Non-linear Dynamics

Program Description

Nonlinear dynamics is concerned with phenomena evolving in time, which are modelled by differential equations (ordinary, partial, or functional differential equations) and difference equations. The issues studied are: the geometrical description of solutions (individually or as a whole), their asymptotic behavior, families of dynamical systems depending on parameters and their bifurcations, the controllability of systems, the sensitivity to perturbations, optimality conditions.

The main purpose of this program is to develop simultaneously some aspects of mathematical dynamics to favour exchanges among researchers in various branches and to offer students a training as complete as possible. These aspects are:

• dynamical systems
• differential equations
• optimization
• ergodic theory
• modelling

Various techniques arise in the program. These include topological methods for proving existence of solutions; algebraic-geometric methods (the study of polynomial vector fields is currently very active); variational methods; the techniques of control theory, both theoretical (for example, non-smooth analysis) and numerical; the theory of fractals with applications to rough surfaces, porous surfaces, different types of aggregations, and percolation theory; ergodic theory and methods of Markov chains. There is emphasis on biological models arising in physiology, epidemiology, population dynamics, and genetics.

Program Members

Students are expected to acquire the fundamentals of analysis, differential equations, and, when required, probability theory. Students are then expected to take a number of more specialized courses offered within the program.

2020-21 Course Listings

Topics in Analysis/Special Topics in Dynamical Systems: Iterated Function Systems, Complex Dynamics and Fractals

Iterated Function Systems:

The mathematical fundations: metric spaces, the Hausdorff space of subsets and its metric, the theorem about the completeness of the Hausdorf space. The symbolic dynamics and “codes” on the attractor of IFS.

Applications: Attractors of the IFS. Fractal approximation. Fractal compression of pictures.

Complex Dynamics:  General introduction to complex dynamics, Julia and Fatou sets, dynamics of polynomials and rational functions, Mandelbrot set.

If time allows additional topics might be covered.

Systèmes dynamiques - Laval

Rappels sur les systèmes linéaires. Systèmes non linéaires : linéarisation et méthode de Lyapounov. Solutions périodiques : application de Poincaré, théorème de Poincaré-Bendixon. Variétés répulsives et attractives. Introduction à la stabilité structurelle et théorème de Peixoto. Variétés neutres, formes normales et application à la théorie locale des bifurcations. Exemple de Smale et bifurcation de points homocliniques.

Dynamical Systems

Dynamical systems, phase space, limit sets. Review of linear systems. Stability. Liapunov functions. Stable manifold and Hartman-Grobman theorems. Local bifurcations, Hopf bifurcations, global bifurcations. Poincare Sections. Quadratic maps: chaos, symbolic dynamics, topological conjugacy. Sarkovskii's theorem, periodic doubling route to chaos. Smale Horseshoe.

Systèmes dynamiques

Flots discrets et continus. Équations différentielles non linéaires, techniques classiques d’analyse de dynamique, existence et stabilité de solutions, variétés invariantes, bifurcations, formes normales, systèmes chaotiques. Applications moderne.

Dynamique bicomplexe et fractales 3D

Approfondir les propriétés de base des nombres réels. Étudier la topologie des espaces métriques. Introduction à l'espace des fractales via les systèmes de fonctions itérées et la dynamique complexe. Exploration des fractales 3D générées à l'aide de la dynamique bicomplexe.