UE Dynamic systems, chaos and applications

Degrees incorporating this pedagocial element :

Description

The theory of dynamical systems examines systems governed by simple elemental, deterministic or stochastic laws, which exhibit complex and unexpected emerging phenomena.

The course will introduce students to basic concepts of nonlinear physics, dynamical systems theory and chaos theory. The aim is for students to acquire analytical methods, a geometric approach, and a knowledge and understanding of striking and diverse practical examples, from physics (fluid physics, astronomy, quantum mechanics, Earth physics), engineering, biology, chemistry, economics and mathematics.

 We will also present several major issues in current research. Indeed, it seems that the major scientific challenges of the 21st century are associated with questions of complex dynamical systems such as: QCD and quark confinement, the Riemann hypothesis, turbulence and Navier-Stokes equations, living things (morphogenesis, brain behaviour, evolution of species), societies (complex interactions, economics).

It is based on the following plan:

  • Introduction: characteristic models of dynamical systems, phase spaces, Poincaré section, examples: damped and maintained pendulum, logistical application, Mandelbrot set, chemical oscillator.
  • Bifurcations in one-dimensional systems, hysteresis loops, oscillators, synchronisation (fireflies, Josephson junctions).
  • Two-dimensional flows and applications: classification of fixed points of linear systems, stability, basins of attraction, limit cycles (Lyapunov function, Poincaré-Bendixson theorem), oscillators (glacial cycles, Duffing equation), Hopf bifurcation and oscillating chemical reactions, quasiperiodicity, frequency locking, Kuramoto model.
  • Infinite-dimensional systems of the field theory type: hydro instabilities, solitary waves, morphogenesis models based on reaction-diffusion.
  • Hyperbolic dynamics through examples: symbolic dynamics, fractals, Haussdorff dimension, ergodicity, mixing, central limit theorem.
  • Presentation of advanced models: Lorenz attractor, Hénon map, Hamiltonian systems, billiards, KAM theorem, Poincaré-Melnikov scenario, rings of Saturn, chaos in the solar system, spatio-temporal chaos and turbulence.

Prerequisites

All the courses from the Physics bachelor degree (L3), in particular analytical mechanics.

Bibliography

Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering, Steven H. Strogatz

"Pattern formation and dynamics in Nonequilibrium Systems". 2009. Michael Cross, Henry Greenside,

"Physique des solitons", Michel Peyrard et Thierry Dauxois. 2004.

Differential Equations, Dynamical Systems, and an Introduction to Chaos, Morris W. Hirsch, Stephen Smale , Robert L. Devaney

L'ordre dans le chaos, Pierre Bergé, Yves Pomeau, Christian Vidal