Course Details in

Module Title	LH Applied Nonlinear Dynamical Systems
School	Mathematics
Department	Mathematics
Module Code	06 27718
Module Lead	Dr Chris Good
Level	Honours Level
Credits	10
Semester	Semester 2
Pre-requisites	LI Differential Equations - (06 25670)
Co-requisites
Restrictions	None
Contact Hours	Lecture-23 hours Tutorial-5 hours Guided independent study-72 hours Total: 100 hours
Exclusions
Description	Smooth dynamical systems are systems whose evolution is described by differential equations that depend continuously on time. Differential equations provide the basis for describing phenomenon observed in many areas of science, for example, biology, meteorology, astronomy and engineering. In the majority of applications, the underlying equations are inherently nonlinear. Nonlinear differential equations do not, in general, have closed-form solutions. The behaviour of the solutions can be rather complex, even unpredictable! However, the solutions can be characterised using an alternative approach to determine their long term qualitative behaviour. For example, do solutions approach a steady state, are they periodic in time or could they be chaotic? This module will introduce the tools used to determine the qualitative behaviour of solutions to nonlinear ordinary differential equations, focusing on the existence and stability of steady states and periodic orbits in two-dimensional phase space. There will also be an introduction to bifurcation theory which describes how the structure of the solutions can change as parameters in the system are varied. In nonlinear systems, small changes in a parameter value (e.g. temperature) can lead to sudden changes in the phase space topology and therefore the behaviour of the system. We will explore the dynamics of systems near simple bifurcations. If time allows, further topics will be discussed. These may include the effect of symmetries in systems of nonlinear ordinary differential equations or dynamics and bifurcations and chaos in discrete time dynamical systems.
Learning Outcomes	By the end of the module students should be able to: analyse simple dynamical systems to find and classify regular behaviour, sketch phase portraits and determine stable and unstable manifolds. determine the existence and stability of periodic orbits in two dimensions using Dulac's criterion, Lyapunov functions, the Poincaré-Bendixson Theorem and Floquet theory. identify and classify codimension one bifurcations of fixed points and sketch bifurcation diagrams. use the Centre Manifold Theorem to reduce the order of a system appropriately and bring the reduced system into normal form.
Assessment	27718-01 : Raw Module Mark : Coursework (100%)
Assessment Methods & Exceptions	90% on one 1.5 hour examination; 10% from coursework and/or class tests.
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