Undergraduate Course: Applied Dynamical Systems (MATH11140)
|School||School of Mathematics
||College||College of Science and Engineering
|Credit level (Normal year taken)||SCQF Level 11 (Year 5 Undergraduate)
||Availability||Available to all students
|Summary||This course presents an overview of the theory of dynamical systems, with the aim of showcasing their widespread applicability in the modelling of biological, medical, and chemical phenomena in areas such as population biology, gene expression, mathematical physiology, enzyme kinetics, and neuronal dynamics. Substantial use will be made of the theory of ordinary differential equations; the course hence builds on Honours Differential Equations (MATH10066).
A suggested syllabus for the course is as follows. Diffeomorphisms and flows. Hyperbolicity and structural stability. Invariant manifolds. Bifurcations and normal forms. Asymptotic theory. Singular perturbations and multiple-scale dynamics. Attractors. Chaos and fractals. Applications from the life sciences.
Dynamical systems theory has a rich and varied history. Established with the invention of differential calculus in the 1600s by Newton and Leibniz, dynamics was at first primarily concerned with the exact solution of differential equations. It was not until the 1800s that Poincaré's work triggered a paradigm shift towards a more qualitative point of view: instead of exact formulae, he emphasised a geometric approach, enquiring for instance about the long-term behaviour of solutions or their dependence on parameters. Poincaré also theorised the possibility of chaos, which is characterised by the sensitive dependence of solutions on their initial conditions, thus making forecasts unreliable.
The theory of dynamical systems has since matured, through its centuries-long history, and has found innumerable applications, not least in the life sciences; the resulting branch of mathematics is often broadly referred to as mathematical biology, and will be the focus of this course. The applications covered may include Michaelis-Menten-type enzyme kine+cs, growth and control of brain tumours, travelling fronts in the Belousov-Zhabotinsky reaction, the dynamics of infectious diseases, waves of pursuit and evasion in predator-prey systems, oscillations in population-based models, and the FitzHugh-Nagumo model for neuronal impulses.
Information for Visiting Students
|Pre-requisites||Prior knowledge of ordinary differential equations is essential. Visiting students are advised to check that they have studied the material covered in the syllabus of each prerequisite course before enrolling.
|High Demand Course?
Course Delivery Information
|Academic year 2022/23, Available to all students (SV1)
|Learning and Teaching activities (Further Info)
Lecture Hours 22,
Seminar/Tutorial Hours 5,
Summative Assessment Hours 2,
Programme Level Learning and Teaching Hours 2,
Directed Learning and Independent Learning Hours
|Assessment (Further Info)
|Additional Information (Assessment)
||Coursework 25%, Examination 75%
||Hours & Minutes
|Main Exam Diet S2 (April/May)||2:00|
On completion of this course, the student will be able to:
- Describe qualitative properties of dynamical systems, such as their bifurcation structure, by constructing invariant manifolds and normal forms.
- Perform stability analyses of simple invariant sets in low-dimensional systems and understand how Krylov subspace methods can be used to both find solutions and assess their stability in very large systems.
- Undertake unsupervised reading of course-relevant material, and demonstrate an understanding of key concepts from dynamical systems theory.
- Adapt standard dynamical models from the biological and physical sciences, and interpret them in the context of the modelled natural phenomena.
- Apply results and techniques from dynamical systems theory to solve unseen assignments that extend concepts and examples studied in the course.
|The course is loosely based on a selection of material from the following books, which students are encouraged to consult for background and further reading: |
Stability, instability & chaos by Glendinning
S.H.Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering (Studies in Nonlinearity), Westview Press, 2001.
P. G. Drazin, Nonlinear Systems, Cambridge University Press, 1992.
L.Edelstein-Keshet, Mathematical Models in Biology (Classics in Applied Mathematics), Society for Industrial and Applied Mathematics, 2005.
J.D. Murray, Mathematical Biology I: An Introduction (Interdisciplinary Applied Mathematics), Springer-Verlag, 2007.
J.D. Murray, Mathematical Biology II: Spatial Models and Biomedical Applications (Interdisciplinary Applied Mathematics), Springer-Verlag, 2008.
|Graduate Attributes and Skills
|Course organiser||Dr Jacob Page
|Course secretary||Mr Martin Delaney
Tel: (0131 6)50 6427