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.
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:
Introduction and Motivating Examples Axioms of chaos, symbolic dynamics. Examples of chaotic and fractal systems.
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.
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.
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
Measures of Chaos and Time-series Analysis Hausdsorff & correlation dimensions, Liapunov exponent, Kolomogrov entropy and information, the power spectrum. Takens theorem and time series.
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.
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)
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.
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:
twodcell2.m simulates chemical waves, life, demons and the time tunnel.
Cellular automata modeling two species of gastropod (collected and modeled by CK).
Assessment:
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%.
Hall Nina (1991) New Scientist Guide to Chaos, Penguin.
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
Keen L. ed. Chaos and Fractals , Proc. Symp. App. Math. A.M.S.
Kusch I, Markus M 1996 Mollusc Shell Pigmentation: Cellular automation simulations and evidence for undecidability J Theor Biol 178 333-340.
Levy Steven (1992) Artificial Life : The Quest for a New Creation Pantheon.
Meinhardt Hans (1995) Algorithmic beauty of sea shells with contributions and images by Przemyslaw Prusinkiewicz and Deborah R. Fowler Berlin ; New York : Springer-Verlag.
Peitgen H.O. & Richter P.H., (1986), The Beauty of Fractals Springer-Verlag, Berlin. DC
Peitgen H.O. et.al. (1988) The Science of Fractal Images New York ; Berlin : Springer-Verlag
Heinz-Otto Peitgen, Hartmut Jurgens, Dietmar Saupe (2004) Chaos and fractals : New frontiers of science New York: Springer
Prusinkiewicz P., Lindenmayer (1990) The Algorithmic Beauty of Plants Springer-Verlag
Schiff Joel L. (2008) Cellular Automata: A Discrete View of the World (Wiley Series in Discrete Mathematics & Optimization).
Schuster H.J., (1986), Deterministic Chaos , Springer-Verlag, Berlin. DC
Stewart I. (1988), Does God Play Dice? Basil Blackwell, Oxford.
Schroeder M. (1993) Fractals, Chaos and Power Laws ISBN 0-7167-2136-8.
Sprott, Julien Clinton (2003) Chaos and Time-Series Analysis. Oxford University Press, Oxford & New York.
Strogartz Stephen Non-linear Dynamics and Chaos; With applications to physics biology chemistry and engineering.
Waldrop Mitchell (1993) Complexity, Penguin.
Matlab tutorials (note you can buy the Matlab student edition CD from the resource center for around $60)
MATHS 745
Chaos, Fractals and Bifurcations
