MPhys Physics

The unit introduces concepts such as nonlinearity, continuous and discrete systems, stability, bifurcations, fractals and chaos. It discusses methods for analysing nonlinear systems, such as graphical analysis, linear stability analysis and numerical methods for solving ordinary differential equations. Emphasis is placed on the framework for analysing and interpreting behaviour of nonlinear systems using examples from classical and non-classical physics.  

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

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.

The core material which includes problem solving is delivered using two one-hour in-person lectures per week. The recordings of these lectures are available on Podcast and linked to the course online page. The lectures are accompanied by scans of notes produced during lectures, slides available in advance, explicit references to the reading material and weekly videos that explains concepts from the unit using modern research examples. This is augmented by an on-line quiz after each lecture and a set of biweekly problems with solutions. A Piazza discussion forum is also provided where students can ask questions with answers provided by other students and the unit lead. 

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. 

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

Mullin T. The nature of chaos (OUP 1993)

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