Undergraduate Course: Applied Dynamical Systems (MATH11140)
Course Outline
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 
SCQF Credits  10 
ECTS Credits  5 
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 multiplescale dynamics. Attractors. Chaos and fractals. Applications from the life sciences. 
Course description 
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 PoincarĂ©'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 longterm behaviour of solutions or their dependence on parameters. PoincarĂ© 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 centurieslong 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 MichaelisMententype enzyme kine+cs, growth and control of brain tumours, travelling fronts in the BelousovZhabotinsky reaction, the dynamics of infectious diseases, waves of pursuit and evasion in predatorprey systems, oscillations in populationbased models, and the FitzHughNagumo model for neuronal impulses.

Entry Requirements (not applicable to Visiting Students)
Prerequisites 
Students MUST have passed:
Honours Differential Equations (MATH10066)

Corequisites  
Prohibited Combinations  
Other requirements  Note that PGT students on School of Mathematics MSc programmes are not required to have taken prerequisite courses, but they are advised to check that they have studied the material covered in the syllabus of each prerequisite course before enrolling. 
Information for Visiting Students
Prerequisites  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? 
Yes 
Course Delivery Information

Academic year 2024/25, Available to all students (SV1)

Quota: None 
Course Start 
Semester 2 
Timetable 
Timetable 
Learning and Teaching activities (Further Info) 
Total Hours:
100
(
Lecture Hours 22,
Seminar/Tutorial Hours 5,
Summative Assessment Hours 2,
Programme Level Learning and Teaching Hours 2,
Directed Learning and Independent Learning Hours
69 )

Assessment (Further Info) 
Written Exam
80 %,
Coursework
20 %,
Practical Exam
0 %

Additional Information (Assessment) 
Coursework 20%, Examination 80% 
Feedback 
Not entered 
Exam Information 
Exam Diet 
Paper Name 
Hours & Minutes 

Main Exam Diet S2 (April/May)   2:00  
Learning Outcomes
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 lowdimensional 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 courserelevant 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.

Reading List
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.EdelsteinKeshet, 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), SpringerVerlag, 2007.
J.D. Murray, Mathematical Biology II: Spatial Models and Biomedical Applications (Interdisciplinary Applied Mathematics), SpringerVerlag, 2008. 
Additional Information
Graduate Attributes and Skills 
Not entered 
Keywords  ADS 
Contacts
Course organiser  Dr Jacob Page
Tel:
Email: Jacob.Page@ed.ac.uk 
Course secretary  Mr Martin Delaney
Tel: (0131 6)50 6427
Email: Martin.Delaney@ed.ac.uk 

