebouh's and Benzeki's course)

Tewfik Sari, Institut national de recherche en sciences et technologies pour l'environnement et l'agriculture (IRSTEA) Montpellier, France : Mathematical models in biology: The Consumer–Resource relationship.

1. The Predator–Prey Model

1.1. The logistic model

1.2. The Lotka–Volterra predator–prey model

1.3. The Gause model

1.4. The Rosenzweig–MacArthur model

1.5. The “ratio-dependent” model

1.6. The chemostat model

2. Competition

2.1. The two-species competition Volterra model

2.2. Competition and the Rosenzweig–MacArthur model

2.3. Competition with ratio-dependent models

2.4. Coexistence through periodic solutions

2.5. Competition in the chemostat

References:

R. Arditi, L. Ginzburg. How species interact : altering the standard view on
thophic ecology, Oxford University Press, Oxford, 2012.

J. Harmand, C. Lobry, A. Rapaport, T. Sari. The Chemostat: Mathematical
Theory of Microorganism Cultures, ISTE-WILEY, 2017

C. Lobry. The Consumer-Resource Relationship: Mathematical Modeling,
ISTE-WILEY, 2018

J. D. Murray. Mathematical Biology, Springer-Verlag, New York, 2005.

 Tounsia Benzekri, USTHB, Algérie : Introduction to bifurcation theory

The aim of this course is to recall some basic notions of dynamical systems and stability theory, in continuous time, to introduce the theory of bifurcations.
Bifurcation theory considers families of systems depending on parameters. A bifurcation is a qualitative change of dynamics and aims to divide the parameter space in regions in which the system has qualitatively similar behaviors.
In this course, the focus is on the simplest case of single-parameter bifurcations, in two-dimensional systems, as saddle-node, transcritical , pitchfork and Hopf bifurcations.