Systèmes Dynamiques-Dynamical Systems-S2

Systèmes Dynamiques / Dynamical Systems

Contact: A. Pocheau & J. Deschamps

Motivations:

This course aims to show, in simple contexts, how determining the possible evolutions of a dynamical system, especially at long time, depending on the system parameters. The approach is thus both global (for all initial conditions) and parametric (for a range of parameters).

The objective is to:

  1. determine states (fixed points, limit cycle) that structure the dynamics ; use them to determine the system’s evolutions.
  2. show how change of parameters can change the structure of the dynamics ; learn how to identify them.

The systems will be modelled by ordinary differential equations. They will address both familiar systems (e.g. linear or non-linear oscillators) or new systems (e.g. population dynamics). Attention will be laid on global approaches (phase portrait) and structural concepts (attractors, stability/instability, bifurcation point).

I] Linear Systems

  • evolution operator, Jordan reduction
  • eigenmodes, secular modes, existence and uniqueness of solutions
  • phase portrait, fixed points, attractors

II] Non-linear Systems

  • Tools and methods (linearization, orbits, invariants, symmetries, perturbative expansions, multiplicity, finite-time divergence …)
  • Phase portraits (pendulum, population dynamics, epidemics, …)
  • Flow dynamics (trapping domain, limit cycle, Poincaré-Bendixson criterion, Dulac criterion, Lyapunov function, gradient systems , …)

III] Introduction to bifurcations

  • Nature of bifurcations
  • Examples of bifurcation (pitchfork , transcritical, …)

This course provides a valuable preparation to the S3-course « Dynamical System and Non-Linear Physics ».