What will I learn on this module?
The module aims to present an introduction to Dynamical Systems and associated transferable skills, providing the students with tools and techniques needed to understand the dynamics of those systems. You will analyse non-linear ordinary differential equations and maps, focusing on autonomous systems, and will learn analytical and computational methods to solve them. This module offers the additional opportunity of research-orientated learning through a hands-on approach to selected research-based problems.
Topics may include (note this is indicative rather than prescriptive):
1. Autonomous linear systems, fixed points and their classification.
2. 1-dimensional non-linear systems: critical points; local linear approximations; qualitative analysis; linear stability analysis; bifurcations.
3. Multi-dimensional non-linear systems: linearisation about critical points, limit cycles, bifurcations.
4. Discrete systems: maps (such as tent map, logistic map, Henon map, standard map).
5. Numerical schemes for ordinary differential equations, such as the embedded Runge-Kutta method.
6. Numerical applications and programming: generation of the orbit of a map, Lorenz map for a dynamical system, orbit diagrams, cobwebs, simple fractals.
7. Elements of Chaos theory: Lyapunov exponents, sensitive dependence on initial conditions, strange attractors, Hausdorff dimension, self-similarity, fractals.
How will I learn on this module?
The learning strategy of this module is based on a combination of lectures and problem-solving/computer-based workshops. Lectures will give you a formal introduction to theoretical aspects of dynamical systems while the workshops offer the opportunity to deepen your knowledge by applying the theory to problems coming from physics, biology, chemistry and engineering. Workshops will be an opportunity to engage with open research problems; they will often address topics with links beyond the discipline, thus also strengthening your transferable skills and employability.
Assessment is by a closed-book assignment worth 30%, and a formal closed-book computer-laboratory-based examination, worth 70%. The presentation will provide an opportunity for you to demonstrate knowledge of particular aspects of dynamical systems and their applications. The examination will cover all aspects of the module and will assess the student’s problem solving abilities when applied to new and unseen problems.
Exam feedback will be provided individually and also generically to indicate where the cohort has a strong or a weaker answer to examination questions. You will receive formal feedback from the presentation and also formative feedback throughout the course, in particular during the problem-solving/computer-based workshops.
Independent study is supported by further technology-enhanced resources provided via the e-learning portal, including short videos, e-lecture notes, e-hand outs, sample problems and past-paper questions.
How will I be supported academically on this module?
Lectures and workshops will be the main point of academic contact, offering you a formal teaching environment for core learning. Workshops will provide opportunities for critical enquiry and exchanges.
Outside formal scheduled teaching, you will be able to contact the module team (module tutor, year tutor, programme leader) either via email or the open door policy operated throughout the programme.
Further academic support will be provided through technology-enhanced resources via the e-learning portal. You will have the opportunity to give their feedback formally through periodic staff-student committees and directly to the module tutor at the end of the semester.
What will I be expected to read on this module?
All modules at Northumbria include a range of reading materials that students are expected to engage with. Online reading lists (provided after enrolment) give you access to your reading material for your modules. The Library works in partnership with your module tutors to ensure you have access to the material that you need.
What will I be expected to achieve?
Knowledge & Understanding:
1. Formulate a linear autonomous system of first-order ordinary differential equations as a matrix system involving state variable, solve this directly, and interpret the results (KU1, KU2, KU3).
2. Describe and critically appraise the qualitative behaviour of dynamical systems in terms of trajectories, attractors, limit cycles, bifurcations, transitions to chaos and complex pattern formation (KU1, KU2, KU3).
3. Compare, select and analyse a variety of approaches and computational tools used to investigate and interpret the behaviour of dynamical systems, and use these to perform appropriate qualitative and quantitative analyses (KU2).
Intellectual / Professional skills & abilities:
4. Perform critical comparisons of computational experiments with theory and derive suitable conclusions (IPSA1, IPSA2, IPSA3).
Personal Values Attributes (Global / Cultural awareness, Ethics, Curiosity) (PVA):
5. Analyse, discuss and report analytical and computational results in a professional manner using appropriate software?(PVA1, PVA2, PVA3).
How will I be assessed?
SUMMATIVE
1. Coursework (30%) – 1, 2, 3
2. Exam (70%) – 1, 2, 3, 4, 5
FORMATIVE
1. Problem-solving/computer-based workshops – 1, 2, 3, 4, 5
Feedback will take several forms, including verbal feedback during the workshops; individual verbal and written comments on the assignment delivered in class and via blackboard; written feedback on the exam.
Pre-requisite(s)
None
Co-requisite(s)
None
Module abstract
‘Dynamical Systems is designed to introduce you to the theory Dynamical Systems and their applications, providing you with tools and techniques needed to understand the time evolution of those systems. You will analyse non-linear ordinary differential equations and maps, focusing on autonomous systems, and will learn analytical and computational methods to solve them.
You will learn through a combination of lectures and problem-solving/computer-based workshops. Lectures give a formal introduction to theoretical aspects of dynamical systems while the workshops offer the opportunity to deepen the knowledge by applying the theory to problems coming from physics, biology, chemistry and engineering. Workshops will be an opportunity to address open research problems; they will often address topics with links beyond the discipline, thus also strengthening your transferable skills and employability.
The module is assessed with a closed book assignment worth 30%, and a formal closed-book computer-laboratory-based examination, worth 70%. The presentation will provide an opportunity to demonstrate your knowledge of particular aspects of dynamical systems and their applications. The examination will cover all aspects of the module and will assess your problem solving abilities when applied to new and unseen problems.
Course info
UCAS Code F2W4
Credits 20
Level of Study Undergraduate
Mode of Study 4 years Full Time or 5 years with a placement (sandwich)/study abroad
Department Mathematics, Physics and Electrical Engineering
Location City Campus, Northumbria University
City Newcastle
Start September 2025
All information is accurate at the time of sharing.
Full time Courses are primarily delivered via on-campus face to face learning but could include elements of online learning. Most courses run as planned and as promoted on our website and via our marketing materials, but if there are any substantial changes (as determined by the Competition and Markets Authority) to a course or there is the potential that course may be withdrawn, we will notify all affected applicants as soon as possible with advice and guidance regarding their options. It is also important to be aware that optional modules listed on course pages may be subject to change depending on uptake numbers each year.
Contact time is subject to increase or decrease in line with possible restrictions imposed by the government or the University in the interest of maintaining the health and safety and wellbeing of students, staff, and visitors if this is deemed necessary in future.
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