- Lecturer: Dan Hill
- Lecturer: Andreas Muench
General prerequisites:
There is no optional Part B course as a formal prerequisite. This course builds on the material, which appears in several mandatory Prelims and Part A courses, including courses on differential equations, calculus, probability, linear algebra, constructive mathematics, computational mathematics, dynamics, metric spaces and analysis.
Course term: Hilary
Course lecture information: 16 lectures
Course weight: 1
Course level: H
Assessment type: Written Examination
Course overview:
This course aims to provide an introduction to the tools and concepts of dynamical systems theory which have become a central tool of both pure and applied mathematics with applications in celestial mechanics, mathematical biology, fluid dynamics, granular media, and social sciences. The course will focus on the geometry of both ordinary differential equations and maps. It will draw examples from appropriate model systems and various application areas. The problem sheets will require basic skills in numerical computation (numerical integration and visualisation of solutions of differential equations).
Learning outcomes:
Students will have developed a sound knowledge and appreciation of some of the tools, concepts, and computations used in the study of dynamical systems. They will also get some exposure to some modern research topics in the field.
Course synopsis:
1. Geometry of linear systems: Basic concepts of stability and linear manifold of solutions. Orbits in phase-space, linear flows, eigenvalues of fixed points.
2. Geometry on nonlinear systems: Notion of flows, invariant sets, asymptotic sets, attractor. Conservative and Non-Conservative systems.
3. Local analysis: Stable manifold theorem, notion of hyperbolicity, centre manifold.
4. Bifurcations: Bifurcation theory, codimension one normal forms (saddle-node, pitchfork, trans-critical, Hopf). Poincare-Lindstedt method.
5. Maps: Poincare sections and first-return maps. Stability and periodic orbits; bifurcations of one-dimensional maps, period-doubling.
6. Chaos: Logistic map, Bernoulli shift map, symbolic dynamics, Smale's Horseshoe Map. Melnikov's method.
2. Geometry on nonlinear systems: Notion of flows, invariant sets, asymptotic sets, attractor. Conservative and Non-Conservative systems.
3. Local analysis: Stable manifold theorem, notion of hyperbolicity, centre manifold.
4. Bifurcations: Bifurcation theory, codimension one normal forms (saddle-node, pitchfork, trans-critical, Hopf). Poincare-Lindstedt method.
5. Maps: Poincare sections and first-return maps. Stability and periodic orbits; bifurcations of one-dimensional maps, period-doubling.
6. Chaos: Logistic map, Bernoulli shift map, symbolic dynamics, Smale's Horseshoe Map. Melnikov's method.