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Faculty Handbook 2022-2023

# Module Description

Please ensure you check the module availability box for each module outline, as not all modules will run each academic year.

## Department: Mathematical Sciences

### MATH3091: DYNAMICAL SYSTEMS III

Type Level Credits Availability Module Cap Open 3 20 Available in 2022/23 Durham

#### Prerequisites

• Complex Analysis II (MATH2011) and Analysis in Many Variables II (MATH2031)

• None.

• 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 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.

Attendance at all activities marked with this symbol will be monitored. Students who fail to attend these activities, or to complete the summative or formative assessment specified above, will be subject to the procedures defined in the University's General Regulation V, and may be required to leave the University