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.
Problem Sheet 0 includes questions covering some relevant background material.
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 applied and pure mathematics with applications including chemical reaction networks, celestial mechanics, mathematical biology, fluid dynamics and social sciences.
The course will focus on 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: The first 8 lectures of this course is part of the core syllabus for the MSc in Mathematical Modelling and Scientific Computing A2 Mathematical Methods.Lectures 1-8: Discrete-time (maps) and continuous-time (differential equations) dynamical systems. Notion of flows, stability of fixed points, Lyapunov function, invariant manifolds, stable manifold theorem, notion of hyperbolicity, center manifold. Chemical reaction networks. Stable, unstable and center subspaces. Poincaré-Bendixson theorem. Periodic solutions, stable and unstable limit cycles. Introduction to bifurcation theory, covering saddle-node, transcritical, supercritical pitchfork and subcritical pitchfork bifurcations. Extended center manifold. Logistic map. Periodic points of maps. Stability of \(N\)-cycles. Period-doubling bifurcation. Sharkovsky’s theorem. Invariant distribution.
Lectures 9-16: Bifurcations of limit cycles, covering supercritical and subcritical Hopf bifurcations, saddle-node bifurcation of cycles, infinite-period (SNIC) bifurcation and homoclinic (saddle-loop) bifurcation. Oscillations in chemical reaction networks. Weekly nonlinear oscillators. Poincaré-Lindstedt method. Conservative and non-conservative systems. Liénard systems, van der Pol oscillator. Hilbert's 16th problem. Lorenz equations. Lorenz map. Poincaré section. Poincaré map. Converse of Sharkovsky's theorem. Bernoulli shift map, symbolic dynamics. Tent map. Dynamics on metric spaces, sensitive dependence on initial conditions, transitivity, conjugate maps, chaotic dynamics.
