Lecturer: Robert MacKay

Term(s): Term 2

Status for Mathematics students: List C for Math

Commitment: 30 lectures

Assessment: 100% 3 hour examination

Formal registration prerequisites: None

Assumed knowledge:

• MA251 Algebra I: Advanced Linear Algebra: Jordan normal form
• MA254 Theory of ODEs
• MA259 Multivariable Calculus: Differentiation in more than one dimension, implicit function theorem, divergence theorem

Useful background:

• MA3J3 Bifurcations, Catastrophes and Symmetry
• MA3K8 Variational Principles, Symmetry and Conservation Laws

Synergies: This module provides a complementary view of dynamical systems theory to others offered by the department. It concentrates on continuous time and aspects relevant to physics and biology. If you want a well rounded training in dynamical systems theory you are recommended to take one of the other modules:

• MA424 Dynamical Systems
• MA427 Ergodic Theory
• MA4N3 Hyperbolic Dynamics

Books on the subject: You may find the following three books useful:

• MW Hirsch, S Smale & RL Devaney, Differential Equations, Dynamical Systems and an Introduction to Chaos
• JD Meiss, Differential Dynamical Systems
• RC Robinson, An Introduction to Dynamical Systems

Content: This course will introduce and develop the notions underlying the geometric theory of dynamical systems and ordinary differential equations. Particular attention will be paid to ideas and techniques that are motivated by applications in a range of the physical, biological and chemical sciences.

The module will be structured around the following topics:

• Review of basic theory: flows, notions of stability, linearization, phase portraits, etc
• Invariant manifold theorems: stable, unstable and center manifolds
• Bifurcation theory from a geometric perspective
• Chaotic dynamics: horsehoes, Melnikov method and discussion of strange attractors

Learning Outcomes: Appreciate the geometric approach to dynamical systems

Additional Resources