Dynamical Systems and Non-Linear Physics-S3

Dynamical Systems and Non-Linear Physics

Contact: Alain Pocheau or Xavier Leoncini


Dynamical Systems and Chaos

  • Basics  : Introduction to the notion of dynamical systems, examples of non-linear systems, Discrete and Continuous time, from one to the other, Poincaré section. Reminders on differential equations and vector fields, and on linear systems and phase portraits.
  • Local properties: Stability of fixed points, equilibrium for maps and flows, linearisation, Lyapunov function, attractors and basin of attraction. Stable and unstable manifolds, Bifurcations
  • Global properties: Cantor sets, and fractals, notion of invariant measure, ergodicity and mixing. Birkhoff Theorem, Lyapunov exponent. Entropy, transport properties.
  • Hamiltonian systems and chaos: Liouville equation, Integrability, actions-angles, variables, perturbation theory, Chririkov criterion, KAM theorem, Arnold diffusion.

Spatio-temporal systems : instabilities and self-organization

  • Instabilities : Nature and method (base system, linearization, eigenmodes, spectrum)
  • Examples (thermoconvection, Turing, …)
  • Non-linearities and central manifold reduction, mode interaction and amplitude saturation
  • From motifs to patterns (multi-scale expansions, Fredholm alternative, envelope equation, Ginzburg-Landau equation, phase diffusion equation, …) Selection mechanisms (structures, wave number, front speeds,…)
  • Roads to chaos in dissipative systems: : Hopf Bifurcations, the Landau approach, Poincaré system, and first return maps, Bernoulli system, logistic map, subharmonic cascade (period doubling)/Feigenbaum, sensitivity to initial conditions, the Lorentz model (mode truncation and closure, bifurcations, strange attractors). Intermittency and roads to chaos.