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MA998 - Applied Dynamical Systems (12 Cats)

Lecturer: Mike Tildesley (Life Sciences)

Module Aims

This is one of 4 core modules for the new MSc in Mathematics of Systems. The module aims to demonstrate some of the techniques used in the modern theory of dynamical systems and illustrate their application to real-world systems, from epidemiology and cancer modelling to population dynamics. By the end of the module, students will be familiar with Nonlinear dynamical systems (continuous and discrete), deterministic chaos, bifurcation theory, system dynamics, partial differential equations (reaction-diffusion models).

Syllabus

1. Introduction to Dynamical Models

  • Differential and difference eq’ns
  • First-order systems, Bifurcations, Flows on the circle
  • Second-order systems (linear and nonlinear), Linear stability analysis, Lyapunov functions

2. Nonlinear Oscillations

  • Limit cycles in two dimensions, Poincaré -Bendixson theorem
  • Linear stability of limit cycle, Poincaré sections, Floquet theory
  • Relaxation & Coupled oscillators

3. Chaos in continuous and discrete systems

  • Strange and chaotic attractors, Lyapunov exponents
  • Fractal boundaries
  • Routes to chaos
  • State space reconstruction
  • Global bifurcations, homoclinic and heteroclinic connections

4. Reaction-diffusion systems

  • Travelling wave solutions
  • Pattern formations

Illustrative Bibliography

  1. Strogatz, S.H. (2000). Nonlinear dynamics and chaos. (Westview Press).
  2. Glendinning, P. (1994). Stability, instability and chaos. (Cambridge University Press).
  3. Alligood, K.T., Sauer, T.D. and Yorke, J. A. (1996). Chaos: An Introduction to Dynamical Systems. (Springer).
  4. Ott, E. (2002). Chaos in dynamical systems. (Cambridge University Press).
  5. Murray, J. D. (2002) Mathematical Biology. (Springer).

Teaching

See main calendar for timetable

  • Per week: 2 x 2 hours of lectures, 2 x 2 hours of classwork
  • Duration: 5 weeks (second half of term 1)

Assessment

For deadlines see Module Resources page

  • Assignment (25%)
  • Class test (25%)
  • Oral examination (50%)