## PH3160 Non-Linear Phenomena and Chaos.

## Aims and Objectives of the Course:

Aims, Course Content, Course Homepage and Books

Aims and Objectives of the Course:To provide an introduction to some of the fundamental ideas underlying the description of non-linear and chaotic systems in physics. To develop an assortment of concepts and techniques which can be used for describing the essential features of non-linear systems. Although physics makes wide use of linear models the physical world is inherently non-linear. This is a mathematical physics course aimed at providing an appreciation of such non-linear systems. We shall illustrate how non-linear differential equations can be used to introduce the geometric (or topological) concepts used to analyze and model a wide range of non-linear dynamics. We shall also study some one- and two-dimensional difference equations (or maps) and see that they are dynamically much richer than ordinary differential equations with the same number of dynamic dimensions and see in turn how this helps lead to an appreciation of how dynamical systems may approach chaotic behaviour.

The course will consist of lectures, example classes and computing exercises.Overview of Lecture content:Introductory overview of non-linear dynamic systems, dissipative and conservative.

Autonomous and non-autonomous first and second order non-linear differential equations (ie. vector fields), deterministic, stochastic...physical examples. Phase space, stability analysis, Poincare sections, phase transformations and period doubling.

Linear and non-linear difference equations (ie. Maps), logistic, area preserving, Henon, Chirikov.

Global behaviour via Poincare maps. Universal behaviour of quadratic maps, self similarity.

Routes to chaos: bifurcations, intermittency, strange attractors (fractal dimension, renormalization-group).

Regular and irregular motion in conservative systems, ergodicity.

Search for quantum chaos.

Computing exercises:These require working through some worksheets using TWO distinct tools. The first uses Maple to illustrate and explore the graphical solution of first and second order ordinary differential equations of both linear and nonlinear type. The second makes use of a purpose built computer package "Chaos Demonstrations" to familiarise you with certain key concepts as illustrated, for example, by the variation of parameters in a variety of different model systems. Be sure to study carefully and understand the content of the material covered in the lectures and example classes.

Course Homepage:This contains an archive of course material and useful links elsewhere.

You can access all course material handouts of notes, exercises etc, from the ph316 Course Homepage on the WorldWideWeb, using say the teaching lab PC's Netscape facility.

The course homepage contains sections on:

- Administrivia ... Aims + Content + Books etc

- Syllabus + Course work schedule + Exam requirements

- Course Newsletters (weekly)

- Outline Notes of lecture content + Glossary of terms

- a series of Self-study assignments for this course.

- Examples + course work exercises

- Links to other useful nonlinear websites

- Past Exam papers

Textbooks:There are an enormous number of books on this subject and new ones are continuing to appear at a high rate. Thus some sort of guidance through the literature for the interested but possibly puzzled beginner is provided:

No doubt many of you will already have read the popular account of this subject in James Gleick's "Chaos ...making a new science" (Viking 1987 paperback). Although this book has almost become of "cult" status it gives a readable account of some aspects of what it is all about and some of its history. However, one dosn't end up with much in the way of solid knowledge useful to a physicist.

When I first gave this course I tried to adopt in broad terms the topic coverage as laid out in G Rowlands "Nonlinear Phenomena in Science and Engineering" (Ellis Horwood 1993 paperback) ...172pp

I considered this book to be a well-written, easy to read text with a respectable amount of physics and with the associated mathematics included. However, despite its modest length some of its topics are in fact too ambitious for this particular course and students did not find the book as rivetting a read as did I. Therefore over the last few years I have developed a fairly full set of introductory style notes which assumed no prior knowledge of the subject matter other than the usual physics background and mathematical tools accumlated by most third year students.

I should draw to your attention a more recently available and superb little book by D Acheson "From Calculus to Chaos: An Introduction to Dynamics" (OUP 1997 pbk) ... 269pp which provides delightful background reading to some of the material of this course.

A. Books to consider buyingR C Hilborn's "Chaos and Nonlinear Dynamics" (OUP 1994 paperback)...654pp This book is a mine of information, perhaps too much so, since its choice of topics seems bewilderingly large at first sight. It can however be profitably used as background reading for selected topics of interest since it abounds in useful detail and is written for physicists from an introductory perspective. It also contains very useful guidance to the wealth of literature up to 1993.

G L Baker and J P Gollub's "Chaotic Dynamics" 2nd Ed (CUP 1996 pbk) ... 255pp is a nice introduction to many of the topics we consider.

Note also item D of Newsletter 7 ... 3 March 1998 via course homepage.

B. Further useful books in College LibrarySome other useful books to consult include:

H T Davis "Intro to Nonlinear Differential and Integral Equations" (Dover 1962) ...566pp

I Percival and D Richards "Intro to Dynamics" (CUP 1982) ...228pp

P Cvitanovic "Universality in Chaos" (Hilger 1984)...511pp

F C Moon "Chaotic and Fractal Dynamics" (J Wiley 1992) ... 504pp

H G Schuster "Deterministic Chaos" (Springer-Verlag 1984) ...220pp

A B Pippard "Response and Stability" (CUP 1985) ...228pp

S H Strogatz "Nonlinear Dyanamics and Chaos" (Addison Wesley 1994)...498pp

J M T Thompson and H B Stewart "Nonlinear Dynamics and Chaos" (Wiley 1986)

D W Jordan and P Smith "Nonlinear Ordinary Differential Equations" (OUP 1987) ...381pp

E Ott "Chaos in Dynamical Systems" (CUP 1993) ...400pp

H D I Abarbanel, M I Rabinovich and M M Sushchik "Intro to Nonlinear Dynamics for Physicists" (World Scientific 1993) ..158pp

P Glendinning, "Stability, instability and chaos : an introduction to the theory of nonlinear differential equations" (CUP 1994) ... 388pp

N Kumar "Deterministic Chaos" (Sangam Books, London 1996) ... 96pp

C. Articles to ConsultEd: N Hall "The New Scientist Guide to Chaos" (Penguin 1992 pbk) ... 223pp

J Ford "What is Chaos?" Ch 11, pp~348-372 of the "New Physics" ed. P Davies (CUP 1989)

M C Gutzwiller "Quantum Chaos" Sci Amer (Jan 1992) pp~26-32.

MW - 3/1/99