# 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.

## 2018-19 Course Listings

### Topics in Applied Mathematics: Delay Differential Equations

Many physical processes are modelled by differential equations which involve delays. This course will provide an introduction to delay differential equations (DDEs) concentrating on the key tools needed to understand the behaviour of these equations, and also some of numerical techniques used to approximate solutions. Throughout we will emphasise the similarities and differences between DDEs and ordinary differential equations (ODEs).

Topics covered will include: DDEs as infinite dimensional dynamical systems, breaking points and smoothing of DDE solutions, continuous Runge-Kutta methods for ODEs and DDEs, linear stability of steady states, bifurcation theory. A selection of more advanced topics will also be covered. The choice of topics will depend on time and the preferences of the participants, but may include state-dependent delays, distributed delays, numerical continuation and bifurcation techniques.

### Nonlinear Differential Equations

This graduate course is an introduction to the treatment of nonlinear differential equations, and more generally to the theory of dynamical systems. The objective is to introduce the student to the theory of dynamical systems and its applications. Firstly, classical dynamics analysis techniques will be presented: continuous and discrete flows, existence and stability of solutions, invariant manifolds, bifurcations and normal forms. Secondly, an introduction to ergodic theory and an overview of modern applications will be presented: chaotic dynamics, strange attractors, dynamic entropy, high-dimensional systems (e.g. networks), driven dynamics and information processing. Particular attention will be paid to computations performed by dynamical systems.

At the end of the course, the student will be able to apply dynamical systems analysis techniques to concrete problems, as well as navigate the modern dynamical systems literature. Several examples and applications making use of numerical simulations will be used. To take this course, the student must master, at an undergraduate level, notions of calculus, linear differential equations, linear algebra and probability.

Course schedule :

Tuesday : 10h30 - 12h00, 5183 Pav. Andre-Aisenstadt
Thursday : 10h30 - 12h00, 5183 Pav. Andre-Aisenstadt