Syllabus
Nonlinear discrete dynamical systems in the form of iterates of maps are important in applications and are used to describe phenomena in population dynamics, ecology, mechanics and celestial dynamics. The purpose of this class is to introduce the fundamental techniques to obtain computer-assisted proofs in the study of finite dimensional nonlinear discrete dynamical systems. More precisely, the students will learn novel computational techniques to obtain computer-assisted proofs of existence of fixed points, periodic orbits, stable and unstable manifolds attached to fixed points and periodic orbits, homoclinic and heteroclinic orbits. Finally, students will learn how to prove existence of chaos in discrete dynamical systems.
• Chapitre 1: Introduction
• Chapitre 2: Existence of Zeros of Functions
• Chapitre 3: Fixed Points and Periodic Orbits
• Chapitre 4: Linear Theory and Stability of Fixed Points
• Chapitre 5: Dynamical Systems
• Chapitre 6: Continuation of Fixed Points
• Chapitre 7: Bifurcations
• Chapitre 8: Power Series
• Chapitre 9: Stable and Unstable Manifolds for Fixed Points
• Chapitre 10: Connecting Orbits
• Chapitre 11: Chaotic Dynamics in Discrete Dynamical Systems
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 discrete dynamical systems.