Dynamical Systems module (MA32011)

What do heart rhythms, weather patterns, predator-prey cycles, and laser systems have in common? We can describe and understood them all by using dynamical systems. These are mathematical models that describe how things change over time.

In this module, you’ll study two types of dynamical systems. Firstly, those based on ordinary differential equations (ODEs). These describe finite-dimensional systems like mechanical motion or chemical reactions.

You'll also study those based on partial differential equations (PDEs). These model more complex or continuous systems like heat flow or fluid dynamics.

You’ll explore the qualitative behaviour of these systems: when they are stable or unstable, how they behave near equilibrium points, and how solutions evolve over time. You’ll also investigate bifurcations, which are sudden changes in system behaviour, and learn how to detect them.

Overall, you'll learn core mathematical techniques to analyse and understand how systems behave, react, and change. This will be of use whether you’re modelling ecosystems, engineering systems, or biological processes.

What you will learn

In this module, you will:

• study the qualitative behaviour of solutions to ODEs and PDEs
• explore stability through linearisation and phase space analysis
• analyse periodic orbits and bifurcations, including Hopf bifurcation
• use mathematical models to describe physical and biological processes

By the end of this module, you will be able to:

• distinguish between finite- and infinite-dimensional dynamical systems
• use qualitative methods to analyse nonlinear differential equations
• apply bifurcation theory and stability tools to model long-term system behaviour
• interpret the solutions of ODEs and PDEs in the context of real-world systems

Assignments / assessment

• Coursework (20%)
• Final written exam, two hours (80%)

Teaching methods / timetable

• Lectures
  • On campus teaching of theory with worked examples
• Tutorials
  • Focused sessions for problem-solving and feedback