Dynamical Systems module (MA51003)

About the module

This module, aimed at the Level 5 student, takes an advanced look at dynamical systems. The time evolution of many biological, chemical, or physical processes, as well as systems considered in engineering or economics, can be described by difference or differential equations. Dynamical systems theory allows us to study these systems of equations and inver information about the behaviour of the corresponding biological, chemical or physical systems. It addresses questions like the existence and stability of solutions, how the behaviour of solutions changes depending on the system parameters, or determines the existence of strange attractors or chaos in the system.

This module may optionally be taken by students on the MMath in Mathematics, or the MSci in Mathematical Biology or Mathematics and Physics degrees. If you have questions about this module, please contact your Advisor of Studies.

Prerequisites

Students taking this module must usually have achieved a pass mark in each of the modules MA31002 and MA32001, or equivalent.

Indicative Content

• One-Dimensional Maps

  Definition, Cobweb Plot: Graphical Representation of an Orbit, Stability of Fixed Points, Periodic Points, Chaos: Lyapunov Exponents.

• Ordinary Differential Equations

  Background, Examples of main Physical and Biological Processes described by Ordinary Differential Equations (ODEs), Existence and uniqueness of solutions of ODEs, Linearised Stability Analysis, Two-dimensional Systems: Hamiltonian and Gradient systems, Periodic solutions: Floquet theory, Poincare Map and Stability of Periodic Orbits, Bifurcation and Chaos.

• Partial Differential Equations

  Definitions, Background, Well-posedness, Maximum Principles, Spectral Theorem for Laplace Equation, Semigroups for Evolution Equations in Banach Spaces, Nonlinear Evolution Equations: Linearised Stability Analysis for Reaction-Diffusion Equations.

Delivery and Assessment

The module is delivered in the form of lectures and workshops/presentation classes and assessed via an exam (60%) and coursework (40%).

Credit Rating

This module is a Scottish Higher Education Level 5 or SCQF level 11 module and is rated as 15 SCOTCAT credits or 7.5 ECTS credits.