The course will present some recent results and tools in the field of research of dynamical systems, a particular attention will be done to the duality "chaos" (irregular) periodic (regular). Several models will be discussed and analysed, among which the the logistic map that will be used to introduce and develop new concepts. Students will also be in touch with fractal theory: their role and links with dynamical systems.
We will closely follow the book "Nonlinear Dynamics And Chaos" by Steven Strogatz. The following topics will be analised.
1. Introduction : beyond regular orbits 2. The logistic map: orbits, fixed points and periodic points. 3. Stability of fixed points and bifurcation diagram. 4. Dependence on initial conditions: example et definition. 5. Chaotic orbits and mappings wiggly. 6. Conjugation of mappings and topological properties. 7. The tent map and the triadic Cantor fractal set. 8. Fractals as fixed points of dynamical systems : Iterated Functions Systems. 9. Fractal dimensions. 10. Systems of planar differential equations: existence of periodic solutions, the theorem of Poincaré-Bendixon. 11. Conditions of non existence of periodic solutions. 12. Existence and uniqueness theorems for limit cycles: the Theorem by Dragiliev, the Theorem by Massera 13. Stability of periodic solutions and the Theorem by Lyapounov-Andronov-Witte
Exercises describe the theoretical part. It's mostly on computer and usual subjects are : numerical integration of differential equations, stationary points, stability, mappings, bifurcation diagrams, fractals, Mandelbrot sets, fractal dimension, limit cycles, conjugaison.
written part : 3 h exercices
K.T. Alligood, T.D. Sauer et J.A. Yorke, Chaos. An introduction to dynamical systems, Springer-Verlag, New York (1996). Cambridge Univ. press (2003). J. Banks, V. Dragan et A. Jones, Chaos, a mathematical introduction, Australian mathematical society Lecture Series 18, Cambridge Univ. press (2003). M. Barnsley, Fractals everywhere, Academic Press London (1988) G.A. Edgar, Measure, Topology and fractal geometry, UTM, Springer-Verlag, New York (1990). H.O. Peitgen, H. J¿urgens et D. Saupe, Chaos and fractals, new frontiers of science, Springer-Verlag, New York (1993) C. Tricot, Courbes et dimension fractale, Springer Berlin (1999). S. H. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry and Engineering, Westview Press V. Arnol'd: Equations différentielles ordinaires L. Pontriaguine: Equations différentielles ordinaires G. Sansone et R. Conti: Non-linear differential equations Z. Zhang: Qualitative theory of differential equations