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 4 Undergraduate)	Availability	Available to all students
SCQF Credits	10	ECTS Credits	5
Summary	Course in Dynamical Systems for (joint) Honours students in Mathematics, with a strong emphasis on applications from the life sciences, such as in population biology, gene expression, mathematical physiology, enzyme kinetics, and neuronal modelling, among other examples. Diffeomorphisms and flows, and their (local) properties. Structural stability and hyperbolicity. Invariant manifold techniques. Bifurcation theory and normal forms. Attractors. Chaos and fractals. Multiple-scale ("fast-slow") dynamics and perturbation techniques. Asymptotic theory of differential equations. Applications and examples from the life sciences.
Course description	A suggested syllabus for this course is as follows: Week 1 : Diffeomorphisms and flows; equilibrium points and their stability; hyperbolicity and structural stability. Week 2 : Stable, unstable, and centre manifolds, and their significance. Week 3 : Examples of dynamical systems in the life sciences, part I Week 4 : Introduction to bifurcation theory, and some standard bifurcations. Week 5 : Normal forms, and their calculation and interpretation. Week 6 : Examples of dynamical systems in the life sciences, part II Week 7 : Attractors and basins of attraction. Week 8 : Chaotic systems and fractals. Week 9 : Introduction to singular perturbation theory and multiple-scale dynamics. Weeks 10 - 11 : Examples of dynamical systems in the life sciences, part III. The examples covered may include ż but will not be restricted to ż 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.

Entry Requirements (not applicable to Visiting Students)
Pre-requisites	Students MUST have passed: Honours Differential Equations (MATH10066) AND Honours Complex Variables (MATH10067)	Co-requisites
Prohibited Combinations	Students MUST NOT also be taking Dynamical Systems (MATH11027)	Other requirements	None

Information for Visiting Students
Pre-requisites	None

Course Delivery Information

Academic year 2015/16, Available to all students (SV1)		Quota: None
Course Start	Semester 1
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 S1 (December)	MATH11140 Applied Dynamical Systems		2:00

Academic year 2015/16, Part-year visiting students only (VV1)		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
No Exam Information

Learning Outcomes
1. Facility in classifying equilibrium points of discrete and continuous systems. 2. Ability to identify hyperbolicity and structural stability. 3. Facility in constructing stable, unstable, and centre manifolds for nonlinear systems. 4. Ability to characterise standard bifurcations, such as saddle-node, Hopf, or flip bifurcations. 5. Ability to calculate and interpret normal forms for discrete and continuous nonlinear systems. 6. Ability to define attractors and basins of attraction. 7. Knowledge of the notions of chaotic systems and fractal geometry. 8. Facility in discriminating between regular and singular perturbations. 9. Knowledge of multiple-scale dynamics and perturbation techniques for fast-slow systems. 10. Knowledge of standard differential equation models from the biological and physical sciences. 11. Ability to formulate differential equation models for simple phenomena from the biological and physical sciences.

Reading List
Recommended reading: 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.