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_Communications in Mathematical Physics_ 134 (1990), pp. 587-617.
2. J. Milnor, Self-Similarity and Hairiness in the Mandelbrot Set, in
_Computers in Geometry and Topology_, M. Tangora (editor), Dekker,
New York, pp. 211-257.
The "external angles" of the Mandelbrot set (see Douady and Hubbard or
brief sketch in "Beauty of Fractals") induce a Fibonacci partition
onto it.
The boundary of the Mandelbrot set and the Julia set of a generic c in
M have Hausdorff dimension 2 and have topological dimension 1. The
proof is based on the study of the bifurcation of parabolic periodic
points. (Since the boundary has empty interior, the topological
dimension is less than 2, and thus is 1.)
Reference:
1. M. Shishikura, The Hausdorff Dimension of the Boundary of the
Mandelbrot Set and Julia Sets, The paper is available from
anonymous ftp: ftp://math.sunysb.edu/preprints/ims91-7.ps.Z
_Q6h_: Is the Mandelbrot set connected?
_A6h_: The Mandelbrot set is simply connected. This follows from a
theorem of Douady and Hubbard that there is a conformal isomorphism
from the complement of the Mandelbrot set to the complement of the
unit disk. (In other words, all equipotential curves are simple closed
curves.) It is conjectured that the Mandelbrot set is locally
connected, and thus pathwise connected, but this is currently
unproved.
Connectedness definitions:
Connected: X is connected if there are no proper closed subsets A and
B of X such that A union B = X, but A intersect B is empty. I.e. X is
connected if it is a single piece.
Simply connected: X is simply connected if it is connected and every
closed curve in X can be deformed in X to some constant closed curve.
I.e. X is simply connected if it has no holes.
Locally connected: X is locally connected if for every point p in X,
for every open set U containing p, there is an open set V containing p
and contained in the connected component of p in U. I.e. X is locally
connected if every connected component of every open subset is open in
X. Arcwise (or path) connected: X is arcwise connected if every two
points in X are joined by an arc in X.
(The definitions are from _Encyclopedic Dictionary of Mathematics_.)
Reference:
Douady, A. and Hubbard, J., "Comptes Rendus" (Paris) 294, pp.123-126,
1982.
_Q6i_: What is the Mandelbrot Encyclopedia?
_A6i_: The Mandelbrot Encyclopedia is a web page by Robert Munafo
about the Mandelbrot Set. It is available
via WWW at .
_Q6j_: What is the dimension of the Mandelbrot Set?
_A6j_: The Mandelbrot Set has a dimension of 2. The Mandelbrot Set
contains and is contained in a disk. A disk has a dimension of 2, thus
so does the Mandelbrot Set.
The Koch snowflake (Hausdorff dimension 1.2619...) does not satisfy
this condition because it is a thin boundary curve, thus containing no
disk. If you add the region inside the curve then it does have
dimension of 2.
The boundary of the Mandelbrot set and the Julia set of a generic c in
M have Hausdorff dimension 2 and have topological dimension 1. The
proof is based on the study of the bifurcation of parabolic periodic
points. (Since the boundary has empty interior, the topological
dimension is less than 2, and thus is 1.) See reference above
_Q6k_: What are the seahorse and the elephant valleys?
_A6k_: The Mandelbrot set being the most famous fractal, its various
regions are well known and many of them have popular names evoking
graphic details found by zooming into them. The seahorse valley is the
limit border of the main cardioid at the negative side of the x axis
(near to x=-0.75, y=0.0). You can see here convoluted and complex buds
looking more or less like seahorses. The elephant valley is near the
symetry plane on the positive side of the x axis (x=0.25, y=0.0).
Spirals protuding from the border evoke trunks of elephants. By
zooming in these regions many interesting structures can be seen.
A nice guide (by Paul Derbyshire) to explore the various regions of
the Mandelbrot set can be found at :
http://chat.carleton.ca/~pderbysh/manguide.htlm
Subject: Julia sets
_Q7a_: What is the difference between the Mandelbrot set and a Julia
set?
_A7a_: The Mandelbrot set iterates z^2 + c with z starting at 0 and
varying c. The Julia set iterates z^2 + c for fixed c and varying
starting z values. That is, the Mandelbrot set is in parameter space
(c-plane) while the Julia set is in dynamical or variable space
(z-plane).
_Q7b_: What is the connection between the Mandelbrot set and Julia
sets?
_A7b_: Each point c in the Mandelbrot set specifies the geometric
structure of the corresponding Julia set. If c is in the Mandelbrot
set, the Julia set will be connected. If c is not in the Mandelbrot
set, the Julia set will be a Cantor dust.
_Q7c_: How is a Julia set actually computed?
_A7c_: The Julia set can be computed by iteration similar to the
Mandelbrot computation. The only difference is that the c value is
fixed and the initial z value varies.
Alternatively, points on the boundary of the Julia set can be computed
quickly by using inverse iterations. This technique is particularly
useful when the Julia set is a Cantor Set. In inverse iteration, the
equation z1 = z0^2 + c is reversed to give an equation for z0: z0 =
ħsqrt(z1 - c). By applying this equation repeatedly, the resulting
points quickly converge to the Julia set boundary. (At each step,
either the positive or negative root is randomly selected.) This is a
nonlinear iterated function system.
In pseudocode:
z = 1 (or any value)
loop
if (random number < .5) then
z = sqrt(z - c)
else
z = -sqrt(z - c)
endif
plot z
end loop
_Q7d_: What are some Julia set facts?
_A7d_: The Julia set of any rational map of degree greater than one is
perfect (hence in particular uncountable and nonempty), completely
invariant, equal to the Julia set of any iterate of the function, and
also is the boundary of the basin of attraction of every attractor for
the map.
Julia set references:
1. A. F. Beardon, _Iteration of Rational Functions : Complex Analytic
Dynamical Systems_, Springer-Verlag, New York, 1991.
2. P. Blanchard, Complex Analytic Dynamics on the Riemann Sphere,
_Bull. of the Amer. Math. Soc_ 11, 1 (July 1984), pp. 85-141.
This article is a detailed discussion of the mathematics of iterated
complex functions. It covers most things about Julia sets of rational
polynomial functions.
Subject: Complex arithmetic and quaternion arithmetic
_Q8a_: How does complex arithmetic work?
_A8a_: It works mostly like regular algebra with a couple additional
formulas:
(note: a, b are reals, _x_, _y_ are complex, _i_ is the square root of
-1)
Powers of _i_:
_i_^2 = -1
Addition:
(a+_i_*b)+(c+_i_*d) = (a+c)+_i_*(b+d)
Multiplication:
(a+_i_*b)*(c+_i_*d) = a*c-b*d + _i_*(a*d+b*c)
Division:
(a+_i_*b) / (c+_i_*d) = (a+_i_*b)*(c-_i_*d) / (c^2+d^2)
Exponentiation:
exp(a+_i_*b) = exp(a)*(cos(b)+_i_*sin(b))
Sine:
sin(_x_) = (exp(_i_*_x_) - exp(-_i_*_x_)) / (2*_i_)
Cosine:
cos(_x_) = (exp(_i_*_x_) + exp(-_i_*_x_)) / 2
Magnitude:
|a+_i_*b| = sqrt(a^2+b^2)
Log:
log(a+_i_*b) = log(|a+_i_*b|)+_i_*arctan(b / a) (Note: log is
multivalued.)
Log (polar coordinates):
log(r e^(_i_*a)) = log(r)+_i_*a
Complex powers:
_x_^y = exp(y*log(x))
de Moivre's theorem:
_x_^n = r^n [cos(n*a) + _i_*sin(n*a)] (where n is an integer)
More details can be found in any complex analysis book.
_Q8b_: How does quaternion arithmetic work?
_A8b_: quaternions have 4 components (a + _i_b + _j_c + _k_d) compared
to the two of complex numbers. Operations such as addition and
multiplication can be performed on quaternions, but multiplication is
not commutative.
Quaternions satisfy the rules
* i^2 = j^2 = k^2 = -1
* ij = -ji = k
* jk = -kj = i,
* ki = -ik = j
See:
Frode Gill's quaternions page
http://www.krs.hia.no/~fgill/quatern.html
Subject: Logistic equation
_Q9_: What is the logistic equation?
_A9_: It models animal populations. The equation is x -> c x (1 - x),
where x is the population (between 0 and 1) and c is a growth
constant. Iteration of this equation yields the period doubling route
to chaos. For c between 1 and 3, the population will settle to a fixed
value. At 3, the period doubles to 2; one year the population is very
high, causing a low population the next year, causing a high
population the following year. At 3.45, the period doubles again to 4,
meaning the population has a four year cycle. The period keeps
doubling, faster and faster, at 3.54, 3.564, 3.569, and so forth. At
3.57, chaos occurs; the population never settles to a fixed period.
For most c values between 3.57 and 4, the population is chaotic, but
there are also periodic regions. For any fixed period, there is some c
value that will yield that period. See _An Introduction to Chaotic
Dynamical Systems_, by R. L. Devaney, for more information.
Subject: Feigenbaum's constant
_Q10_: What is Feigenbaum's constant?
_A10_: In a period doubling cascade, such as the logistic equation,
consider the parameter values where period-doubling events occur (e.g.
r[1]=3, r[2]=3.45, r[3]=3.54, r[4]=3.564...). Look at the ratio of
distances between consecutive doubling parameter values; let delta[n]
= (r[n+1]-r[n])/(r[n+2]-r[n+1]). Then the limit as n goes to infinity
is Feigenbaum's (delta) constant.
Based on computations by F. Christiansen, P. Cvitanovic and H.H. Rugh,
it has the value 4.6692016091029906718532038... _Note_: several books
have published incorrect values starting 4.6692016_6_...; the last
repeated 6 is a _typographical error_.
The interpretation of the delta constant is as you approach chaos,
each periodic region is smaller than the previous by a factor
approaching 4.669...
Feigenbaum's constant is important because it is the same for any
function or system that follows the period-doubling route to chaos and
has a one-hump quadratic maximum. For cubic, quartic, etc. there are
different Feigenbaum constants.
Feigenbaum's alpha constant is not as well known; it has the value
2.50290787509589282228390287272909. This constant is the scaling
factor between x values at bifurcations. Feigenbaum says,
"Asymptotically, the separation of adjacent elements of period-doubled
attractors is reduced by a constant value [alpha] from one doubling to
the next". If d[a] is the algebraic distance between nearest elements
of the attractor cycle of period 2^a, then d[a]/d[a+1] converges to
-alpha.
References:
1. K. Briggs, How to calculate the Feigenbaum constants on your PC,
_Aust. Math. Soc. Gazette_ 16 (1989), p. 89.
2. K. Briggs, A precise calculation of the Feigenbaum constants,
_Mathematics of Computation_ 57 (1991), pp. 435-439.
3. K. Briggs, G. R. W. Quispel and C. Thompson, Feigenvalues for
Mandelsets, _J. Phys. A_ 24 (1991), pp. 3363-3368.
4. F. Christiansen, P. Cvitanovic and H.H. Rugh, "The spectrum of the
period-doubling operator in terms of cycles", _J. Phys A_ 23, L713
(1990).
5. M. Feigenbaum, The Universal Metric Properties of Nonlinear
Transformations, _J. Stat. Phys_ 21 (1979), p. 69.
6. M. Feigenbaum, Universal Behaviour in Nonlinear Systems, _Los
Alamos Sci_ 1 (1980), pp. 1-4. Reprinted in _Universality in
Chaos_, compiled by P. Cvitanovic.
Feigenbaum Constants
http://www.mathsoft.com/asolve/constant/fgnbaum/fgnbaum.html
Subject: Iterated function systems and compression
_Q11a_: What is an iterated function system (IFS)?
_A11a_: If a fractal is self-similar, you can specify mappings that
map the whole onto the parts. Iteration of these mappings will result
in convergence to the fractal attractor. An IFS consists of a
collection of these (usually affine) mappings. If a fractal can be
described by a small number of mappings, the IFS is a very compact
description of the fractal. An iterated function system is By taking a
point and repeatedly applying these mappings you end up with a
collection of points on the fractal. In other words, instead of a
single mapping x -> F(x), there is a collection of (usually affine)
mappings, and random selection chooses which mapping is used.
For instance, the Sierpinski triangle can be decomposed into three
self-similar subtriangles. The three contractive mappings from the
full triangle onto the subtriangles forms an IFS. These mappings will
be of the form "shrink by half and move to the top, left, or right".
Iterated function systems can be used to make things such as fractal
ferns and trees and are also used in fractal image compression.
_Fractals Everywhere_ by Barnsley is mostly about iterated function
systems.
The simplest algorithm to display an IFS is to pick a starting point,
randomly select one of the mappings, apply it to generate a new point,
plot the new point, and repeat with the new point. The displayed
points will rapidly converge to the attractor of the IFS.
Interactive IFS Playground (Otmar Lendl)
http://www.cosy.sbg.ac.at/rec/ifs/
Frank Rousell's hyperindex of IFS images
http://www.cnam.fr/fractals/ifs.html
_Q11b_: What is the state of fractal compression?
_A11b_: Fractal compression is quite controversial, with some people
claiming it doesn't work well, and others claiming it works
wonderfully. The basic idea behind fractal image compression is to
express the image as an iterated function system (IFS). The image can
then be displayed quickly and zooming will generate infinite levels of
(synthetic) fractal detail. The problem is how to efficiently generate
the IFS from the image. Barnsley, who invented fractal image
compression, has a patent on fractal compression techniques
(4,941,193). Barnsley's company, Iterated Systems Inc
(http://www.iterated.com/), has a line of products including a Windows
viewer, compressor, magnifier program, and hardware assist board.
Fractal compression is covered in detail in the comp.compression FAQ
file (See "compression-FAQ").
ftp://rtfm.mit.edu/pub/usenet/comp.compression .
One of the best online references for Fractal Compress is Yuval
Fisher's Fractal Image Encoding page
(http://inls.ucsd.edu/y/Fractals/) at the Institute for Nonlinear
Science, University for California, San Diego. It includes references
to papers, other WWW sites, software, and books about Fractal
Compression.
Three major research projects include:
Waterloo Montreal Verona Fractal Research Initiative
http://links.uwaterloo.ca/
Groupe FRACTALES
http://www-syntim.inria.fr/fractales/
Bath Scalable Video Software Mk 2
http://dmsun4.bath.ac.uk/bsv-mk2/
Several books describing fractal image compression are:
1. M. Barnsley, _Fractals Everywhere_, Academic Press Inc., 1988.
ISBN 0-12-079062-9. This is an excellent text book on fractals.
This is probably the best book for learning about the math
underpinning fractals. It is also a good source for new fractal
types.
2. M. Barnsley and L. Anson, _The Fractal Transform_, Jones and
Bartlett, April, 1993. ISBN 0-86720-218-1. Without assuming a
great deal of technical knowledge, the authors explain the
workings of the Fractal Transform(TM).
3. M. Barnsley and L. Hurd, _Fractal Image Compression_, Jones and
Bartlett. ISBN 0-86720-457-5. This book explores the science of
the fractal transform in depth. The authors begin with a
foundation in information theory and present the technical
background for fractal image compression. In so doing, they
explain the detailed workings of the fractal transform. Algorithms
are illustrated using source code in C.
4. Y. Fisher (Ed), _Fractal Image Compression: Theory and
Application_. Springer Verlag, 1995.
5. Y. Fisher (Ed), _Fractal Image Encoding and Analysis: A NATO ASI
Series Book_, Springer Verlag, New York, 1996 contains the
proceedings of the Fractal Image Encoding and Analysis Advanced
Study Institute held in Trondheim, Norway July 8-17, 1995. The
book is currently being produced.
Some introductary articles about fractal compression:
1. The October 1993 issue of Byte discussed fractal compression. You
can ftp sample code:
ftp://ftp.uu.net/published/byte/93oct/fractal.exe .
2. A Better Way to Compress Images," M.F. Barnsley and A.D. Sloan,
BYTE, pp. 215-223, January 1988.
3. "Fractal Image Compression," M.F. Barnsley, Notices of the
American Mathematical Society, pp. 657-662, June 1996.
(http://www.ams.org/publications/notices/199606/barnsley.html)
4. A. E. Jacquin, Image Coding Based on a Fractal Theory of Iterated
Contractive Image Transformation, _IEEE Transactions on Image
Processing_, January 1992.
5. A "Hitchhiker's Guide to Fractal Compression" For Beginners by
E.R. Vrscay
ftp://links.uwaterloo.ca/pub/Fractals/Papers/Waterloo/vr95.ps.gz
Andreas Kassler wrote a Fractal Image Compression with WINDOWS package
for a Fractal Compression thesis. It is available at
http://www-vs.informatik.uni-ulm.de/Mitarbeiter/Kassler/papers.htm
Other references:
Fractal Compression Bibliography
http://www.dip.ee.uct.ac.za/imageproc/compression/fractal/fract
al.bib.html
Fractal Video Compression
http://inls.ucsd.edu/y/Fractals/Video/fracvideo.html
Many fractal image compression papers are available from
ftp://ftp.informatik.uni-freiburg.de/documents/papers/fractal
A review of the literature is in Guide.ps.gz.
ftp://ftp.informatik.uni-freiburg.de/documents/papers/fractal/R
EADME
Subject: Chaotic demonstrations
_Q12a_: How can you make a chaotic oscillator?
_A12a_: Two references are:
1. T. S. Parker and L. O. Chua, Chaos: a tutorial for engineers,
_Proceedings IEEE_ 75 (1987), pp. 982-1008.
2. _New Scientist_, June 30, 1990, p. 37.
_Q12b_: What are laboratory demonstrations of chaos?
_A12b_: Robert Shaw at UC Santa Cruz experimented with chaos in
dripping taps. This is described in:
1. J. P. Crutchfield, Chaos, _Scientific American_ 255, 6 (Dec.
1986), pp. 38-49.
2. I. Stewart, _Does God Play Dice?: the Mathematics of Chaos_, B.
Blackwell, New York, 1989.
Two references to other laboratory demonstrations are:
1. K. Briggs, Simple Experiments in Chaotic Dynamics, _American
Journal of Physics_ 55, 12 (Dec 1987), pp. 1083-1089.
2. J. L. Snider, Simple Demonstration of Coupled Oscillations,
_American Journal of Physics_ 56, 3 (Mar 1988), p. 200.
See sci.nonlinear FAQ and the sci.nonlinear newsgroup for further
information.
Subject: L-Systems
_Q13_: What are L-systems?
_A13_: A L-system or Lindenmayer system is a formal grammar for
generating strings. (That is, it is a collection of rules such as
replace X with XYX.) By recursively applying the rules of the L-system
to an initial string, a string with fractal structure can be created.
Interpreting this string as a set of graphical commands allows the
fractal to be displayed. L-systems are very useful for generating
realistic plant structures.
Some references are:
1. P. Prusinkiewicz and J. Hanan, _Lindenmayer Systems, Fractals, and
Plants_, Springer-Verlag, New York, 1989.
2. P. Prusinkiewicz and A. Lindenmayer, _The Algorithmic Beauty of
Plants_, Springer-Verlag, NY, 1990. ISBN 0-387-97297-8. A very
good book on L-systems, which can be used to model plants in a
very realistic fashion. The book contains many pictures.
_________________________________________________________________
More information can be obtained via the WWW at:
L-Systems Tutorial by David Green
http://life.csu.edu.au/complex/tutorials/tutorial2.html
http://www.csu.edu.au/complex_systems/tutorial2.html
Graphics Archive at the Center for the Computation and Visualization
of Geometric Structures contains various fractals created from
L-Systems.
http://www.geom.umn.edu/graphics/pix/General_Interest/Fractals/
Subject: Fractal music
_Q14_: What are sources of fractal music?
_A14_: One fractal recording is "The Devil's Staircase: Composers and
Chaos" on the Soundprint label. A second is "Curves and Jars" by Barry
Lewis. You can contact MPS Music & Video for further information:
Rosegarth, Hetton Road, Houghton-le-Spring, DH5 8JN, England or online
at CDeMUSIC (http://www.emf.org/focus_lewisbarry.html).
Does anyone know of others? Mail me at fractal-faq@mta.ca.
Some references, many from an unpublished article by Stephanie Mason,
are:
1. R. Bidlack, Chaotic Systems as Simple (But Complex) Compositional
Algorithms, _Computer Music Journal_, Fall 1992.
2. C. Dodge, A Musical Fractal, _Computer Music Journal_ 12, 13 (Fall
1988), p. 10.
3. K. J. Hsu and A. Hsu, Fractal Geometry of Music, _Proceedings of
the National Academy of Science, USA_ 87 (1990), pp. 938-941.
4. K. J. Hsu and A. Hsu, Self-similatrity of the '1/f noise' called
music., _Proceedings of the National Academy of Science USA_ 88
(1991), pp. 3507-3509.
5. C. Pickover, _Mazes for the Mind: Computers and the Unexpected_,
St. Martin's Press, New York, 1992.
6. P. Prusinkiewicz, Score Generation with L-Systems, _International
Computer Music Conference 86 Proceedings, _1986, pp. 455-457.
7. _Byte_ 11, 6 (June 1986), pp. 185-196.
Online resources include:
Well Tempered Fractal v3.0 by Robert Greenhouse
http://www-ks.rus.uni-stuttgart.de/people/schulz/fmusic/wtf/
A fractal music C++ package is available at
http://hamp.hampshire.edu/~gpzF93/inSanity.html
The Fractal Music Project (Claus-Dieter Schulz)
http://www-ks.rus.uni-stuttgart.de/people/schulz/fmusic
Chua's Oscillator: Applications of Chaos to Sound and Music
http://www.ccsr.uiuc.edu/People/gmk/Projects/ChuaSoundMusic/Chu
aSoundMusic.html
Fractal Music Lab
http://members.aol.com/strohbeen/fml.html
Fractal Music - Phil Thompson
http://easyweb.easynet.co.uk/~cenobyte/
fractal music in MIDI format by Jose Oscar Marques
http://midiworld.com/jmarques.htm
Don Archer's fractal art and music contains several pieces of fractal
music in MIDI format.
http://www.dorsai.org/~arch/
LMUSe, a DOS program that generates MIDI music and files from 3D
L-systems.
http://www.interport.net/~dsharp/lmuse.html
There is now a Fractal Music mailing list. It's purposes are:
1. To inform people about news, updates, changes on the Fractal Music
Projects WWW pages.
2. To encourage discussion between people working in that area.
The Fractal Music Mailinglist: fmusic@kssun7.rus.uni-stuttgart.de
To subscribe to the list please send mail to
fmusic-request@kssun7.rus.uni-stuttgart.de
Subject: Fractal mountains
_Q15_: How are fractal mountains generated?
_A15_: Usually by a method such as taking a triangle, dividing it into
3 sub-triangles, and perturbing the center point. This process is then
repeated on the sub-triangles. This results in a 2-d table of heights,
which can then be rendered as a 3-d image. This is referred to as
midpoint displacement. Two references are:
1. M. Ausloos, _Proc. R. Soc. Lond. A_ 400 (1985), pp. 331-350.
2. H.O. Peitgen, D. Saupe, _The Science of Fractal Images_,
Springer-Velag, 1988
Available online is an implementation of fractal Brownian motion (fBm)
such as described in _The Science of Fractal Images_. Lucasfilm became
famous for its fractal landscape sequences in _Star Trek II: The Wrath
of Khan_ the primary one being the _Genesis_ planet transformation.
Pixar and Digital Productions are have produced fractal landscapes for
Hollywood.
Fractal landscape information available online:
EECS News: Fall 1994: Building Fractal Planets by Ken Musgrave
http://www.seas.gwu.edu/faculty/musgrave/article.html
Gforge and Landscapes (John Beale)
http://www.best.com/~beale/
Java fractal landscapes :
Fractal landscapes (applet and sources) by Chris Thornborrow
http://www-europe.sgi.com/Fun/free/java/chris-thornborrow/index
.html
Subject: Plasma clouds
_Q16_: What are plasma clouds?
_A16_: They are a Fractint fractal and are similar to fractal
mountains. Instead of a 2-d table of heights, the result is a 2-d
table of intensities. They are formed by repeatedly subdividing
squares.
Robert Cahalan has fractal information about Earth's Clouds including
how they differ from plasma clouds.
Fractal Clouds Reference by Robert F. Cahalan
(cahalan@clouds.gsfc.nasa.gov)
http://climate.gsfc.nasa.gov/~cahalan/FractalClouds/
Also some plasma-based fractals clouds by John Walker are available.
Fractal generated clouds
http://ivory.nosc.mil/html/trancv/html/cloud-fract.html
The Center for the Computation and Visualization of Geometric
Structures also has some fractal clouds.
http://www.geom.umn.edu/graphics/pix/General_Interest/Fractals/
Two articles about the fractal nature of Earth's clouds:
1. "Fractal statistics of cloud fields," R. F. Cahalan and J. H.
Joseph, _Mon. Wea.Rev._ 117, 261-272, 1989
2. "The albedo of fractal stratocumulus clouds," R. F. Cahalan, W.
Ridgway, W. J. Wiscombe, T. L. Bell and J. B. Snider, _J. Atmos.
Sci._ 51, 2434-2455, 1994
Subject: Lyapunov fractals
_Q17a_: Where are the popular periodically-forced Lyapunov fractals
described?
_A17a_: See:
1. A. K. Dewdney, Leaping into Lyapunov Space, _Scientific American_,
Sept. 1991, pp. 178-180.
2. M. Markus and B. Hess, Lyapunov Exponents of the Logistic Map with
Periodic Forcing, _Computers and Graphics_ 13, 4 (1989), pp.
553-558.
3. M. Markus, Chaos in Maps with Continuous and Discontinuous Maxima,
_Computers in Physics_, Sep/Oct 1990, pp. 481-493.
_Q17b_: What are Lyapunov exponents?
_A17b_: Lyapunov exponents quantify the amount of linear stability or
instability of an attractor, or an asymptotically long orbit of a
dynamical system. There are as many Lyapunov exponents as there are
dimensions in the state space of the system, but the largest is
usually the most important.
Given two initial conditions for a chaotic system, a and b, which are
close together, the average values obtained in successive iterations
for a and b will differ by an exponentially increasing amount. In
other words, the two sets of numbers drift apart exponentially. If
this is written e^(n*(lambda) for _n_ iterations, then e^(lambda) is
the factor by which the distance between closely related points
becomes stretched or contracted in one iteration. Lambda is the
Lyapunov exponent. At least one Lyapunov exponent must be positive in
a chaotic system. A simple derivation is available in:
1. H. G. Schuster, _Deterministic Chaos: An Introduction_, Physics
Verlag, 1984.
_Q17c_: How can Lyapunov exponents be calculated?
_A17c_: For the common periodic forcing pictures, the Lyapunov
exponent is:
lambda = limit as N -> infinity of 1/N times sum from n=1 to N of
log2(abs(dx sub n+1 over dx sub n))
In other words, at each point in the sequence, the derivative of the
iterated equation is evaluated. The Lyapunov exponent is the average
value of the log of the derivative. If the value is negative, the
iteration is stable. Note that summing the logs corresponds to
multiplying the derivatives; if the product of the derivatives has
magnitude < 1, points will get pulled closer together as they go
through the iteration.
MS-DOS and Unix programs for estimating Lyapunov exponents from short
time series are available by ftp: ftp://inls.ucsd.edu/pub/ncsu/
Computing Lyapunov exponents in general is more difficult. Some
references are:
1. H. D. I. Abarbanel, R. Brown and M. B. Kennel, Lyapunov Exponents
in Chaotic Systems: Their importance and their evaluation using
observed data, _International Journal of Modern Physics B_ 56, 9
(1991), pp. 1347-1375.
2. A. K. Dewdney, Leaping into Lyapunov Space, _Scientific American_,
Sept. 1991, pp. 178-180.
3. M. Frank and T. Stenges, _Journal of Economic Surveys_ 2 (1988),
pp. 103- 133.
4. T. S. Parker and L. O. Chua, _Practical Numerical Algorithms for
Chaotic Systems_, Springer Verlag, 1989.
Subject: Fractal items
_Q18_: Where can I get fractal T-shirts, posters and other items?
_A18_: One source is Art Matrix, P.O. box 880, Ithaca, New York,
14851, 1-800-PAX-DUTY.
Another source is Media Magic; they sell many fractal posters,
calendars, videos, software, t-shirts, ties, and a huge variety of
books on fractals, chaos, graphics, etc. Media Magic is at PO Box 598
Nicasio, CA 94946, 415-662-2426.
A third source is Ultimate Image; they sell fractal t- shirts,
posters, gift cards, and stickers. Ultimate Image is at PO Box 7464,
Nashua, NH 03060-7464.
Yet another source is Dave Kliman (516) 625-2504 dkliman@pb.net, whose
products are distributed through Spencer Gifts, Posterservice,
1-800-666-7654, and Scandecor International., this spring, through JC
Penny, featuring all-over fractal t-shirts, and has fractal umbrellas
available from Shaw Creations (800) 328-6090.
Cyber Fiber produces fractal silk scarves, t-shirts, and postcards.
Contact Robin Lowenthal, Cyber Fiber, 4820 Gallatin Way, San Diego, CA
92117.
Chaos MetaLink website
(http://www.industrialstreet.com/chaos/metalink.htm) also has
postcards, CDs, and videos.
Free fractal posters are available if you send a self-addressed
stamped envelope to the address given on
http://www.xmission.com/~legalize/gift.html. For foreign requests
(outside USA) include two IRCs (international reply coupons) to cover
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