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MATH3091: DYNAMICAL SYSTEMS III

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Type	Open
Level	3
Credits	20
Availability	Available in 2023/24
Module Cap
Location	Durham
Department	Mathematical Sciences

Prerequisites

Complex Analysis II (MATH2011) and Analysis in ManyVariables II (MATH2031)

Corequisites

None.

Excluded Combinations of Modules

None.

Aims

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

Content

Smooth ODEs: existence and uniqueness ofsolutions.
Autonomous ODEs: orbits, equilibrium and periodicsolutions.
Linearisation: Hartman-Grobman, stable-manifold theorems,phase portraits for non-linear systems, stability ofequilibrium.
Flow, Fixed points: Brouwer's Theorem, periodic solutions,Poincare-Bendixson and related theorems, orbitalstability.
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 solvenovel and/or complex problems in Dynamical Systems.
have a systematic and coherent understanding of theoreticalmathematics in the field of Dynamical Systems.
have acquired a coherent body of knowledge of these subjectsdemonstrated through one or more of the following topic areas: (mostlysecond-order) non-linear ODE's applied to the following:
a smooth finite dimensional dynamical system as a directionfield on a manifold.
critical points and cycles as attractors, and theirinteraction via local bifurcations of co-dimension one.
Local linearization, Lyapunov functions, the Poincare andBendixson theorems of plane topology, and the Hopf bifurcationtheorem.

Subject-specific Skills:

In addition students will have specialised mathematicalskills 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 theapplication of the theory to practical examples.
Assignments for self-study develop problem-solving skills andenable students to test and develop their knowledge andunderstanding.
Formatively assessed assignments provide practice in theapplication of logic and high level of rigour as well as feedback forthe students and the lecturer on students' progress.
The end-of-year examination assesses the knowledge acquiredand the ability to solve predictable and unpredictableproblems.

Teaching Methods and Learning Hours

Activity	Number	Frequency	Duration	Total
Lectures	42	2 per week for 20 weeks and 2 in term 3	1 Hour	42
Problems Classes	8	Four in each of terms 1 and 2	1 Hour	8
Preparation and Reading				150
Total				200

Summative Assessment

Component: Examination		Component Weighting: 100%
Element	Length / Duration	Element Weighting	Resit Opportunity
Written examination	3 Hours	100

Formative Assessment

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

More information

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