MPhys Physics

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Introduction to Nonlinear Physics (M)

To introduce the concepts required for understanding 'real world' nonlinear phenomena using a variety of mathematical and laboratory models.

Learning outcomes

On completion successful students will be able to:

1. describe the key concepts of nonlinear dynamics.
2. analyse simple one and two-dimensional nonlinear systems.
3. apply the basic numerical methods relevant to nonlinear systems
4. 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. 

Recommended texts:
Strogatz, S.H. Nonlinear Dynamics and Chaos, (Addison Wesley 1994).

Useful references:
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

Supplementary reading:
Gleick, J. Chaos:  Making a New Science, (Heinmann 1998)
Stewart, I. Does God play Dice?  The Mathematics of Chaos, (Penguin 1990)
 

. *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