Study information

Computational Nonlinear Dynamics - 2024 entry

MODULE TITLEComputational Nonlinear Dynamics CREDIT VALUE15
MODULE CODEMTH3039 MODULE CONVENERProf Jan Sieber (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.
 
In this module you will gradually assemble a toolbox of small programs  and then use these programs to study nonlinear dynamical systems with complicated behaviour (for example, chaos) and transitions between behaviours  as parameters are changed. In the end you will have solved some 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) the existence of chaos in models, for example, for the forced pendulum or neuron models. Half of the contact time will be supervised lab sessions during which you will get support to get started on the problems.  Additional programming experience is recommended for students taking this module.
 
The module will introduce you to Julia, a modern programming language specifically designed for scientific computing. All programming tasks will be based on Julia.
 
Pre-requisites: MTH2005 OR MTH1003   

 

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, neuroscience or electronic circuitry.
 
The problems will all come from applications encountered in academic research (e.g., lasers, mechanical systems, population dynamics, fluid dynamics, Earth Science, climate modelling). 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);

- [*] Finding attractors and basis of attraction generically in multistable dynamical systems (+ introduction to DynamicalSystems.jl)
 
- [*] Global stability analysis and global stability continuation, and introduction to “tipping points” (transitions between attractors)

- [*] 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 33 Guided Independent Study 117 Placement / Study Abroad 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      

 

SUMMATIVE ASSESSMENT (% of credit)
Coursework 100 Written Exams 0 Practical Exams 0
DETAILS OF SUMMATIVE ASSESSMENT
Form of Assessment % of Credit Size of Assessment (e.g. duration/length) ILOs Assessed Feedback Method
Coursework 1 40% 300 lines of code, 500 words documentation (including graphs etc) 1-5 Ongoing during lab sessions, written after marking
Coursework 2 60% 600 lines of code, 1000 words documentation (including graphs etc) 1-5 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 1 1-5 Referral/Deferral Period
Coursework 2 Coursework 2 1-5 Referral/Deferral Period

 

RE-ASSESSMENT NOTES
Reassessment will be by coursework in the failed or deferred element only.  Deferral – if you have been deferred for any assessment, you will be expected to complete relevant deferred assessments as determined by the Mitigation Committee. The mark given for re-assessment taken as a result of deferral will not be capped and will be treated as it would be if it were your first attempt at the assessment. 
 
Referral – if you have failed the module overall (i.e. a final overall module mark of less than 40%) you will be required to undertake re-assessments as described in the table above for any of the original assessments that you failed. The mark given for a re-assessment taken as a result of referral will be capped at 40%.
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

Basic reading:

Strogatz, S. Nonlinear dynamics and Chaos. Springer. 978-0-387-21906-6

Allgower EL and Georg K. Introduction to Numerical Continuation Methods. Springer. 089871544X
 
Kelley CT. Solving Nonlinear Equations with Newtons Method. SIAM. 0-89871-546-6
 
Datseris , G, Parlitz U. Nonlinear Dynamics: a concise introduction interlaced with code. Springer     2022. 978-3-030-91032-7 https://link.springer.com/book/10.1007/978-3-030-91032-7.

Web based and Electronic Resources:

ELE

http://www.dynamicalsystems.org/tu/cm/

Introduction to Julia workshop: Julia Zero2Hero 
https://github.com/Datseris/Zero2Hero-JuliaWorkshop

Reading list for this module:

There are currently no reading list entries found for this module.

CREDIT VALUE 15 ECTS VALUE 7.5
PRE-REQUISITE MODULES MTH2005, MTH1003
CO-REQUISITE MODULES
NQF LEVEL (FHEQ) 6 AVAILABLE AS DISTANCE LEARNING No
ORIGIN DATE Tuesday 10th July 2018 LAST REVISION DATE Monday 4th March 2024
KEY WORDS SEARCH None Defined

Please note that all modules are subject to change, please get in touch if you have any questions about this module.