Course overview
This course is available for students taking an honours degree in Mathematical Sciences. The course will cover an advanced topic in pure mathematics. For details of the topic offered this year please refer to the Course Outline.
Course learning outcomes
- This year, the topic of this course is Nonlinear Analysis of Dynamical Systems.SynopsisThe course is at the interface of applied and pure mathematics, and thus earns either a pure or an applied contribution towards your honours or higher degree. Nonlinear dynamics aspects will be considered from both the perspective of theory (establishing conditions on existence/uniqueness/bifurcations/chaos which appear in the form of theorems), and applications and intuition (will consider applied problems which link to such theory). These two aspects respectively are usually interpreted as pure and applied mathematics, but this course will emphasise the connections between them, demonstrating that the division is somewhat artificial. The theory will include the implicit function theorem; Gronwall's inequality; Lyapunov functions; Lasalle's invariance principal; stable/unstable manifolds; Hartman-Grobman theorem; Poincare-Bendixson theorem; saddle-node, transcritical, pitchfork, period-doubling and Hopf bifurcations; symbolic dynamics; Smale horseshoe map; Smale-Birkhoff theorem; and Melnikov theory. Applications will be in areas such as vibrations, fluid mechanics, invasion waves, Hamiltonian dynamics and mathematical modelling; however, the emphasis will be on applying methods to an application, rather than focussing principally on the application itself. The course will be run mainly on active learning principles, with the delivery designed to be student-centred. There will be no lectures in the traditional form, and the class meetings will be all in workshop format. The workshops will be based on a range of activities: worksheets which students will work on in small groups for discovery-based learning, discussions based on assigned recordings or readings, and student presentations on assigned topics or problems.Learning Outcomes. Appreciate the connections between the pure and applied aspects of nonlinear dynamical systems;
- Apply relevant theorems to identify and predict the presence of entities (such as stable and unstable manifolds) whose characteristics govern global flow patterns;
- Be cognisant of computational tools for detection of the above entities to empower an understanding of long-term behaviour of nonlinear dynamical systems;
- Explain the mathematical concept of chaos, and be able to apply both analytical and computational tools to investigate and detect chaotic behaviour;
- .Employ a suite of mathematical and computational tools for analysing dynamical systems emerging from a variety of applications.