Study information

Computational Nonlinear Dynamics - 2025 entry

MODULE TITLEComputational Nonlinear Dynamics CREDIT VALUE15
MODULE CODEMTH3039 MODULE CONVENERDr George Datseris (Coordinator)
DURATION: TERM 1 2 3
DURATION: WEEKS 11 0 0
Number of Students Taking Module (anticipated) 19
DESCRIPTION - summary of the module content

Nonlinear dynamical systems are used in almost all disciplines: from applied mathematics to physics of any kind, to biology, chemistry, sociology, ecology, economics, engineering, and computer science. Their nonlinearity makes them so successful, however, it comes at a price: practically nothing about nonlinear systems can be estimated analytically.

Computational nonlinear dynamics is the process of studying nonlinear dynamical systems by devising and running numerical algorithms. Throughout this module we will be discussing many interesting aspects of nonlinear dynamical systems, such as multistability, deterministic chaos, and critical transitions (see Syllabus). For each aspect, we will be devising algorithms that can identify it for arbitrary dynamical systems. In the coursework we will be creating computer programs that apply these algorithms to dynamical systems. Sometimes we may have data obtained directly from some real-world source instead of a dynamical system, but the process will be the same.

As such, this module will not only teach you how nonlinear dynamical systems behave, and how to understand them, but also how to design computer algorithms that fulfil a certain goal. This is an invaluable experience for your future employability in a world increasingly reliant on programming.

During the course, we will be applying nonlinear dynamics to study and understand real world phenomena, such as: climate tipping points, neural dynamics and excitability, chaos in planetary motion and the three body problem, ecosystem dynamics and extinction events, complexity of the stock market, and more.

The module will introduce you to Julia, a modern open-source programming language specifically designed for scientific computing. The course will also teach you how to use the DynamicalSystems.jl software library, the largest and most accessible software for computational nonlinear dynamics ever created. (2 lecture hours will be dedicated to Julia and DynamicalSystems.jl).

This module welcomes students from all disciplines, mathematics and beyond, provided they have had introductory courses in differential equations, linear algebra, and programming.

Pre-requisite modules: MTH2005 Modelling Theory and Practice OR MTH1003 Mathematical Modelling.

AIMS - intentions of the module

You will learn to combine your previously acquired knowledge from stages 1 and 2, and the new knowledge obtained in this module, as well as your programming skills to solve problems related with nonlinear dynamics that help better understand the real world, for example, climate, ecology, neuroscience, and more. You will also learn how to use a state-of-the-art tool for computational nonlinear dynamics: DynamicalSystems.jl.

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. understand how nonlinear dynamical systems behave in different computational and real-world scenarios

  2. develop and numerically solve computational algorithms for dynamical systems

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

Discipline Specific Skills and Knowledge:

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

  2. have a plethora of computational and mathematical tools that you can use to analyse dynamical systems approximating various real-world scenarios

Personal and Key Transferable/ Employment Skills and Knowledge:

  1. apply computational and programming skills to problem-solving

  2. develop a project independently and with appropriate time management

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

Each year this course repeats there is a fixed skeleton of topics, and a second larger pool of topics from which a small selection is taught, varying each year.

Fixed skeleton:

  • Introduction to dynamical systems and the two most common types: differential and difference equations. The state space.

  • Introduction to Julia and DynamicalSystems.jl. Computation and visualization of trajectories of dynamical systems.

  • Multistability and basins of attraction. Stability of dynamical systems: local and nonlocal.

  • Variation of parameter(s). Bifurcations. Global Continuation: tracking any type of stability of a dynamical system as a parameter is varied.

  • Deterministic chaos. Lyapunov exponents.

  • Applying nonlinear dynamics in a real-world example (e.g., climate, neuroscience, ecology, lasers, …). Example varying each year.

Auxiliary topics (small selection each year):

  • Nonlinear timeseries analysis: entropy and complexity.

  • Delay Coordinate Embedding and more nonlinear timeseries analysis.

  • Fractals. Fractal dimensions. Numeric estimation of fractal dimensions.

  • Nonlinear dynamics on networks. Complex systems. Agent based modelling.

  • Periodically driven oscillators. Coupled dynamical systems. Synchronization.

  • Spatiotemporal nonlinear dynamics. Numerical differentiation of partial differential equations. Spatiotemporal chaos.

  • Linear continuation (traditional bifurcation analysis). Linear continuation of limit cycles and connecting orbits.

  • Non-autonomous dynamical systems. Critical transitions, tipping points. Tipping points in the real world, in climate and ecology.

  • Parameter sensitivity. Important system parameters. Fitting parameters to observed data. Data assimilation.

  • Natural measures. Conservative dynamical systems. Billiards.

  • Recurrences. Extreme events. Identifying and analysing extremes in timeseries data.

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 the assessment e.g. duration/length

ILOs assessed 

Feedback method

Coursework 1

33

4 weeks (~16 hours of study)

1-4

Ongoing during lab sessions, written after marking

Coursework 2

33

4 weeks (~16 hours of study)

1-4

Ongoing during lab sessions, written after marking

Coursework 3

34

4 weeks (~16 hours of study)

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 Feedback method
Coursework 1 Coursework 1 (4 weeks (~16 hours of study), 33%) 1-4 Referral/deferral period Ongoing during lab sessions, written after marking
Coursework 2 Coursework 2 (4 weeks (~16 hours of study), 33%) 1-4 Referral/deferral period Ongoing during lab sessions, written after marking
Coursework 3 Coursework 3 (4 weeks (~16 hours of study), 34%) 1-5 Referral/deferral period Ongoing during lab sessions, written after marking

 

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:

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

Other resources:

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 24th March 2025
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.