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MA4H0 Applied Dynamical Systems

Lecturer: Vassili Gelfreich

Term(s): Term 1

Status for Mathematics students: List C for Math

Commitment: 30 lectures

Assessment: 100% 3 hour examination

Formal registration prerequisites: None

Assumed knowledge:

MA254 Theory of ODEs

MA251 Algebra I: Advanced Linear Algebra:

  • Jordan normal form

MA259 Multivariable Calculus:

  • Differentiation in more than one dimension, implicit function theorem, divergence theorem

Useful background: Read one of the following two books:

  • 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.

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 others plus this one.

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. In particular, motivating examples will be taken from chemical reaction network theory, climate models, fluid motion, celestial mechanics and neuronal dynamics.

The module will be structured around the following topics:

  • Review of basic theory: flows, notions of stability, linearization, phase portraits, etc
  • `Solvable' systems: integrability and gradient structure, applications in celestial mechanics and chemical reaction networks
  • Invariant manifold theorems: stable, unstable and center manifolds
  • Bifurcation theory from a geometric perspective
  • Compactification techniques: flow at infinity, blow-up, collision manifolds
  • Chaotic dynamics: horsehoes, Melnikov method and discussion of strange attractors
  • Singular perturbation theory: averaging and normally hyperbolic manifolds

Additional Resources