Module code: PHYM038

Module Overview

This module provides an introduction to the theory of nonlinear physics and to chaos theory, with a discussion of relevant examples in several branches of science such as physics, biology or engineering.  The material is then studied in depth through extensive coursework problems.

Module provider


Module Leader

IZZARD Robert (Physics)

Number of Credits: 15

ECTS Credits: 7.5

Framework: FHEQ Level 7

JACs code: F300

Module cap (Maximum number of students): N/A

Overall student workload

Independent Learning Hours: 117

Lecture Hours: 33

Module Availability

Semester 2

Prerequisites / Co-requisites

Fundamental Newtonian mechanics; Basic Calculus

Module content

Linear and nonlinear dynamics

• Fixed points, bifurcations in one dimension

• Fixed points, bifurcations, and limit cycles in two-dimensional systems

• Linear analysis in phase space

• Liapunov Stability and Poincaré-Bendixon theorems

• Fractals, and concepts of fractal dimensions

• Attractors, strange attractors and chaos

• Lorenz and Rössler attractors

• Measures of nonlinearity and quantification by Liapunov exponents and dimensional analysis

• Nonlinear maps and cascade route to chaos

• Unimodal maps, renormalization and Feigenbaum constants

• Examples of oscillator in two and three dimensions (van der Pol, Duffing, etc…)

• Other examples from physics, engineering and biology


Assessment pattern

Assessment type Unit of assessment Weighting

Alternative Assessment


Assessment Strategy

The assessment strategy is designed to provide students with the opportunity to demonstrate their insight into the physic of nonlinear and complex phenomena. The coursework assigned during the semester will give the occasion to put in practice the concepts learned in class, therefore gradually building a deeper understanding of nonlinear phenomena and mastering the techniques used for their analysis. The final exam, at the end of the semester, will assess the overall understanding of the fundamental concepts and theory of complex phenomena.


Thus, the summative assessment for this module consists of:

  • One coursework to be completed during the semester requiring about 5 weeks. The coursework will have to be completed at the end of the semester.

  • One exam at the end of the semester lasting 1.5 hours (with 2 questions to be answered out of 3).


Formative assessment and feedback

Students will have the opportunity to receive continuous feedback through discussions during class time. Assessment of the submitted coursework will be returned to the students with feedback on their performance.


Module aims

  • To provide a sound grounding and the basic theorems, methods and applications of the theory of nonlinear physics. To gain computational and mathematical skill to characterise the qualitative complex systems in physics, finance, and other disciplines.
  • To then encourage the in-depth investigation of aspects of nonlinearity through extended coursework problems.

Learning outcomes

Attributes Developed
001 Upon successful completion of the module the student will be able to appreciate the implications of nonlinearity in physics and identify its basic mechanisms. They will be able to analyse and classify the motion of complex nonlinear systems by identifying their qualitative features and by categorizing them in terms of periodic, quasi-periodic or chaotic behaviour.  They will be able to recognize effects of nonlinearity in situations of everyday life.
002 On successful completion of the coursework part of the module, they will be able to apply fundamental (analytical and numerical) methods of nonlinear dynamics to analyse chaotic systems and will appreciate the concept and the benefits of chaos control.

Attributes Developed

C - Cognitive/analytical

K - Subject knowledge

T - Transferable skills

P - Professional/Practical skills

Methods of Teaching / Learning

28 hours of theory lectures plus 5 hours of class exercises session (33 hours of lectures in total) and open-ended study for the coursework problems.

Exercises will be held in the form of tutorial sessions and are meant to demonstrate specific examples and applications of the material taught in class.

The coursework will focus on the study of specific nonlinear system(s) that give the student with opportunities to gain knowledge of advanced topics and to prove independent thinking and investigation skills. The assignment is handed out at mid semester and typically requires the use of computational techniques, with the results to be presented in report form at the end of the semester (usually in week 12).

There are two coursework assignments to be completed for this module, along with a final examination of 1.5h duration, in which 2 questions from 3 should be answered.


Indicated Lecture Hours (which may also include seminars, tutorials, workshops and other contact time) are approximate and may include in-class tests where one or more of these are an assessment on the module. In-class tests are scheduled/organised separately to taught content and will be published on to student personal timetables, where they apply to taken modules, as soon as they are finalised by central administration. This will usually be after the initial publication of the teaching timetable for the relevant semester.

Reading list
Upon accessing the reading list, please search for the module using the module code: PHYM038

Programmes this module appears in

Programme Semester Classification Qualifying conditions
Physics MSc 2 Optional A weighted aggregate mark of 50% is required to pass the module
Physics MPhys 2 Optional A weighted aggregate mark of 50% is required to pass the module
Mathematics and Physics MPhys 2 Optional A weighted aggregate mark of 50% is required to pass the module
Mathematics and Physics MMath 2 Optional A weighted aggregate mark of 50% is required to pass the module
Physics with Quantum Technologies MPhys 2 Optional A weighted aggregate mark of 50% is required to pass the module
Physics with Nuclear Astrophysics MPhys 2 Optional A weighted aggregate mark of 50% is required to pass the module
Physics with Astronomy MPhys 2 Optional A weighted aggregate mark of 50% is required to pass the module

Please note that the information detailed within this record is accurate at the time of publishing and may be subject to change. This record contains information for the most up to date version of the programme / module for the 2019/0 academic year.