Although we do not yet know how long the social distancing related to the Covid-19 pandemic will last, and regardless of the changes that had to be made in the evaluation of the June 2020 session in relation to what is provided for in this learning unit description, new learnig unit evaluation methods may still be adopted by the teachers; details of these methods have been - or will be - communicated to the students by the teachers, as soon as possible.
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 contribution of this Teaching Unit to the development and command of the skills and learning outcomes of the programme(s) can be accessed at the end of this sheet, in the section entitled “Programmes/courses offering this Teaching Unit”.
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.
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.
- 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).