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MATH3091: DYNAMICAL SYSTEMS III

Please ensure you check the module availability box for each module outline, as not all modules will run in each academic year. Each module description relates to the year indicated in the module availability box, and this may change from year to year, due to, for example: changing staff expertise, disciplinary developments, the requirements of external bodies and partners, and student feedback. Current modules are subject to change in light of the ongoing disruption caused by Covid-19.

Type Open
Level 3
Credits 20
Availability Available in 2024/2025
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

ActivityNumberFrequencyDurationTotalMonitored
Lectures422 per week for 20 weeks and 2 in term 31 Hour42 
Problems Classes8Four in each of terms 1 and 21 Hour8 
Preparation and Reading150 
Total200 

Summative Assessment

Component: ExaminationComponent Weighting: 100%
ElementLength / DurationElement WeightingResit Opportunity
Written examination3 Hours100 

Formative Assessment

Eight assignments to be submitted.

More information

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