# Mathematics

## MTH3039 - Computational Nonlinear Dynamics (2019)

MODULE TITLE CREDIT VALUE Computational Nonlinear Dynamics 15 MTH3039 Dr James Rankin (Coordinator)
DURATION: TERM 1 2 3
DURATION: WEEKS 11 0 0
 Number of Students Taking Module (anticipated) 19
DESCRIPTION - summary of the module content

Most mathematical problems in engineering and science lead to systems of nonlinear equations that cannot be solved with pencil and paper, and where a numerical approach does not give a complete answer. In this module you will use theory and mathematical methods from Stages 1 and 2 (calculus, dynamics, differential equations, numerics and scientific computing) to solve realistic problems as they occur in nonlinear dynamics in engineering and science.

You will gradually assemble a toolbox of small programs (in MATLAB) and then use these programs to study nonlinear dynamical systems with complicated behaviour (for example, chaos). In the end you will have solved some fairly complex problems from scratch, using tools developed and written by yourself during the module. For example, you will have proved (or, at least given robust numerical evidence for) chaos in the forced pendulum. Half of the contact time will be supervised lab sessions during which you will get support to get started on the problems.

Pre-requisites: MTH1003 Mathematical Modelling or equivalent programming experience.

AIMS - intentions of the module

You will learn to combine your previously acquired knowledge from Stages 1 and 2 (specifically calculus and modelling) and your programming skills to solve nonlinear problems as they occur in real-world applications, for example, in mechanics, lasers, climate, ecology, chemistry, biology, or electronic circuitry.

The problems will all come from applications encountered in academic research (e.g., lasers, mechanical systems, population dynamics, fluid dynamics). The module does NOT intend to teach you  how to use particular state-of-the-art research tools (such as AUTO), but will rather guide you to develop idealized versions of these tools from scratch. The module will give you the opportunity to solve problems that are beyond the reach of exam-based assessment but short of individual research projects.

INTENDED LEARNING OUTCOMES (ILOs) (see assessment section below for how ILOs will be assessed)

On successful completion of this module you should be able to:

Module Specific Skills and Knowledge

1. solve high-dimensional nonlinear systems that depend on parameters

2. apply mathematical and computational methods previously learned to study dynamical systems from applications

Discipline Specific Skills and Knowledge

3.  solve mathematical problems of medium complexity (that is, requiring combination of a range of computational and mathematical techniques)

Personal and Key Transferable / Employment Skills and Knowledge

4. apply computational and programming skills to problem-solving

5. develop a project independently and with appropriate time management.

SYLLABUS PLAN - summary of the structure and academic content of the module

- Implicit function theorem and Newton iteration in arbitrary dimensions;

- Numerical differentiation and numerical solution of initial-value problems of ODEs in arbitrary dimensions;

- Solution of parameter-dependent nonlinear problems;

- Computation and visualisation of phase portraits and their structural stability;

- Finding and tracking singularities (bifurcations: saddle-node and Hopf bifurcations);

- [*] Regularity and discretisation of ODE boundary-value problems;

- [*] Tracking of periodic orbits, starting from a Hopf bifurcation, and some of their bifurcations;

- [*] computation of basins of attraction for equilibria of autonomous systems or periodic points of periodically forced systems;

- [*] Lyapunov exponents, stable and unstable manifolds of periodic points and detection of their homoclinic tangles in periodically forced systems.

[*] only a selection of these topics will be covered, varying each year

LEARNING AND TEACHING
LEARNING ACTIVITIES AND TEACHING METHODS (given in hours of study time)
 Scheduled Learning & Teaching Activities Guided Independent Study Placement / Study Abroad 33 117 0
DETAILS OF LEARNING ACTIVITIES AND TEACHING METHODS
 Category Hours of study time Description Scheduled learning and teaching activity 18 Lectures Scheduled learning and teaching activity 15 Computer lab sessions for work on problems Guided independent study 87 Independent work on problems Guided independent study 30 Study of notes and wider reading

ASSESSMENT
FORMATIVE ASSESSMENT - for feedback and development purposes; does not count towards module grade
Form of Assessment Size of Assessment (e.g. duration/length) ILOs Assessed Feedback Method
n/a (but see feedback for Summative Assessment)

SUMMATIVE ASSESSMENT (% of credit)
 Coursework Written Exams Practical Exams 100 0 0
DETAILS OF SUMMATIVE ASSESSMENT
Form of Assessment % of Credit Size of Assessment (e.g. duration/length) ILOs Assessed Feedback Method
Coursework 1 35 300 lines of code, 500 words documentation (including graphs etc) All Ongoing during lab sessions, written after marking
Coursework 2 65 600 lines of code, 1000 words documentation (including graphs etc) All Ongoing during lab sessions, written after marking

DETAILS OF RE-ASSESSMENT (where required by referral or deferral)
Original Form of Assessment Form of Re-assessment ILOs Re-assessed Time Scale for Re-assessment
Coursework 1 Coursework All Ref/Def Period
Coursework 2 Coursework All Ref/Def Period

RE-ASSESSMENT NOTES

All referred/deferred assessments will normally be by assignment.

RESOURCES
INDICATIVE LEARNING RESOURCES - The following list is offered as an indication of the type & level of
information that you are expected to consult. Further guidance will be provided by the Module Convener

ELE: http://vle.exeter.ac.uk/

Web based and Electronic Resources:

Other Resources:

Reading list for this module:

Type Author Title Edition Publisher Year ISBN Search
Set Kuznetsov Y Elements of Applied Bifurcation Theory Springer 978-0-387-21906-6 [Library]
Set Allgower EL and Georg K Introduction to Numerical Continuation Methods Springer 089871544X [Library]
Set Kelley CT Solving Nonlinear Equations with Newtons Method SIAM 0-89871-546-6 [Library]
CREDIT VALUE ECTS VALUE 15 7.5
PRE-REQUISITE MODULES MTH1003
NQF LEVEL (FHEQ) AVAILABLE AS DISTANCE LEARNING 6 No Tuesday 10 July 2018 Tuesday 02 July 2019
KEY WORDS SEARCH None Defined