This is an archived handbook for a previous year. The current undergraduate and postgraduate handbooks can be viewed using the links above.

Undergraduate Handbook 2010-2011 (archived)

Module MATH3091: DYNAMICAL SYSTEMS III

Department: Mathematical Sciences

MATH3091: DYNAMICAL SYSTEMS III

Type	Level	Credits	Availability	Module Cap	Location
Open	3	20	Available in 2010/2011	None.	Durham

Pre-requisites

Complex Analysis II (MATH2011) and Analysis in Many Variables II (MATH2031); OR Contours and Symmetries II (MATH2111) and Analysis in Many Variables II (MATH2031); OR Contours and Hyperbolic Geometry II (MATH2121) and Analysis in Many Variables II (MATH2031); OR Contours and Actuarial Mathematics II (MATH2171) and Analysis in Many Variables II (MATH2031); OR Contours and Probability II (MATH2561) and Analysis in Many Variables II (MATH2031).

Corequisites

None.

Excluded Combinations

None.

Aims

To provide an introduction to modern analytical methods for nonlinear ordinary differential equations in real variables.

Content

Smooth ODEs: existence and uniqueness of solutions.
Autonomous ODEs: orbits, equilibrium and periodic solutions.
Linearisation: Hartman-Grobman, stable-manifold theorems, phase portraits for non-linear systems, stability of equilibrium.
Flow, Fixed points: Brouwer's Theorem, periodic solutions, Poincare-Bendixson and related theorems, orbital stability.
Hopf and other local bifurcations from equilibrium, bifurcations from periodic solutions.

Learning Outcomes

Subject-specific Knowledge

By the end of the module students will: be able to solve novel and/or complex problems in Dynamical Systems.
have a systematic and coherent understanding of theoretical mathematics in the field of Dynamical Systems.
have acquired a coherent body of knowledge of these subjects demonstrated through one or more of the following topic areas: (mostly second-order) non-linear ODE's applied to the following:
a smooth finite dimensional dynamical system as a direction field on a manifold.
critical points and cycles as attractors, and their interaction via local bifurcations of co-dimension one.
Local linearization, Lyapunov functions, the Poincare and Bendixson theorems of plane topology, and the Hopf bifurcation theorem.

Subject-specific Skills

In addition students will have specialised mathematical skills in the following areas which can be used with minimal guidance: Modelling.

Key Skills

Modes of Teaching, Learning and Assessment and how these contribute to the learning outcomes of the module

Lectures demonstrate what is required to be learned and the application of the theory to practical examples.
Assignments for self-study develop problem-solving skills and enable students to test and develop their knowledge and understanding.
Formatively assessed assignments provide practice in the application of logic and high level of rigour as well as feedback for the students and the lecturer on students' progress.
The end-of-year examination assesses the knowledge acquired and the ability to solve predictable and unpredictable problems.

Teaching Methods and Learning Hours

Activity	Number	Frequency	Duration	Total Hours
Lectures	40	2 per week for 19 weeks and 2 in term 3	1 Hour	40
Preparation and Reading				160
Total				200

Summative Assessment

Component	Component Weighting	Element	Length/Duration	Element Weighting	Resit Opportunity
Examination	100%	three hour written examination		100%

Formative Assessment

Four written assignments to be assessed and returned. Other assignments are set for self-study and complete solutions are made available to students.

Students who do not attend monitored activities shown under Teaching Methods and Learning Hours, or who fail to complete the summative or formative assessment(s) specified above, may be subject to the Academic Progress procedures defined in the University's General Regulation V, and may be required to leave the University.