Ordinary Differential Equations (27 hours)

The course will alternate lectures and active participation of students; it contains a mix of theory, computations and numerical simulations.
This course is part of an experimental COIL (Collaborative Online International Learning) initiative which includes in particular participation of students from Morroco, China and United States. For Le Havre students, tutorials are given by N. Verdière.

Chapters

  1. General Introduction
  2. Theoretical Results.
  3. Steady states, qualitative analysis and stability of solutions.
    4. Students will be asked to work on a specific non trivial problem (project), this will be part of the grade. The work will be done at home and in class.

Schedule (expected)

  1. Lectures 1-3: examples and exercices: 2nd Newton's Law, Radioactivity, electricity, Hodgkin Huxley Equations, examples:Hodgkin Huxley Equations (exercices), FitzHugh-Nagumo Equations, Lotka-Volterra, Oregonator, Lorenz.Projects
  2. Lectures 4-6: the Cauchy problem, exercices. Maximal solutions, global solutions. Regularity of solutions.Projects. Integral Equation and the Cauchy Problem. Approximate solutions Ascoli Theorem. Projects.
    3. Theorem of Existence (Cauchy, Arzela, Peano). Theorem of Existence and Uniqueness (Cauchy-Lipschitz). The Gronwall Lemma. Convergence of Euler method toward exact solutions.
  3. Lectures 7-9: Qualitative Analysis of 2d linear systems: saddle, sources, sinks, centers, spirals, Repeated eigenvalues. Non Linear Systems. Proof of stability for sinks. Examples.
  4. January 14 (to be confirmed): Dissertation.