One was the face of Nature; if a face:
Rather a rude and indigested mass:
A lifeless lump, unfashion'd, and unfram'd,
Of jarring seeds; and justly Chaos nam'd.
from Ovid , Metamorphoses Book 1, translated by John Dryden (the Garth translation).
Many nonlinear dynamical systems exhibit behaviour which is popularly described as `chaotic' and studies of such behaviour have blossomed in the last three decades. A single course cannot hope to cover all aspects of this work. The aim of this course is to develop an understanding of some of the basic concepts of chaos, both in the context of the abstract theory of dynamical systems and in the particular context of the simplest systems generating chaos: the iteration of real-valued functions of a single real variable, and, in particular, the iteration of quadratic maps. A facility with "epsilon-delta" analysis is a desirable prerequisite.
This is a half module (10 credits) consisting of 20 lectures in semester 1. This page gives the formal `aims and objectives' of the course and details of the organization, content and recommended books, together with notices posted during the course.
See the SoMaS information page for this module at http://maths.dept.shef.ac.uk/maths/module_info.php?module_code=PMA324.
Two lectures per week in the first semester : Wednesdays at 10 in Hick Lecture Theatre 9 and Thursdays at 4 in F24. Duplicated notes will be distributed during the course, after the corresponding lectures. Homework will be set each week. Students will be encouraged to use the computer program FRACTINT (available on the network) for background experience.
All definitions, theorems and proofs in the duplicated notes are examinable, except for the following.
You will be expected to have understood the solutions to the exercises set during the course.
In general, past examination papers are a good guide to the style of questions set.
The past papers for January 2008 and January 2009 are available in pdf form.
Solutions to past papers are available in PDF form (though without all the diagrams): January 2008, January 2009, January 2010.
D. Gulick,``Encounters with chaos'' (McGraw-Hill Education - Europe, 1992) £26.99 ISBN 0071129278 ML 531.3(G) [also recommended for the Fractals course.] [B]
Saber N. Elaydi,``Discrete chaos'' (CRC Press, 1999) paperback £39.99, ISBN 1584880023 [B]
R. L. Devaney,``An introduction to chaotic dynamical systems'' second edition (Addison-Wesley, 1989) £35.99 ML 531.3(D) ISBN 0201130467 [B]
R. M. Crownover,``Introduction to fractals and chaos'' (Jones and Bartlett, 1995) £24.95, ISBN 0867204648 [also recommended for the Fractals course.] [B]
P.~Cvitanovic, (ed.) ``Universality in chaos'' 2/e (Adam Hilger,1989) (a reprint selection). ML (1 OWL copy) 531.3(U) ISBN 0 85274 259 2 \pounds 18.50 [C]
J.~Gleick ``Chaos'' (Minerva, 1997) ISBN 0749386061 \pounds 7.99, ML 531.3(G) [C]
I.~N.~Stewart ``Does God play dice? The new mathematics of chaos.'' (Penguin, 2004) ML 531.3(S), ISBN 0140256024 \pounds9.99 (\pounds 6.99 + postage at Amazon) [C]
R.~L.~Devaney and L.~Keen ``Chaos and fractals: the mathematics behind the computer graphics'' (American Mathematical Society, Proceedings of Symposia in Applied Mathematics vol.39, 1989) ISBN 0-8218-0137-6 \pounds21.00 (A collection of articles introducing the subject; technical in places.) ML 3 PER 510.5 [C]
W. de Melo and S. van Strein, ``One-dimensional dynamics'' (Springer, 1993) [advanced]. ML 3 PER 510.5 [C]
Link to Blackwell's reading lists site.
The following are links to the software, some of which may be used for demonstrations during the course. Use the right mouse button and `save link' option to save the software to your hard disk and then unzip it if necessary using PKUNZIP.
The very simple musical iteration program described in the Appendix is CH_MUSIC.BAS. It runs in QBASIC.
Graphical analysis can be demonstrated using ITERATE (the "read me" file for the program is here) with the following additional function files:
Unfortunately, this, being a DOS program, does not work well with modern computers --- it depends on the graphics set-up and it is hard to predict whether or not it will work on any given machine. Consequently, most of our study of the logistic equation will be done using the Java web-based program Chaos lab (alternative URL here)--- you use this online rather than downloading it; be patient as it takes a little while to load the Java program.
The QBASIC program BIFURC.BAS demonstrates how the bifurcation diagram for F_mu is built up line-by-line. Unfortunately, it is currently lacking documentation. Essentially: the number pad cursor control keys move the on-screen cursor; F1 starts an iteration; F2 or F3 starts the plotting of iterates (F2 is supposed to be slower than F3); iterating stops when the cursor is moved; and F10 stops the program. Again, this is a DOS program which depends on having suitable graphics set-up.
Later in the course we shall use FRACTINT, a very extensive program for drawing Mandelbrot and Julia sets. Fractint 20.0 is a DOS program, but there is also an (earlier) Windows version WINFRACT version 18.21. For the bifurcation diagram for F_mu with mu between 2 and 4, use the parameter file BIF.PAR.
For the relation between Mandelbrot sets, Julia sets and iterations of Q_c, we use the Java program Mandelbrot and Julia sets from the Yale collection mentioned above.
The following are Maple worksheets used in lectures. To view, save them to disk and then click on the filename in Windows Explorer, which should open Maple. Links to solution will be dead until the solutions are given out.
Notes and Problem Sheets
Here you will find PDF files of the notes and problem sheets and solutions as the course progresses. The sketch graphs for the solution of Question Sheet 3 question 1 are in this jpeg (incorrectly labelled Question Sheet 2).
Finally, here is a Chaos icon for your Windows desktop.
23 students.
Office hours: see my appointment diary.
Telephone extension 23775 (prefix by 22 from outside the University
and further prefix by 0114 from outside Sheffield).
E-mail: P.Dixon with the usual extension: @sheffield.ac.uk