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BSc Physics with Theoretical Physics

Year of entry: 2021

Course unit details:
Introduction to Non-linear Physics

Unit code PHYS30471
Credit rating 10
Unit level Level 3
Teaching period(s) Semester 1
Offered by Department of Physics & Astronomy
Available as a free choice unit? No

Overview

Introduction to Nonlinear Physics (M)

Pre/co-requisites

Unit title Unit code Requirement type Description
Dynamics PHYS10101 Pre-Requisite Compulsory
Vibrations & Waves PHYS10302 Pre-Requisite Compulsory
Mathematics 2 PHYS10372 Pre-Requisite Compulsory
Mathematics of Waves and Fields PHYS20171 Pre-Requisite Compulsory

Aims

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

Learning outcomes

This course unit detail provides the framework for delivery in 20/21 and may be subject to change due to any additional Covid-19 impact.  Please see Blackboard / course unit related emails for any further updates

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

 

Syllabus

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.

 

Assessment methods

Method Weight
Other 33%
Written exam 67%

Feedback methods

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 reading

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)
 

Study hours

Scheduled activity hours
Assessment written exam 1.5
Lectures 22
Independent study hours
Independent study 76.5

Teaching staff

Staff member Role
Fedor Bezrukov Unit coordinator

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