PMA 324 Chaos 2009-10

by Dr. P. G. Dixon

A pretty picture of a mini-Mandelbrot set

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.     

Aims, Learning Outcomes, etc.

See the SoMaS information page for this module at

Teaching methods

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.

Examinable syllabus

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.


Past Examination Papers

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 2008January 2009, January 2010.

Recommended Books (A=core text, B=secondary text, C=background reading)

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.

Online Sources

  1. The search engine Yahoo has a page of, mainly popular, chaos sites, and Google has a page on Chaos and Fractals.
  2. The Dynamical Systems and Technology Project at Boston University has lots of exciting stuff. For example, Robert Devaney's expository papers on the Mandelbrot set, the Mandelbrot Set Explorer, the Fractal Movie Theater (including an animation entitled The Fibonacci sequence and the Mandelbrot set.), etc.
  3. David Wright's Dynamical Systems and Fractals Lecture Notes are useful, though more elementary than PMA324.
  4. The Fractal Geometry page at Yale has a certain amount about chaos and includes some good Java software.
  5. An wide-ranging article by Dyson in the February 2009 issue of the Notices of the American Mathematical Society described the Li-Yorke
    paper as 'one of the immortal gems in the literature of mathematics'.    However, it provoked a correspondence about the proper use of the term 'chaos', with David Ruelle in the June Letters to the Editor objecting to the use of the term in this context. The Letters in the November issue include both a response from James Yorke and a rejoinder from David Ruelle.


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:

  1. 1A.TWD for f(x) = ax with a=1.5 (repelling fixed point at x = 0);
  2. 1B.TWD for f(x) = ax with a=0.7 (attracting fixed point at x = 0);
  3. 1C.TWD for f(x) = x^3 ((super)attracting fixed point at x = 0, repelling fixed points at x = +1, -1);
  4. EX3-5.TWD for f(x) = -x^3 ((super)attracting fixed point at x = 0, repelling periodic points order 2 at x = +1, -1)
  5. EX4-2.TWD for f(x) = (x + x^3)/2 (attracting fixed point at x = 0, repelling fixed points at x = +1, -1)
  6. EX4-2.TWD for f(x) = -(x + x^3)/2 (attracting fixed point at x = 0, repelling periodic points order 2 at x = +1, -1)
  7. EX4-6.TWD for f(x) = x + x^2 (non-hyperbolic fixed point at x = 0, which might be described as "weakly attracting on the left, weakly repelling on the right");
  8. EX4-6.TWD for f(x) = x + x^3 (non-hyperbolic, weakly repelling, fixed point at x = 0);
  9. EX4-6.TWD for f(x) = x - x^3 (non-hyperbolic, weakly attracting, fixed point at x = 0);
  10. EX4-7.TWD for f(x) = x^2 + a;
  11. LOGISTIC.TWD for f(x) = ax(1-x), the logistic function, on the unit interval;
  12. LOGIST2.TWD for f(x) = a(ax(1-x))(1-ax(1-x)), the second iterate of the logistic function, on the unit interval;
  13. LOGIST3.TWD for f(x) = ax(1-x), the logistic function, on the unit [-1.5,+1.5];
  14. SINE.TWD for f(x) = a sin x, this family behaves rather like the logistic family;
  15. TWO.TWD loads the sine family and the logistic family together.

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.

  1. Solutions to Question Sheet 1, (in due course): Question 1 used two worksheets, QS1E1AB.MWS for the first two functions and QS1E1C.MWS for the last two.
  2. The proof that F^2 has only two fixed points for mu between 1 and 3 used the Maple worksheet F2FPTS.MWS.
  3. Animations showing the graphs of various iterates of F_mu as mu increases FMU3.MWS. This is a large file 1,750KB; a zipped version (745 KB) may be downloaded here.
  4. The Maple solutions to Question Sheet 5, QS5.MWS.
  5. The cardioid. This worksheet is easily modified, by changing the three variables at the beginning, to show epicycloids, hypocycloids and their relatives --- but not cycloids --- so, for the sake of completeness, here is a worksheet for the cycloid.

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.

Class size

23 students.

Lecturer's Office Hours, Telephone, E-mail.

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:

Peter Dixon's home page

Dept. of  Pure Maths

School of Mathematics and Statistics (SoMaS)