References on Chaos and Fractals

for PMath 370 at the University of Waterloo

References on Chaos and Fractals
for PMath 370 at the University of Waterloo

Websites on Chaos and Fractals

General References

Tutorials

The Chaos Hypertextbook
Fractal Geometry, a Yale course by Michael Frame, Benoit Mandelbrot, and Nial Neger.
Dynamical Systems and Technology Project at Boston U
Fractint L-Systems Tutorial
A sketch book on L systems - a student project from Spain.
An Introduction to Lindenmayer Systems by Gabriela Ochoa
Chaos, Fractals and Dynamics;University of Toronto course with applets, project suggestions, old exams, and more resource links.
The Fract-ED Information Pages, an introductory fractal tutorial for high school or college entry-level students.
The Encyclopedia of the Mandelbrot Set by Robert P. Munafo.
Fractals and other things by Joseph Thomas.
The Fractal Music Lab by David Strohbeen.

Newgroups

Further Links

Internet Fractal Databasecontaining links to many fractal galleries.
Math Archives Fractal Links
Math Archives Dynamical Systems Links
Math Archives Nonlinear Dynamics Links
Chaos-Fraktale-Fun in German
Yahoo Fractal Links
Internet Resources from the Chaos Hypertextbook

Books on Chaos and Fractals

Fractals and chaos : an illustrated course: AUTHOR: Paul S. Addison.
PUBLISHER: Bristol, UK : Institute of Physics Pub., 1997.
CALL NUMBER: QA614.86.A23 1997
A history of complex dynamics : from Schröder to Fatou and Julia: AUTHOR: Daniel Alexander
PUBLISHER: Braunschweig : F. Vieweg, c1994.
CALL NUMBER: QA845 .A54x 1994
Chaos : an introduction to dynamical systems: AUTHOR: Kathleen T. Alligood, Tim D. Sauer, and James A. Yorke.
PUBLISHER: New York : Springer, c1997.
CALL NUMBER: QA614.8.A44 1997
Dynamical systems : differential equations, maps, and chaotic behaviour: AUTHOR: D.K. Arrowsmith and C.M. Place.
PUBLISHER: Chapman & Hall, 1992.
CALL NUMBER: QA614.8.A774 1992
Chaotic dynamics : an introduction: AUTHOR: Gregory L. Baker and Jerry P. Gollub.
PUBLISHER: 2nd ed.; Cambridge University Press, 1996.
CALL NUMBER: QA862.P4B35 1996
Fractals everywhere: AUTHOR: Michael Barnsley
PUBLISHER: Boston : Academic Press Professional, c1993.
CALL NUMBER: QA614.86.B37 1993
[A classic book mainly concentrating on Iterated Function Systems.]
Fractal image compression: AUTHOR: Michael Barnsley and Lyman P. Hurd
PUBLISHER: Boston : Jones and Bartlett, 1992.
CALL NUMBER: TA1632.B353 1992
Iteration of rational functions : complex analytic dynamical systems: AUTHOR: Alan F. Beardon.
PUBLISHER: New York : Springer-Verlag, 1991; Graduate texts in mathematics ; 132
CALL NUMBER: QA297.8.B43 1991
[A beginning graduate level book.]
Dynamical systems and fractals : computer graphics experiments in Pascal: AUTHOR: Karl-Heinz Becker and Michael Dprfler
PUBLISHER: Cambridge University Press, 1989.
CALL NUMBER: T385.B4413 1989
Turbulent Mirror : An Illustrated Guide to Chaos Theory and the Science of Wholeness: AUTHOR: John Briggs and F. David Peat
PUBLISHER: New York : Harper & Row, c1989.
CALL NUMBER: Q172.5.C45B75 1989
[A popular book written for a fairly general audience.]
Fractals and disordered systems: AUTHOR: Armin Bunde, Shlomo Havlin, eds.
PUBLISHER: 2nd rev. and enl. ed.; Springer, c1996.
CALL NUMBER: QA614.86.F72 1996
Over and over again: AUTHOR: Gengzhe Chang and Thomas W. Sederberg
PUBLISHER: Mathematical Association of America, c1997.
CALL NUMBER: QA601 .C53 1997
[A general book on iterations of all kinds, with a chapter on Sharkovskii's Theorem.]
Applications of fractals and chaos : the shape of things: AUTHOR: A.J. Crilly, R.A. Earnshaw, H. Jones, editors.
PUBLISHER: Springer-Verlag, c1993.
CALL NUMBER: QA614.86.A67 1993
Introduction to fractals and chaos: AUTHOR: Crownover, Richard M.
PUBLISHER: Jones and Bartlett, c1995.
CALL NUMBER: QA614.86.C74 1995
Chaos, fractals, and dynamics : computer experiments in mathematics: AUTHOR: Devaney, Robert L., 1948-
PUBLISHER: Menlo Park, Calif. ; Don Mills, Ont. : Addison-Wesley Pub. Co., c1990.
CALL NUMBER: QA614.86.D48 1990
[Written to be understandable by a high school student.]
An introduction to chaotic dynamical systems: AUTHOR: Devaney, Robert L., 1948-
EDITION: 2nd ed. -
PUBLISHER: 2nd ed.: Addison-Wesley, c1989.
CALL NUMBER: QA614.8.D48 1989
[Covers a lot of the same material on iterations of dynamical systems as Gulick does, but is a little more mathematically sophisticated.]
Complex dynamical systems : the mathematics behind the Mandelbrot and Julia sets: AUTHOR: Robert L. Devaney, editor
PUBLISHER: American Mathematical Society, 1994.
CALL NUMBER: QA614.8.C645 1994
[A follow-up to "Chaos and fractals : the mathematics behind the computer graphics", but more mathematically technical.]
Chaos and fractals : the mathematics behind the computer graphics: AUTHOR: Robert L. Devaney and Linda Keen, editors
PUBLISHER: American Mathematical Society, 1989.
CALL NUMBER: T385.C454 1989
[Fairly technical, with the emphasis on the mathematics, but still quite readable.]
Measure, topology, and fractal geometry: AUTHOR: Edgar, Gerald A., 1949-
PUBLISHER: Springer-Verlag, c1990.
CALL NUMBER: QA614.86.E34 1990
Discrete chaos: AUTHOR: Saber N. Elaydi
PUBLISHER: Chapman & Hall/CRC, c2000.
CALL NUMBER: QA614.8 .E53 2000
Fractal geometry : mathematical foundations and applications: AUTHOR: Falconer, K. J., 1952-
PUBLISHER: Wiley, c1990.
CALL NUMBER: QA614.86.F35 1990
[A good source for the more measure-theoretical aspects of Hausdorff and other dimensions.]
Symmetry in chaos : a search for pattern in mathematics, art and nature: AUTHOR: Mike Field and Martin Golubitsky
PUBLISHER: Oxford University Press, 1992.
CALL NUMBER: Q172.5.S95F54 1992
[A colourful book on iterating two dimensional real functions with various symmetries imposed.]
Fractal image compression : theory and application.: AUTHOR: Yuval Fisher
PUBLISHER: New York ; Berlin : Springer-Verlag, c1995.
CALL NUMBER: TA1637.F73 1995
The computational beauty of nature : computer explorations of fractals, chaos, complex systems, and adaptation.: AUTHOR: Flake, Gary William
PUBLISHER: Cambridge, Mass : MIT Press, c1998.
CALL NUMBER: QA76.6.F557 1998
Fractals, graphics, and mathematics education: AUTHOR: Michael Frame and Benoit Mandelbrot
PUBLISHER: Mathematical Association of America, c2002.
CALL NUMBER: QA614.86 .F68424 2002
Chaos : making a new science: AUTHOR: Gleick, James
PUBLISHER: Viking, 1987.
CALL NUMBER: Q172.5.C45G54 1987 (PORTER)
[A very readable general book about the history of chaos and fractals.]
Encounters with chaos: AUTHOR: Gulick, Denny.
PUBLISHER: McGraw-Hill, c1992.
CALL NUMBER: Q172.5.C45G85 1992
[The text for this course.]
Exploring chaos : a guide to the new science of disorder: AUTHOR: Nina Hall, editor
PUBLISHER: W.W. Norton, 1993.
CALL NUMBER: Q172.5.C45E98 1993
[General essays on a variety of topics.]
Fractals : a user's guide for the natural sciences: AUTHOR: Hastings, Harold M.
PUBLISHER: Oxford University Press, 1993.
CALL NUMBER: QA614.86.H37 1993
Chaos and nonlinear dynamics : an introduction for scientists and engineers: AUTHOR: Hilborn, Robert C.
PUBLISHER: Oxford University Press, 1994.
CALL NUMBER: Q172.5.C45H55 1994
A first course in discrete dynamical systems: AUTHOR: Holmgren, Richard A.
PUBLISHER: Springer-Verlag, c1994.
CALL NUMBER: QA614.8.H65 1994
Controlling chaos : theoretical and practical methods in non-linear dynamics: AUTHOR: Tomasz Kapitaniak
PUBLISHER: Academic Press, c1996.
CALL NUMBER: Q172.5.C45K372x 1996
Understanding nonlinear dynamics: AUTHOR: Kaplan, Daniel.
PUBLISHER: Springer-Verlag, c1995.
CALL NUMBER: QA845.K36 1995
Chaos & complexity : discovering the surprising patterns of science and technology: AUTHOR: Brian Kaye
PUBLISHER: Weinheim ; New York : VCH, c1993
CALL NUMBER: Q172.5.C45K39 1993
A random walk through fractal dimensions: AUTHOR: Brian Kaye
PUBLISHER: Weinheim : VCH, 1989 or 1994.
CALL NUMBER: QA614.86.K39x 1989
Invariant factors, Julia equivalences, and the (abstract) Mandelbrot set: AUTHOR: Karsten Keller
PUBLISHER: Lecture notes in mathematics, 1732. Springer, c2000
CALL NUMBER: QA3 .L28 no. 1732
The Ubiquity of chaos: AUTHOR: Saul Krasner
PUBLISHER: Washington, D.C. : American Association for the Advancement of Science, c1990.
CALL NUMBER: Q172.5.C45U25 1990
Fractals : endlessly repeated geometrical figures: AUTHOR: Lauwerier, H.
PUBLISHER: Princeton University Press, 1991.
CALL NUMBER: QA614.86.L3813 1991
[Written for a general audience with a high school level of mathematics.]
The essence of chaos: AUTHOR: Edward N. Lorenz
PUBLISHER: Seattle : University of Washington Press, c1993.
CALL NUMBER: Q172.5.C45L67 1993
Dynamical systems with applications using MAPLE: AUTHOR: Stephen Lynch
PUBLISHER: Boston : BirkhYuser, c2001.
CALL NUMBER: QA614.8.L96 2001
Fractal Imaging: AUTHOR: Ning Lu
PUBLISHER: Academic Press, 1997.
CALL NUMBER: TA1637 .L8 1997
The fractal geometry of nature: AUTHOR: Benoit B. Mandelbrot
PUBLISHER: New York : W.H. Freeman, c1983.
CALL NUMBER: QA447.M357 1983
[The most recent edition of the classic book that introduced fractals in English.]
Dynamics in one complex variable : introductory lectures: AUTHOR: John Milnor
PUBLISHER: Braunschweig : Vieweg, 2000.
CALL NUMBER: QA331.7 .M575x 2000
Chaotic and fractal dynamics : an introduction for applied scientists and engineers: AUTHOR: Francis C. Moon
PUBLISHER: Wiley, c1992.
CALL NUMBER: Q172.5.C45.M66 1992
Dynamics : numerical explorations: AUTHOR: Nusse, Helena E. & Yorke, James A.
PUBLISHER: Springer-Verlag, 1994.
CALL NUMBER: QA1.A647 v. 101
Chaos in dynamical systems: AUTHOR: Ott, Edward
PUBLISHER: Cambridge University Press, 1993.
CALL NUMBER: Q172.5.C45O87 1993
Chaos under control : the art and science of complexity: AUTHOR: David Peak and Michael Frame.
PUBLISHER: New York : W.H. Freeman, c1994.
CALL NUMBER: Q172.5.C45P43 1994
Chaos and Fractals: new frontiers of science: AUTHOR: Heinz-Otto Peitgen et al.
PUBLISHER: Springer-Verlag, c1992
CALL NUMBER: QA614.86.P43 1992
[A large encyclopedic book on Chaos and Fractals.]
Fractals for the classroom, Volumes 1 and 2: AUTHOR: Heinz-Otto Peitgen et al.
PUBLISHER: Springer-Verlag, c1991-
CALL NUMBER: QA614.86.P45 1991 and QA614.86.P45 1991 v.2
Newton's method and dynamical systems: AUTHOR: Heinz-Otto Peitgen, editor
PUBLISHER: Kluwer Academic, c1989.
CALL NUMBER: QA614.8.N495 1989
The beauty of fractals : images of complex dynamical systems: AUTHOR: Heinz-Otto Peitgen, P.H. Richter.
PUBLISHER: Springer-Verlag, c1986.
CALL NUMBER: QA447.P45 1986
[Many pictures of Julia and Mandelbrot sets.]
The Science of fractal images: AUTHOR: Heinz-Otto Peitgen, Dietmar Saupe, editors
PUBLISHER: New York : Springer-Verlag, c1988.
CALL NUMBER: QA614.86.S35 1988
[Based on courses given at SIGGRAPH in 1987.]
Fractal horizons : the future use of fractals: AUTHOR: edited by Clifford A. Pickover
PUBLISHER: St. Martin's Press, 1996.
CALL NUMBER: QA614.86.F6845 1996
The chaos cookbook : a practical programming guide: AUTHOR: Joe Pritchard
PUBLISHER: Oxford, England : Butterworth-Heinemann, 1996.
CALL NUMBER: Q172.5.C45P75 1996
The algorithmic beauty of plants (online): AUTHOR: Przemyslaw Pruskinkiewicz, Aristid Lindenmayer
PUBLISHER: Springer-Verlag, c1990.
CALL NUMBER: QK46.5.M3P78x 1990
[A mathematical theory of plant development using L-systems.]
An introduction to dynamical systems : continuous and discrete: AUTHOR: Robinson, R. Clark
PUBLISHER: Prentice Hall, c2004.
CALL NUMBER: QA614.8.R65 2004
Space-filling curves: AUTHOR: Sagan, Hans
PUBLISHER: New York : Springer-Verlag, c1994.
CALL NUMBER: QA643.S12 1994
Discrete dynamical systems : theory and applications: AUTHOR: Sandefur, James T.
PUBLISHER: Oxford University Press, 1990.
CALL NUMBER: QA614.8.S25 1990
Invitation to Dynamical Systems: AUTHOR: Scheinerman, Edward R.
PUBLISHER: Prentice Hall, N.J. 1996.
CALL NUMBER: QA614.8.S34 1996
[A textbook with an emphasis on dynamical systems in the plane, but also covering fractals, iterated function systems, and the iteration of complex functions.]
Explaining chaos: AUTHOR: Peter Smith
PUBLISHER: Cambridge University Press, 1998.
CALL NUMBER: Q172.5.C45 S55 1998
Fractal programming in C: AUTHOR: Stevens, Roger T.
PUBLISHER: Redwood City, Calif. : M & T Books, 1989.
CALL NUMBER: QA614.86.S74 1989
Does God play dice? : the mathematics of chaos: AUTHOR: Ian Stewart
PUBLISHER: Oxford, UK : B. Blackwell, 1989.
CALL NUMBER: Q172.5.C45S74 1989
Nonlinear dynamics and Chaos : with applications in physics, biology, chemistry, and engineering: AUTHOR: Strogatz, Steven H. (Steven Henry)
PUBLISHER: Addison-Wesley Pub., c1994.
CALL NUMBER: Q172.5.C45S767 1994
Sync : the emerging science of spontaneous order: AUTHOR: Strogatz, Steven H. (Steven Henry)
PUBLISHER: Hyperion Press, c2003.
CALL NUMBER: Q172.5.S96 S87 [Guelph]
[A readable general book explaining nonlinear dynamics and chaos.]
Fractal geometry in digital imaging: AUTHOR: Martin J. Turner, Jonathan M. Blackledge, Patrick R. Andrews.
PUBLISHER: San Diego, Calif. : Academic Press, c1998.
CALL NUMBER: TA1637 .T88x 1998
Chaos theory tamed: AUTHOR: Williams, Garnett P.
PUBLISHER: Washington, D.C. : Joseph Henry Press, 1997.
CALL NUMBER: Q172.5.C45W55x 1997

Articles on Chaos and Fractals

You will need to be logged in from uwaterloo.ca [or other institution] to gain access to the links of some journals.

S. Bailey, T. Kim and R.S. Strichartz,"Inside the Levy Dragon", Amer. Math. Monthly 109 (2002) pp.689-703.
S. Bassein, "The Dynamics of a Family of One-Dimensional Maps", Amer. Math. Monthly 105 (1998) pp.118-130.
R. Corless, "Continued Fractions and Chaos", Amer. Math. Monthly 99 (1992) pp.203-215.
R. L. Devaney, "The Mandelbrot Set, the Farey Tree, and the Fibonacci Sequence", Amer. Math. Monthly 106 (1999) pp.289-302.
R. L. Devaney, "Cantor and Sierpinski, Julia and Fatou: Complex Topology Meets Complex Dynamics", in Notices of the Amer. Math. Soc. 51#1 (January 2004) pp.9-15.
W. Derrick and J. Eidwick, "Continued Fractions, Chebychev Polynomials and Chaos", Amer. Math. Monthly 102#4 (April 1995) pp.337-344.
B. I. Epureanu and H. S. Greenside, "Fractal Basins of Attraction associated with a damped Newton's method", Chaotic Dynamics arXiv.org (Dec 1996)
M. Frame, B. Johnson and J. Sauerberg, "Fixed Points and Fermat: A Dynamical Systems Approach to Number Theory", Amer. Math. Monthly 107#5 (May 2000) pp.422-428.
J. T. Sandefur, "Using Self-Similarity to Find Length, Area, and Dimension", Amer. Math. Monthly 103 (1996) pp.107-120.
P. Touhey, "Yet Another Definition of Chaos", Amer. Math. Monthly 104#5 (May 1997) pp.411-414.
Roger L. Kraft, "Chaos, Cantor Sets, and Hyperbolicity for the Logistic Maps", Amer. Math. Monthly 106 (1999) pp.400-408.
G. Strang, "A Chaotic search for i", College Math. Journal 22 (1991) pp.3-12.
W. Ditto and L. Pecora, "Mastering Chaos," in Scientific American (1993), Vol. 269, pp. 78.
Benoit B. Mandelbrot, "A Multifractal Walk Down Wall Street," in Scientific American (February 1999), Vol. 280#2, pp. 70-73.
R.D. Neidinger and R.J. Annen"The Road to Chaos is Filled with Polynomial Curves,", Amer. Math. Monthly 103#8 (October 1996) pp.640-653.
S. Bedding and K. Briggs"Iteration of Quaternion Functions,", Amer. Math. Monthly 103#8 (October 1996) pp.654-664.
R. Barton and K. Burns"A Simple Special Case of Sharkovskii's Theorem," in Amer. Math. Monthly 107#10 (December 2000) pp.932-933.
W.J. Gilbert"Fractal Geometry derived from Complex Bases," in The Mathematical Intelligencer 4#2 (1982) pp.78-86.
I. Stewart"Four Encounters with Sierpinski's Gasket," in The Mathematical Intelligencer 17#1 (Winter 1995) pp.52-64.
Tien-Yien Li; James A. Yorke"Period Three Implies Chaos,", Amer. Math. Monthly 82#10 (December 1975) pp.985-992.
M. Lyubich"The Quadratic family as a Qualitatively Solvable Model of Chaos," in Notices of the Amer. Math. Soc. 47#9 (October 2000) pp.1042-1052.
W. Willinger and V. Paxson"Where Mathematics Meets the Internet," in Notices of the Amer. Math. Soc. 45#8 (September 1998) pp.961-970. (Fractal scaling phenomena in Internet traffic)
L. Yau and A. Ben-Israel"The Newton and Halley Methods for Complex Roots,", Amer. Math. Monthly 105#9 (November 1997) pp.806-818.
[see also E. R. Vrscay and W. J. Gilbert"Extraneous Fixed Points, Basin Boundaries and Chaotic Dynmaics for Schröder and König Rational Iteration Functions," in Numerische Mathematik 52 (1988) pp.1-16.]
Computers and Graphics [CALL NUMBER: Per T385.C64x and online access from UW computers] has a Chaos and Graphics section in each issue, with many articles that you should be able to understand.
MathSciNet Search of research papers (works from uwaterloo.ca computers; will only work if your institution subscribes to this service)

Videos on Chaos and Fractals

Search the WATMEDIA Database for films and videos obtainable at Waterloo (from UW and other Ontario universities).

Here are some videos I have or know about [and where there are copies]:

The Beauty and Complexity of the Mandelbrot Set, with John Hubbard, Science Television, 73 mins, 1989. [W. Gilbert and U of Waterloo Media centre]
Chaos, Fractals and Dynamics: Computer Experiments in Mathematics, with Robert Devaney, Science Television, 63 mins, 1989. [W. Gilbert and U of Waterloo Media centre]
Transition to Chaos: the Orbit Diagram and the Mandelbrot Set, with Robert Devaney, Science Television, 1990. [W. Gilbert and U of Waterloo Media centre]
Mandelbrot and Julia Sets, by John Hubbard, Art Matrix, 120 mins, 1990. [W. Gilbert and U of Ottawa]
Fractals: The Colors of Infinity, narrated by Arthur C. Clarke, directed by Nigel Lesmoir-Gordon, 52 mins, 1994. [W. Gilbert]
Strange New Science of Chaos, Nova, 58 mins, 1988. [U of Waterloo Media centre]
Chaos: The New Science Of The Unexpected, CBC. [U of Toronto]
Chaos: An Emerging Science, University of Maryland, 287 mins, 1989. (A satellite teleconference on chaos theory.) [U of Toronto]

Computer Programs on Chaos and Fractals

Fractint: Fractint is a very popular freeware (not public domain) fractal generator. There are DOS, Windows, and Unix/X versions. There are plans to port to Windows NT and the Mac. See Fractint WWW pages to get the latest versions.
Fractal Creations, Second Edition is the definitive Fractint book.: AUTHOR: Tim Wegner, Bert Tyler
PUBLISHER: Corte Madera, CA : Waite Group Press , c1993.
CALL NUMBER: QA614.86 .W44 1993 [The library copy lacks accompanying materials like Disk and CD, as it was bought second-hand. However most of the information is available on the web from the Index to Fractint documentation.]
Ultra Fractal is a shareware Windows program for creating beautiful images of various Mandelbrot and Julia set fractals in 256 colours, high-color or true-color.: AUTHOR: Frederik Slijkerman
Fractal Image Compression Software for Windows.: AUTHOR: Giovambattista Pulcini
Dynamics is a free program for Mac OS X written to illustrate the connection between Julia Sets and the Mandelbrot Set. Source code is available.: AUTHORS: Richard Koch and Pierre d'Herbemont
ChaosPro is a freeware fractal generator for the PC.: AUTHOR: Martin Pfingstl
Gaston: Gaston provides interactive, stereoscopic real-time 3D rendering of 4D quaternion Julia set fractals on a Mac. Download via VersionTracker; AUTHOR: Leo Fink
Filmer fractal animation system A program for creating fractal animation from Fractint.: AUTHOR: Julian Haight
XaoS real-time interactive fractal zoomer for various platforms.: AUTHOR: Thomas Marsh and Jan Hubicka
Fractal Forge is a freeware fractal generator for the PC.: AUTHOR: Uberto Barbini
Google Chaos and Fractals software links
Fractal SoftWare Links from Paul N. Lee
MORE PC LINKS from Download.com
MORE MAC LINKS from VersionTracker.com for OS X
Java demos shown in class.

Back to PMath 370 This page was last updated on

References on Chaos and Fractals for PMath 370 at the University of Waterloo