Chaos, Fractals and Bifurcations

Lecturer: Chris King

Before retiring, I set up this course to be available as an electronic teaching resource complete with downloads and applications. If you are a student studying in this area, you can easily read through the key sections and gain a good understanding of the area using the notes, applications and toolbox functions.

  1. Content
  2. Course Outline
  3. Electronic Course Resources
  4. Live Applications and Toolboxes
  5. My Research Papers
  6. Assessment

Content: Chaos, fractals and bifurcation, and their application to wide areas including commerce, medicine, biological and physical sciences. The course focuses on discrete iterations, including the classical fractals of computer science and art such as Julia and Mandelbrot sets, iterated function systems and higher-dimensional strange attractors. Quantum chaos and complexity theory are emerging frontier areas discussed in the course.

Java Applet (Requires Java Runtime):
To Zoom in, drag a rectangle over part of the Mandelbrot Set image >>

The aim is to provide a course in non-linear discrete dynamics and it's applications to many fields which will give graduate students, in addition to a knowledge of chaotic and fractal dynamics, opportunities to apply their mathematical expertise to a variety of areas which may also provide research and employment opportunities in other disciplines.

A core of lecture material on the central ideas lead into applications to a variety of sciences and some areas of economics and the humanities. The aim is for this to be complemented at the end by a series of short seminar talks by the class members about a mini-project in a related area.

Course Outline:

  1. Introduction and Motivating Examples Axioms of chaos, symbolic dynamics. Examples of chaotic and fractal systems.
  2. Iterations Bifurcations and the Development of Chaos. The logistic map and its properties including period doubling, the tangent bifurcation. Feigenbaum numbers, crises, intermittency, topological conjugacy, Sarkovski's theorem etc. The three classical routes to chaos. The circle map, mode-locking and the devil's staircase. Structural stability, bifurcation theory, Morse-smale systems homoclinic points, kneading.
  3. Fractals. Metric spaces and affine mappings. Code space. The space H(X). Iterated function systems. Fixed points. Random and deterministic algorithms. The fractal basis of natural forms. Fractal image compression. Complex iterations. Computer methods for Julia sets and the Mandelbrot set. Complex analytic dynamics, normal families and exceptional points, the geometry of Julia sets.
  4. Continuous Flows and Higher Dimensional Systems The Lorentz & Rossler flows, Henon-Heiles and double scroll system. The dynamics of linear maps. Attractors: The Smale horseshoe and the solenoid. Stable and unstable manifolds. The Henon map. Conservative flows, elliptic and hyperbolic points and cantori. Homoclinic & heteroclinic points
  5. Measures of Chaos and Time-series Analysis Hausdsorff & correlation dimensions, Liapunov exponent, Kolomogrov entropy and information, the power spectrum. Takens theorem and time series.
  6. Applications of Chaos and Bifurcations to physics, chemistry, biology, medicine, geography, and economics. Chaos in the electroencephalogram, chemical oscillations, heartbeats, land forms, crustal movements, astronomy.
  7. Complex systems, Symmetry, Controlling chaos, Edge-of-chaos, anti-chaos. Complex systems, digital systems: cellular automata, chaos and complexity. Symmetric chaos. Quantum chaos.

Flash Video of the Exploding Julia set of a Complex Cosine Function (Video and audio generated and uploaded by CK)
Note that music is also fractal, following a 1/f noise distribution characteristic of many chaotic processes.

Electronic Course Resources:

A: Handouts: (Password protected resources use the course number as password)

  1. Study Guide
  2. Exploding the Dark Heart of Chaos 8th March 2009
    This paper, with its associated graphical software and movies, is an investigation of the universality of the cardioid at the centre of the cyclone of chaotic discrete dynamics, the quadratic 'heart' forming the main body of the classic Mandelbrot set.
  3. The 3n+1 problem as an Introduction to complex fractal dynamics: Collatz Handout Collatz article (Wikipedia) Collatz article (Mathworld) Collatz M-files
  4. Terminology: Dynamical System (Wikipedia) Dynamical System (Mathworld)
  5. 1-D Iterations notes (Key introduction to 1D dscrete dynamics), Proposition 5.3 proof 2+root5 proof Cantor Set (Wikipedia) Logistic M-files
  6. Critical Curves, critical.m Matlab M-file, Logistic differential equation
  7. Sarkovski's Theorem
  8. Brain Paper (used to demonstrate elementary chaos theory as well as applications to neurodynamics)
  9. Circle Maps, Mediants and Mode-Locking (Introduces the twisting circular motion that will appear in complex dynamics)
  10. Mandelbrot and Julia Sets, Additional Julia notes, M-file, X-code Projects
  11. Fractal Dimension
  12. Metric Space Introduction, IFS Notes, IFS M-file, Fractal Image Compression
  13. Multifractals 1, Multifractals 2
  14. Strange Attractors Solenoid, Horseshoe, Horseshoe2, Henon Map (Wikipedia), Henon M-file
  15. Lorenz Attractor, Lorenz M-files, Rossler attractor (Wikipedia)
  16. Conservative systems, Henon-Heiles M-files
  17. Chaos in Electronic Circuits
  18. Neural Net Methods for Generating Functions
  19. Sounds and Brownian Motion, Self-similarity, Fractional Calculus (Wikipedia), Sounds and Noises M-files
  20. Cellular Automaton (Wikipedia) Complex system Encyclopedia Complex system (Wikipedia)
  21. Rule 110 (Wikipedia) Rule 30 (Wikipedia), CA M-files
  22. 1-D Cell Auto Lambda Java Applet Illustrates Langton's lambda (Use New to generate new rules)
  23. Sea Shells and CAs (Matlab m-file shell.m)
  24. Chemical Waves and Infection CA (Matlab m-file twodcell2.m)
  25. LifeLab, Mac software which will run the life CPU simulation
  26. Mastering Chaos, Quantum Chaos, Fractal Cosmic Inflation
  27. Why the Universe is a Fractal - Chris King
  28. Antichaos, Complexity1, Complexity2
  29. Freeman Brain Chaos, Fractal Physiology
  30. Music and 1/f noise

Quantum chaos: Scarring of the chaotic quantum stadium wave function (a) Classical (b) Quantum scars surround classically repelling orbits (d).
Semiclassical simulation (c) gives similar results to the quantum (g). A realization in carefullly placed iron atom on a copper sheet (f).
(e) Scarring on repelling orbits of absorption peaks of an excited atom in a magnetic field.

C: Mac XCode Applications and Matalb Zeta Toolbox:

The most up-to-date downloadable releases of the major XCode applications, which have been tested for both Tiger and Snow Leopard are as follows:

  1. Riemann Zeta Viewer: Application - Source - RZ Flight Manual
  2. MATLAB Zeta Collatz Toolbox with XCode viewers: Download - Manual
  3. Dark Heart Viewer: Application - Source - DH Flight Manual
  4. Wave Function Method Viewer: Application - Source - WF Flight Manual
  5. Modified Inverse Iteration Viewer: Application - Source
  6. Herman Ring 4D parameter Viewer: Application - Source
  7. Eight Critical Point csin(z)/z Viewer: Application - Source
  8. Collatz 3n+1 Complex Map Viewer: Application - Source

Matlab m-files:

  1. Logistic
    1. logistic.m Matlab m file for making a web plot of the Logistic population quadratic
    2. feigenbaum.m Matlab m file for the bifurcation diagram of the Logistic quadratic
    3. critical.m Matlab m file for plotting the critical curves of the logistic function
  2. Collatz
    1. collatziter.m Matlab m file for making a plot or web plot of the real Collatz problem for a given value
    2. cruncher.m Matlab m file for iterating real Collatz sequences to plot orbit length and period
    3. ccollatz.m Matlab m file for making a level set plot of a region of the complex Collatz problem
    4. ccollatziter.m Matlab m file for iterating specific values of the complex Collatz problem
  3. Mandelbrot (see also xcode projects below)
    1. msetlsm.m Matlab m-file for drawing and saving level set depictions of the Mandelbrot set
  4. ifs4.m Matlab m-file for portraying an IFS based on a matrix of values
  5. henon.m Matlab file to display the Henon map
  6. Henon-Heiles
    1. henonphase.m Matlab file to plot Henon-Heiles phase portraits
    2. henonorb.m Matlab m-file to plot Henon-Heiles orbits
  7. Lorenz Attractor
    1. lorenzdyn.m Matlab m file for picturing sensitive dependence in the Lorenz attractor
    2. lorenzfour.m Matlab Frequency Spectrum of Lorenz. Uses lorenzo.m
    3. lorlambda.m Matlab estimate of Lyapunov exponent of Lorenz. Uses lorenzo.m
    4. d2lor.m Matlab Correlation dimension calculation for Lorenz
  8. Fractal and Chaotic Noises and Sounds
    1. ahenon.m, alogistic.m bifurcation sound generators
    2. muscwt.m generate wavelet or Fourier transforms from a wave file Needs
    3. alogisticwave.m generates Logistic bifurcation wave file
    4. pink.m, halfint.m, quadtest.m 1/f generators usign relaxation fractional integration and chaotic intermittency
    5. fracsound1213.m generates an endlessly rising cyclic fractal sound wave
  9. Cellular Automata
    1. cell1.m 2-state 3 parent 1-D cellauto
    2. cellwolf.m multistate additive cell auto
    3. shell.m sea shell simulator
    4. twodcell2.m simulates chemical waves, life, demons and the time tunnel.

Cellular automata modeling two species of gastropod (collected and modeled by CK).


One class test 50%, two assignments 10% each, , including one or two computer simulations of a dynamical system, a mini-project on any related area you find interesting 15%, which will also be the subject of a 20 minute talk 15%.

Lectures: 2008 Semester 1 M, Tu, W 4 pm B08.

Assignments and Solutions:

  1. Assignment 1 Solutions
  2. Assignment 2 Solutions

Previous Class Test:


My Research Papers:

  1. The Ising Model of Spin Interactions as an Oracle of Self-Organized Criticality, Fractal Mode-Locking and Power Law Statistics in Neurodynamics Aug 2009
  2. Exploring Quantum and Classical Chaos in the Stadium Billiard 9 July 2009
  3. Experimental Observations on the Riemann Hypothesis, and the Collatz Conjecture 22 May 2009
  4. Exploding the Dark Heart of Chaos 8th March 2009

Source References:

  1. Abraham R. (1978) Foundations of Mechanics (2nd ed.) Benjamin/Cummings, Reading, Mass.
  2. Arnold V.I. (1986) Catastrophe Theory (2nd ed.) Springer-Verlag, Berlin.
  3. Barnsley M. (1988) Fractals Everywhere Academic Press, New York.
  4. Barnsley M. Fractal Image Compression.
  5. Beck C. Schogl F. 1993 Thermodynamics of Chaotic Systems, Cambridge Univ. Pr.
  6. Devaney R.L. (1986) An Introduction to Chaotic Dynamical Systems Benjamin/Cummings, Menlo Park.
  7. Falconer Kenneth (1990) Fractal Geometry John Wiley and sons.
  8. Gilmore Robert, Letellier Christophe (2007) The Symmetry of Chaos Oxford University Press.
  9. Glieck James (1987) Chaos: Making a New Science, Penguin Viking
  10. Gutzwiller, M.C. (1992). Quantum chaos. Sci. Am. 266, 78 - 84.
  11. Hall Nina (1991) New Scientist Guide to Chaos, Penguin.
  12. Jaap A. Kaandorp, Janet E. Kubler (2001) Algorithmic beauty of seaweed, sponges, and corals NY Springer Jen Erica ed. (1990) 1989 Lectures in Complex Systems Santa Fe Inst. Addison-Wesley
  13. Keen L. ed. Chaos and Fractals , Proc. Symp. App. Math. A.M.S.
  14. Kusch I, Markus M 1996 Mollusc Shell Pigmentation: Cellular automation simulations and evidence for undecidability J Theor Biol 178 333-340.
  15. Levy Steven (1992) Artificial Life : The Quest for a New Creation Pantheon.
  16. Meinhardt Hans (1995) Algorithmic beauty of sea shells with contributions and images by Przemyslaw Prusinkiewicz and Deborah R. Fowler Berlin ; New York : Springer-Verlag.
  17. Peitgen H.O. & Richter P.H., (1986), The Beauty of Fractals Springer-Verlag, Berlin. DC
  18. Peitgen H.O. (1988) The Science of Fractal Images New York ; Berlin : Springer-Verlag
  19. Heinz-Otto Peitgen, Hartmut Jurgens, Dietmar Saupe (2004) Chaos and fractals : New frontiers of science New York: Springer
  20. Prusinkiewicz P., Lindenmayer (1990) The Algorithmic Beauty of Plants Springer-Verlag
  21. Schiff Joel L. (2008) Cellular Automata: A Discrete View of the World (Wiley Series in Discrete Mathematics & Optimization).
  22. Schuster H.J., (1986), Deterministic Chaos , Springer-Verlag, Berlin. DC
  23. Stewart I. (1988), Does God Play Dice? Basil Blackwell, Oxford.
  24. Schroeder M. (1993) Fractals, Chaos and Power Laws ISBN 0-7167-2136-8.
  25. Sprott, Julien Clinton (2003) Chaos and Time-Series Analysis. Oxford University Press, Oxford & New York.
  26. Strogartz Stephen Non-linear Dynamics and Chaos; With applications to physics biology chemistry and engineering.
  27. Waldrop Mitchell (1993) Complexity, Penguin.

Matlab tutorials (note you can buy the Matlab student edition CD from the resource center for around $60)

  1. Elementary stage 1 introduction to Matlab symbolic manipulation
  2. 162 Matlab notes
  3. 363 matlab primer
  4. official matlab getting started
  5. additional tutorial

Web Links on Fractals and Complexity:

  1. Fractal Images
  2. Buddhabrot fractal method
  3. Olivier Langlois's Fractal Art
  4. Julius Ruis Fractal Science
  5. Open Directory Fractal Art
  6. chaotic_attractors
  7. resources
  8. Dynamical Systems and Technology Project
  9. Bob Devaney's Home Page
  10. The Fractal Microscope
  11. chaos chap7_1.pdf
  12. Cantor sets
  13. Nowhere Dense -- From MathWorld
  14. Online Course - Systems Sciences
  15. chaos
  16. Stephen Wolfram: A New Kind of Science | Online - Table of Contents
  17. The Julia sets trip with orbit
  19. Christopher G. Langton
  20. Fractal Geometry
  21. Algorithmic Botany: Publications
  22. Modular Origami - fractal models folded by Michael Kosmulski
  23. Fractal World Gallery Thumbnails : cosmic recursive fractal flames or flame fractals
  24. Matlab M-Files Database - Files
  25. Fractional Calculus
  26. 1-D Cell Auto Examples Yale
  27. Wolfram on Universality