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:
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.
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.
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.