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 4 Undergraduate)
||Availability||Available to all students
|Summary||The course presents an overview of the theory of dynamical systems for (joint) Honours students in Mathematics, with a strong emphasis on applications from the life sciences. Substantial use will be made of the theory of ordinary differential equations; the course hence builds on Honours Differential Equations (MATH10066). The mathematical theory of dynamical systems is applied in areas such as population biology, gene expression, mathematical physiology, enzyme kinetics, and neuronal modelling, among other examples, with the aim of showcasing their widespread applicability in the modelling of biological, medical, and chemical phenomena.
Diffeomorphisms and flows. Hyperbolicity and structural stability. Invariant manifolds. Bifurcation theory and normal forms. Attractors. Chaos and fractals. Singular perturbation theory and multiple-scale dynamics. Asymptotic theory of differential equations. Applications and examples from the life sciences.
Dynamical systems theory has a rich and varied history. Established with the invention of differential calculus in the 1600s, dynamics was at first primarily concerned with the exact solution of differential equations. The breakdown of this quantitative point of view came with the realisation that the so-called three-body problem could not be solved exactly: while Newton had calculated the motion of the earth around the sun, subsequent generations of scientists did not succeed in extending his solution to describe the motion of the sun, the earth, and the moon.
It was not until the 1800s that Poincaré's work triggered a paradigm shift towards a more qualitative view: instead of trying to determine exact solution formulae, he emphasised a geometric approach, enquiring, for instance, about the long-time behaviour of such solutions, or their dependence on system parameters. Poincaré also theorised the possibility of chaos, which is characterised by the sensitive dependence of a system on its initial conditions, thus rendering long-term forecasts impossible. Chaos theory received an immense boost through the invention of high-speed computing in the 1950s, which allowed for the first time for the systematic numerical simulation of complex dynamical systems. Computers were also instrumental in the popularisation of fractals in the 1970s, making these beautiful mathematical objects accessible to a wide audience.
Perhaps most importantly, the theory of dynamical systems has matured, through its centuries-long history, from an appendage of celestial mechanics to a field in its own right. In the process, it 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 in this course. The examples covered may include, but will not be restricted to, Michaelis-Menten-type enzyme kinetics, growth and control of brain tumours, travelling fronts in the Belousov-Zhabotinskii 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 neural impulses.
The course will be delivered through two weekly lecture hours and one bi-weekly workshop hour. It is intended to employ Inquiry-Based Learning techniques, in particular in workshop sessions; examples may include group-based activities and presentation by students. Workshops will serve to cement ideas and concepts, and to elaborate on applications introduced in lectures. The course will be assessed 80% on a final examination and 20% on coursework.
A suggested syllabus for the course is as follows. Dynamical systems: diffeomorphisms and flows. Equilibrium points: hyperbolicity and (structural) stability. Invariant manifold theory: stable, unstable, and centre manifolds. Bifurcation theory: introduction and examples. Normal forms: calculation and interpretation. Attractors and basins of attraction. Chaotic systems and fractals. Singular perturbation theory and multiple-scale dynamics. Asymptotic theory of differential equations. Dynamical systems in the life sciences: applications and examples.
Information for Visiting Students
|High Demand Course?
Course Delivery Information
|Academic year 2017/18, 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 20%, Examination 80%
||Hours & Minutes
|Main Exam Diet S1 (December)||MATH11140 Applied Dynamical Systems||2:00|
On completion of this course, the student will be able to:
- Identify defining characteristics of discrete and continuous dynamical systems, such as equilibria and their hyperbolicity, attractors and their basins of attraction, and their bifurcations.
- Construct and interpret structures crucial to the qualitative analysis of discrete and continuous nonlinear systems, such as their normal forms and invariant manifolds.
- Identify the potential for multiple-scale dynamics in dynamical systems, discriminate between regular and singular perturbations, and apply perturbation techniques as appropriate.
- Identify, critique, and adapt standard differential equation models from the biological and physical sciences, and formulate and validate such models to describe simple phenomena therefrom.
- Apply and adapt results and techniques from the course both co-operatively and individually to solve unseen assignments that extend concepts and examples studied in class.
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:
S.H.Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering (Studies in Nonlinearity), Westview Press, 2001.
L.Edelstein-Keshet, Mathematical Models in Biology (Classics in Applied Mathematics), Society for Industrial and Applied Mathematics, 2005.
D.K. Arrowsmith and C.M. Place, An Introduction to Dynamical Systems,
Cambridge University Press, 1990.
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 Nikola Popovic
Tel: (0131 6)51 5731
|Course secretary||Mrs Alison Fairgrieve
Tel: (0131 6)50 5045