- UCAS course code
- UCAS institution code
MMath&Phys Mathematics and Physics
Year of entry: 2023
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Course unit details:
Introduction to Non-linear Physics
|Unit level||Level 3|
|Teaching period(s)||Semester 1|
|Available as a free choice unit?||No|
Introduction to Nonlinear Physics (M)
|Unit title||Unit code||Requirement type||Description|
|Vibrations & Waves||PHYS10302||Pre-Requisite||Compulsory|
|Mathematics of Waves and Fields||PHYS20171||Pre-Requisite||Compulsory|
To introduce the concepts required for understanding 'real world' nonlinear phenomena using a variety of mathematical and laboratory models.
On completion successful students will be able to:
- describe the key concepts of nonlinear dynamics.
- analyse simple one and two-dimensional nonlinear systems.
- apply the basic numerical methods relevant to nonlinear systems
- explain the origin and key features of chaotic behaviour
1. Introduction - overview of the course introducing some of the basic ideas. (1 lecture)
General introduction and motivation; examples of linearity and nonlinearity in physics and the other sciences; modelling systems using iterated maps or differential equations.
2. General features of dynamical systems - the structures that may arise in the analysis of ordinary differential equations. (10 lectures)
Systems of differential equations with examples; control parameters; fixed points and their stability; phase space; linear stability analysis; numerical methods for nonlinear systems; properties of limit cycles; nonlinear oscillators and their applications; the impossibility of chaos in the phase plane; bifurcations: their classification and physical examples; spatial systems, pattern formation and the Turing mechanism; strange attractors and chaotic behaviour.
3. The logistic map - period doubling and chaos in a simple iterated map. (4 lectures)
Linear and quadratic maps; graphical analysis of the logistic map; linear stability analysis and the existence of 2-cycles; numerical analysis of the logistic map; universality and the Feigenbaum numbers; chaotic behaviour and the determination of the Lyapunov exponent; other examples of iterated maps.
4. Fractals - complex geometrical objects of which strange attractors are examples. (4 lectures)
How long is the coastline of Britain? Artificial fractals: the Cantor set and von Koch curve; fractal dimensions; iterations of the complex plane and the Mandelbrot set; how fractals arise in the description of dynamical systems.
5. Further aspects of chaotic dynamics - exploring the basic ingredients of chaos. (3 lectures)
Fractal structures in simple maps; how strange attractors come about; the evolution of phase space volumes in chaotic and non-chaotic systems; mixing and information entropy.
While students will not be required to hand in solutions to example sheets, I will give feedback on written solutions, should students wish to hand in work. Model answers will be issued. One or two Question & Answer sessions may be arranged.
Strogatz, S.H. Nonlinear Dynamics and Chaos, (Addison Wesley 1994).
Baker, G.L. & Gollub, J.P. Chaotic Dynamics: An Introduction, (CUP 1996), Second edition
Jordan, D.W. & Smith, P. Nonlinear Ordinary Differential Equations, (OUP 1999), Third edition
Gleick, J. Chaos: Making a New Science, (Heinmann 1998)
Stewart, I. Does God play Dice? The Mathematics of Chaos, (Penguin 1990)
|Scheduled activity hours|
|Assessment written exam||1.5|
|Independent study hours|
|Draga Pihler-Puzovic||Unit coordinator|
. *Note - scheduled teaching hours:
11 weeks of teaching times 2 lectures plus the exam.
All material for the unit, such as videos, example scripts and notes, is available online via Blackboard