Module code: PHYM038

Module Overview

Non-linear physics introduces the relevant theory with examples in physics, astrophysics, chemistry, biology and engineering. The material is developed through lectures and examples classes, and applied in the coursework.

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

Module Availability

Semester 2

Prerequisites / Co-requisites

Newtonian mechanics, vectors and vector calculus, complex analysis. Familiarity with writing computer programs to solve differential equations, e.g. in C, python or similar. Ability to make professional-level scientific data plots, e.g. with tools like gnuplot or python's matplotlib, and write short scientific reports.

Module content

• Linear and non-linear dynamics

• Fixed points, bifurcations and limit cycles in one and two dimensions

• Phase space analysis

• Lyapunov Stability and the Poincaré-Bendixon theorem

• Fractals, and concepts of fractal dimensions

• Attractors, strange attractors and chaos

• Lorenz and Rössler attractors

• Measures of non-linearity and quantification by Lyapunov exponents

• Non-linear maps, the cascade route to chaos and the 0-1 test for chaos

• Unimodal maps, renormalization and Feigenbaum constants

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

• Other examples from physics, astrophysics, chemistry, 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 non-linear physics and complex phenomena. The coursework assigned during the semester allows the student to put into practice the concepts learned in class, therefore building a deeper understanding of non-linear phenomena and mastering the techniques used in their analysis. The final written examination, at the end of the semester, assesses the overall understanding of the fundamental concepts and theory of complex phenomena.


Thus, the summative assessment for this module consists of:

  • One piece of coursework (30%) to be completed during the semester requiring approximately five weeks. The coursework must be completed by the end of the semester.

  • One two-hour examination at the end of the semester (70%), with a section A of compulsory questions and a section B of two questions chosen from three. In Part A answer all questions (20 points). In Part B answer two questions out of three (20 points each). If all three questions in Part B are attempted only the best two will be counted.


Formative assessment and feedback

Students receive continuous feedback through discussions during class time. Assessment of the submitted coursework is 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 non-linear 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 non-linearity 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 non-linearity in physics and identify its basic mechanisms. They will be able to analyse and classify the motion of complex non-linear 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 non-linearity 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 non-linear 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

Overall student workload

Independent Study Hours: 117

Lecture Hours: 33

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 focuses on specific non-linear system(s) that give the student with opportunities to gain knowledge of advanced topics and to prove independent thinking, investigation and computational skills. The assignment is handed out mid-semester and typically requires the use of computational techniques such as non-linear differential equation solving, with the results to be presented in report form at the end of the semester (usually in week 12).

There final examination is 90 minutes in duration and two out of three questions 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 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
Physics MSc 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 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

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 2020/1 academic year.