Many problems in Applied Mathematics are nonlinear and described by nonlinear ordinary (or partial) differential equations. This course aims to introduce students to the tools and techniques needed to understand the dynamics that might be found in such systems. The emphasis will be on concepts and examples rather than theorems and proofs, and will include a brief survey of useful numerical methods and packages. Students will be invited to submit examples of their own for possible discussion.
No prerequisites information is available yet.
* Definition of a flow (ordinary differential equation), limit sets
* Equilibria, stability of equilibria, Hartman-Grobman Theorem
* Local structural stability, local bifurcation theory
* Continuation and the implicit function theorem
* Limit cycles, planar systems, Floquet multipliers
* Centre manifold theorem, normal form transformation
* Examples, including an introduction to numerical methods for ODEs and a mention of packages available
* Brief discussion of dynamics of dissipative PDEs (partial differential equations)
* Mention how the role of symmetry and how it changes everything
* Brief discussion of the Poincare map and the connection between flows and maps near periodic orbits and global bifurcations, leading on to Dynamical Systems II (maps)