Syllabus
Continuous dynamical systems in the form of nonlinear ordinary differential equations (ODE) are at the core of mathematical modeling and are widely used to describe complicated phenomena in fields as broad as biology, physics, chemistry, meteorology and epidemiology. A fundamental difficulty arising in studying nonlinear ODE is the absence of closed-form expressions for the solutions, which almost inevitably forces the scientists to use numerical methods to study the models.
Traditionally, purely pen and paper mathematics (e.g. functional analysis, topological methods, nonlinear analysis) and the tools of scientific computing were used separately to study the models. The purpose of this class is to change perspective, and to combine the strength of pure and applied mathematics by introducing a state-of-the-art mathematical machinery which leads to computer-assisted proofs of existence of dynamical objects in ODE. More precisely, the students will learn novel rigorous computational techniques to prove existence (in a constructive fashion) of steady states, periodic orbits, homoclinic and heteroclinic orbits, solutions of initial and boundary value problems, and to compute rigorously stable and unstable manifolds attached to steady states and periodic orbits. Finally, students will learn how to prove existence of chaos.
• Chapitre 1: Motivation
• Chapitre 2: Banach Spaces
• Chapitre 3: Radii Polynomial Approach on Banach Spaces
• Chapitre 4: Fundamental Results: Existence and Uniqueness
• Chapitre 5: Taylor Methods
• Chapitre 6: Periodic Orbits
• Chapitre 7: Boundary Value Problems via Fourier Series
• Chapitre 8: Initial Value Problems via Chebyshev Series
• Chapitre 9: Stable and Unstable Manifolds for Equilibria
• Chapitre 10: Linear Theory for Periodic Orbits
• Chapitre 11: Stable and Unstable Manifolds for Periodic Orbits
• Chapitre 12: Connecting Orbits
• Chapitre 13: Dynamical Aspects of ODEs
• Chapitre 14: Chaotic Dynamics
Annotation
Course by Visiting Professor J.-P. Lessard.
Students will learn to use novel computer-assisted techniques to prove existence of different types of dynamical objects in nonlinear continuous dynamical systems.