At the end of this learning unit, the student is able to :
a. Contribution of the teaching unit to the learning outcomes of the programme (PHYS2MA)
1.1, 1.3, 1.4, 2.1, 3.1, 3.2, 3.3, 3.4, 3.5, 3.6
b. Specific learning outcomes of the teaching unit
At the end of this teaching unit, the student will be able to :
1. use mathematical tools to characterise the properties of discrete and continuous non-linear systems;
2. characterise the chaotic dynamics of a system.
The following topics are covered by the teaching unit:
1. Basic concepts: definition of a dynamical system, examples of continuous and discrete dynamic systems, hyperbolic points of equilibrium and stability, bifurcations.
2. Discrete chaotic systems: chaos and sensitivity to initial conditions, itineraries, topological conjugation, Lyapunov exponents, the logistic map.
3. Linearisation, stable and unstable manifolds: the dynamics of linear systems, classification of two-dimensional fixed points, linearisation around hyperbolic fixed points, stable and unstable manifolds, perturbative analysis;
4. The horseshoe map: intersections of stable and unstable manifolds, homoclinic points, horseshoe and chaos, Cantor sets;
5. The Poincaré-Bendixon theorem: trapping regions, limit cycles and limit sets, the Poincaré map, the Poincaré-Bendixon theorem, applications (existence of periodic orbits, Liénard systems).
6. Ergodic theory: the concept of ergodicity, relations with statistical mechanics, Poincaré's reccurrence theorem, ergodic theorems, examples and applications.
Due to the COVID-19 crisis, the information in this section is particularly likely to change.The learning activities consist of lectures and exercise sessions.
The lectures introduce fundamental concepts of the theory of nonlinear systems and their motivation through concrete examples from various scientific disciplines.
The main objective of the exercise sessions is the application of the theory to concrete examples.
Due to the COVID-19 crisis, the information in this section is particularly likely to change.The evaluation is based on a written exam and a continuous assessment during the semester.
The written exam deals with the application of the theory of non-linear systems to concrete examples. It tests the student's knowledge and his understanding of the notions seen in the theoretical course, the mastery of calculation techniques and the coherent presentation of this analysis.
The result of the continuous assessment will be used for each session and cannot be represented.
The evaluation methods may be adapted and modified according to the evolution of the Covid-19 pandemic.
- K.T. Alligood, T.D. Sauer, J.A. Yorke, Chaos. An introduction to dynamical systems. Springer-Verlag (2008).
- M.W. Hirsch, S. Smale, R.L. Devaney, Differential equations, dynamical systems, and an introduction to chaos. Academic Press (2013).
- S.H. Strogatz, Nonlinear dynamics and chaos. Westview Press (2015).
- M. Tabor, Chaos and integrability in non-linear dynamics : an introduction. J. Wiley & Sons (1989).