Fractals FAQ
Date: 02-27-92 (05:32) Number: 885 of 902
To: ALL Refer#: NONE
From: KEN SHIRRIFF Read: (N/A)
Subj: ALT.FRACTALS FAQ Status: PUBLIC MESSAGE
Conf: ANSWERS (1446) Read Type: TEXT SCAN (+)
ú Newsgroup: news.answers
ú Message-ID: <ogtu5INNhe8@agate.berkeley.edu>
ú Subject: alt.fractals FAQ
Archive-name: fractal-faq
Last-modified: February 26, 1992
This file is a frequently asked questions file for alt.fractals.
The purpose of this file is to collect common fractal questions and answers
into a convenient file.
I am happy to receive more information to add to this file. Also, if you
can correct mistakes you find, let me know. Remember, this file depends
on your feedback and contributions.
Please send additions, comments, errors, etc. to Ken Shirriff
(shirriff@sprite.Berkeley.EDU).
Q1: What is fractint?
A1: Fractint is a freeware (not public domain) fractal generator.
There are DOS, Windows, OS/2, and Unix/X versions.
The DOS version is the original version, and is the most up-to-date.
The Unix version is still an alpha version with lots of bugs.
Q2: Where can I get a copy of fractint?
Fractint is on many ftp sites. Some are listed:
DOS: ftp to wuarchive.wustl.edu
The source is in the file /mirrors/msdos/graphics/frasr171.zip
The executable is in the file /mirrors/msdos/graphics/frain171.zip
Windows: ftp to wuarchive.wustl.edu
The source is in the file /mirrors/msdos/graphics/winfr171.zip
The executable is in the file /mirrors/msdos/graphics/winsr171.zip
OS/2: available on Compuserve in its COMART forum in Library 12
The files are PM*.ZIP.
Unix: ftp to sprite.berkeley.edu [128.32.150.27] (note: unreliable machine)
The source is in the file xfract100.shar.Z
For European users, these files are available from ftp.uni-koeln.de
If you can't use ftp, see the mail server info in Q3a.
Q3: Where can I obtain other software packages to generate fractals of various
types?
A3: Some programs are:
For X windows:
xmntn and xlmntn: these generate fractal mountains
They can be obtained from uunet.uu.net in
/usenet/comp.sources.x/volume8/xmntns.
xfroot: generates a fractal root window.
xmartin: generates a Martin hopalong root window.
xmandel: generates Mandelbrot/Julia sets.
xfroot, xmartin, xmandel are part of the X11 distribution.
For Mac:
fractal, L-System, 3DL-System, IFS, FracHill are available from
ftphost.aukuni.ac.nz [130.216.1.5] in the architec directory.
Distributed X systems:
MandelSpawn: computes Mandelbrot/Julia sets on a network of machines.
Ftp from: funic.funet.fi:/pub/X11/contrib/mandelspawn-0.06.tar.Z
Please let me know of any other programs you know of.
Q3a: What if I can't use ftp?
A3a: There is allegedly a ftp mail server: send mail to
"ftpmail@decwrl.dec.com" with "help" in the body. Let me know if this
works.
Q4: Where is alt.fractals.pictures archived?
A4: The pictures are available via anonymous FTP from csus.edu
[130.86.90.1] in /pub/alt.fractals.pictures.
Q5: I want to learn about fractals. What should I read first?
A5: There is a book list at the end. (Please send me suggestions on
which books are best to start with. I'd also like brief descriptions
of the books.) _Chaos_ is a good book to get a general overview and
history. _Fractals Everywhere_ is a textbook on fractals that describes
what fractals are and how to generate them. _Chaos, Fractals, and
Dynamics_ is also a good start.
Q6: What is the Mandelbrot set?
A6: The Mandelbrot set is the set of all complex c such that iterating
z -> z^2+c does not go to infinity (starting with z=0).
Q7: How is the Mandelbrot set actually computed?
A7: The basic algorithm is:
For each c, start with z=0. Repeat z=z^2+c up to N times, exiting if
the magnitude of z gets large.
If you finish the loop, the point is probably inside the Mandelbrot set.
If you exit, the point is outside and can be colored according to how
many iterating were completed. You can exit if |z|>2, since if z gets
this big it will go to infinity. The maximum number of iterations, N,
can be selected as desired, for instance 100. Larger N will give sharper
detail but takes longer.
Q8: How can I speed up Mandelbrot set generation?
A8: Look at R. Rojas: "A tutorial on efficient computer graphic representations
of the Mandelbrot set", Computers and Graphics, 1991, V15 N1 p91-100.
There are a bunch of text files on fast integer math available by
ftp to cs.dal.ca [129.173.4.5] in the pub directory.
Q9: What is the difference between the Mandelbrot set and a Julia set?
A9: 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.
Q10: What is the connection between the Mandelbrot set and Julia sets?
A10: 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.
Q11: How is a Julia set actually computed?
A11: The Julia set can be computed by iteration similar to the Mandelbrot
computation.
Alternatively, points on the boundary of the Julia set can be computed
quickly by using inverse iterations.
Q12: How does complex arithmetic work?
A12: It works mostly like regular algebra with a couple additional formulas:
(note: a,b are reals, x,y are complex, i is the root of -1)
i^2 = -1
exp(a+i*b) = exp(a)(cos(b)+i*sin(b))
From these:
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)
Sine: sin(x) = (exp(i*x)-exp(-i*x))/(2*i)
Cos: 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)
Complex powers: x^y = exp(y*log(x))
More details can be found in any complex analysis book.
Q13: What is an iterated function system?
A13: If a fractal is self-similar, you can specify various mappings that map
the whole onto the parts. By taking a point and repeatedly applying
these mappings you end up with a collection of points on the fractal.
Iterated function systems can be used to make fractal ferns and trees
and are also used in fractal image compression.
_Fractals Everywhere_ by Barnsley is mostly about iterated function
systems.
Q14: How are fractal mountains generated?
A14: Usually by a method such as taking a triangle, dividing it into 3
subtriangles, and perturbing the center point. This process is then
repeated on the subtriangles. This results in a 2-d table of heights,
which can then be rendered as a 3-d image.
(Does anyone have references on this, to save me digging them up?)
Q15: What are plasma clouds?
A15: 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.
Q16: Where are the popular Lyapunov fractals described?
A16: Scientific American September 1991, and Computers and Graphics, Vol 13,
Number 4.
Q17: What is the logistic equation?
A17: 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.
For larger c, the population will oscillate between two values, then four
values, eight, sixteen, etc.
For still larger c (between 3.57 and 4), the population behavior is
chaotic (for most c values).
Q18: I want to know what is meant by fractal, chaos, etc.
A18: Matthew Bernhardt did a good job of collecting definitions of these terms.
The rest of this file is from his collected definitions. Note that
many of these definitions are very technical.
Q19: What is chaos?
A19: -An attractor is chaotic if at least one of its Lyapunov exponents
is positive. Chaos results from the existence of a chaotic attractor.
-Chaos is the recurrent behavior of a deterministic dynamical system
in which the phase-space divergence of nearby trajectories at an
exponential rate results in a limited predictability horizon.
-In chaotic iterated systems of the form x_{i+1}=f(x_i), the result
after iteration is extremely sensitive to the initial value such that
f^n(x_0+(epsilon)) is nowhere near f^n(x_0).
-Chaos results from our inability to predict the future behavior of a
deterministic system from initial conditions because of its great
sensitivity to initial conditions.
-Apparently unpredictable behavior arising in a deterministic system.
-Unpredictable yet nonrandom behavior of a system.
-Stochastic behavior occurring in a deterministic system.
(Ian Stewart's _Does God Play Dice?_)
Chaos References:
1. Devaney - An Introduction to Chaotic Dynamical Systems
(second edition) Addison-Wesley: Redwood City, CA, 1989
2. _Chaos, Fractals, and Dynamics: Computer Experiments in Mathematics_,
Devaney. in Mathematics, Addison-Wesley: Menlo Park, CA, 1990
This is elementary and readable, yet rigorous.
Continued in the next message...
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Date: 02-27-92 (05:32) Number: 886 of 902
To: ALL Refer#: NONE
From: KEN SHIRRIFF Read: (N/A)
Subj: ALT.FRACTALS FAQ Status: PUBLIC MESSAGE
Conf: ANSWERS (1446) Read Type: TEXT SCAN (+)
ú Newsgroup: news.answers
ú Message-ID: <ogtu5INNhe8@agate.berkeley.edu>
ú Subject: alt.fractals FAQ
(Continued from the previous message)
Q20: What is nonlinearity? What are nonlinear equations?
A20: -Nonlinear maps fail to satisfy the condition that
f(ax+by)=af(x)+bf(y)
where x and y are vectors, and a and b are scalars.
e.g. f(x)=ax is linear.
f(x)=x^2 is nonlinear.
Nonlinearity is a map or term that is not linear.
Nonlinear equations are equations containing nonlinearities.
e.g. f(x)=ax+x^2 and f(x)=ax^2+bx+c are nonlinear.
-A nonlinear system gives an output which is not proportional to the
corresponding input. Nonlinear dynamical systems possess nonlinear
dynamical laws, which are functions of the system's state variables.
-In linear systems, dy/dx is a constant, while in nonlinear systems
dy/dx=some nonconstant function of x.
-Nonlinear equations fail to exhibit linear superimposability.
Nonlinear equations can be categorized by differentiability,
discontinuity, and "memory" (e.g. hysteresis in an electric circuit),
etc. This can be important to some types of nonlinear analysis such
as the Popov hyperstability criterion.
Nonlinearity References:
1. Zelinsky, D. "A First Course in Linear Algebra" (1973) p. 65. Academic
Press.
2. W.A. Brock and E.G. Baek "Some theory of statistical inference for
nonlinear science." (1991) Review of Economic Studies 58(4), 697-716.
3. Guckenheimer, John and Philip Holmes. _Nonlinear Oscillations,
Dynamical Systems, and Bifurcations of Vector Fields_. Springer-
Verlag, New York, 1983.
Q21: What is a fractal? What are some examples of fractals?
A21: -A set of points whose fractal dimension exceeds its topological
dimension.
-Examples of fractals:
Sierpinski triangle, Koch snowflake, Peano curve, Mandlebrot set.
Fractal References:
1. Rudy Rucker's _Mind Tools_
Plus many of the books mentioned later.
Q22: What is fractal dimension? How is it calculated?
-Fractal dimension is the Hausdorff-Besikovich Dimension.
-Roughly, fractal dimension can be calculated by taking the limit of the
quotient of the log change in object size and the log change in
measurement scale, as the measurement scale approaches zero.
e.g.
Consider a straight line. Now blow up the line by a factor of two.
The line is now twice as long as before. Log 2 / Log 2 = 1,
corresponding to dimension 1.
Consider a square. Now blow up the square by a factor of two. The
square is now 4 times as large as before (i.e. 4 original squares
can be placed on the original square). Log 4 / log 2 = 2,
corresponding to dimension 2 for the square.
Consider a snowflake curve formed by repeatedly replacing ___ with _/\_,
where each of the 4 new lines is 1/3 the length of the old line.
Blowing up the snowflake curve by a factor of 3 results in a snowflake
curve 4 times as large (one of the old snowflake curves can be placed
on each of the 4 segments _/\_). Log 4 / log 3 = 1.261... Since the
dimension 1.261 is larger than the dimension 1 of the lines making up
the curve, the snowflake curve is a fractal.
Fractal Dimension References:
1. Barnsley et. al., "Science of Fractal Images", Springer Verlag, 1985.
2. Falconer, K.J. (1985) "The Geometry of Fractal Sets" Cambridge Univ.
Press.
3. T.S. Parker and L.O. Chua (1989) "Practical Numerical Algorithms for
Chaotic Systems." Springer Verlag, Ch. 7.
4. J.P. Eckmann and D. Ruelle (1985) Reviews of Modern Physics 57(3)
617-656
5. J. Theiler (1990) Physical Review A 41, 3038-3051.
6. Grassberger and Procaccia (1983) Physica D 9, 189-208.
Q23: What is a strange attractor?
A23: -An indecomposable closed invariant set that "attracts" the points about
it which contains a transversal homoclinic orbit. (This orbit accounts
for the strangeness.)
-The limit set of a chaotic trajectory.
-A phase space locus of a bounded long-term dynamical behavior which has
a nonzero probability of being observed - its basin of attraction has
positive measure - and contains not a smooth manifold structure, but
rather a self-similar or fractal structure.
Note: While all chaotic attractors are strange, not all strange
attractors are chaotic. (Grebogi: et. al. (1984) "Strange Attractors
that are not chaotic." Physica D13, 261-268.)
-Consider a volume in phase space defined by all the initial conditions
a system may have. For a dissipative system, this volume will shrink as
the system evolves in time (Liouville's Theorem). If the system is
sensitive to initial conditions, the trajectories of the points defining
initial conditions will move apart in some directions, closer in others,
but there will be a net shrinkage in volume. Ultimately, all points will
lie along a fine line of zero volume. This is the strange attractor.
All initial points in phase space which ultimately land on the attractor
form a Basin of Attraction.
Note: A strange attractor results if a system is sensitive to initial
conditions and is not conservative.
-The surfaces which the state of a chaotic system will be confined to,
given time for transients to die out.
Q24: What are Lyapunov exponents? How are they calculated?
A24: -A generalization of an eigenvalue.
-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
in one iteration. Lambda is the Lyapunov exponent. At least one
Lyapunov exponent must be positive in a strange attractor. A simple
derivation is available in "Deterministic Chaos" by Heinz Georg Schuster,
Physics Verlag, 1984, p. 18.
Lyapunov Exponent References:
1. Frank, M. and T. Stenges (1988) Journal of Economic Surveys 2, 103-133.
2. T.S. Parker and L.O. Chua (1989) "Practical Numerical Algorithms for
Chaotic Systems." Springer Verlag, Ch. 3
3. Scientific American, Sept. 1991
References on self-affine fractals: (Anyone know what these are?)
Bedford, Math. Proc. Cambridge Phil. Soc. 106 (1989) 325
Falconer, Math. Proc. Cambridge Phil. Soc. 103 (1988) 339
McMullen, Nagoya Math. J. 96 (1984) 1
Przytycki & Urbanski, Studia Math. 93 (1989) 155
Miscellaneous References:
_Chaos_, James Gleick (1987)
_Does God Play Dice?_, Ian Stewart (1990)
_The Beauty of Fractals_, Peitgen and Saupe.
_The Fractal Geometry of Nature_, Benoit Mandlebrot
_Fractals Everywhere_, Michael Barnsley
Scientific American
Algorithm
Mandala
Proceedings of the Santa Fe Institute's Complex Systems Summer School (1988)
Official Title: Volume I "Lectures in the Sciences of Complexity"
ed. Dan Stein
see esp. the first article by David Campbell
"Introduction to nonlinear phenomena."
_Chaos and Fractals: The Mathematics Behind the Computer Graphics_, R.
Devaney and L. Keen. This book contains detailed mathematical
descriptions of chaos, the Mandelbrot set, etc.
_Computers, pattern, chaos, and beauty_, C. Pickover. This book
contains a bunch of interesting explorations of different fractals.
"Chaos and Graphics" section in the quarterly journal _Computers and
Graphics_. This contains recent work in fractals from the graphics
perspective.
_Fractal Creations_. This is the book on the fractint program.
_Fractals for the Classroom_, Peitgen, Jurgens, Saupe. This book
is aimed at advanced secondary school students, has lots of examples,
and BASIC programs.
Acknowledgements -- thanks to:
Michael A. Rolenz, Murray Frank, Bruce Stewart, Kai-Mikael J, Leon Katz,
Miriam Nadel, Erik Boman, Abhijit V. Deshmukh, Tom Parker, Matt Perry,
Guy Metcalfe, Robert Hood, Gordon Erlebacher, Gerald Edgar, Duncan Foster,
Tim Scavio, and others.
Special thanks to Matthew J. Bernhardt (mjb@acsu.buffalo.edu) for collecting
the chaos definitions.
Copyright 1992 Ken Shirriff (shirriff@sprite.Berkeley.EDU)
---
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From: shirriff@sprite.berkeley.edu (Ken Shirriff)
Newsgroups: alt.fractals,news.answers
Subject: alt.fractals FAQ
Keywords: fractals
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Date: 26 Feb 1992 20:39:33 GMT
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ú Newsgroup: news.answers
ú Message-ID: <ogtu5INNhe8@agate.berkeley.edu>
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Archive-name: fractal-faq
Last-modified: February 26, 1992
This file is a frequently asked questions file for alt.fractals.
The purpose of this file is to collect common fractal questions and answers
into a convenient file.
I am happy to receive more information to add to this file. Also, if you
can correct mistakes you find, let me know. Remember, this file depends
on your feedback and contributions.
Please send additions, comments, errors, etc. to Ken Shirriff
(shirriff@sprite.Berkeley.EDU).
Q1: What is fractint?
A1: Fractint is a freeware (not public domain) fractal generator.
There are DOS, Windows, OS/2, and Unix/X versions.
The DOS version is the original version, and is the most up-to-date.
The Unix version is still an alpha version with lots of bugs.
Q2: Where can I get a copy of fractint?
Fractint is on many ftp sites. Some are listed:
DOS: ftp to wuarchive.wustl.edu
The source is in the file /mirrors/msdos/graphics/frasr171.zip
The executable is in the file /mirrors/msdos/graphics/frain171.zip
Windows: ftp to wuarchive.wustl.edu
The source is in the file /mirrors/msdos/graphics/winfr171.zip
The executable is in the file /mirrors/msdos/graphics/winsr171.zip
OS/2: available on Compuserve in its COMART forum in Library 12
The files are PM*.ZIP.
Unix: ftp to sprite.berkeley.edu [128.32.150.27] (note: unreliable machine)
The source is in the file xfract100.shar.Z
For European users, these files are available from ftp.uni-koeln.de
If you can't use ftp, see the mail server info in Q3a.
Q3: Where can I obtain other software packages to generate fractals of various
types?
A3: Some programs are:
For X windows:
xmntn and xlmntn: these generate fractal mountains
They can be obtained from uunet.uu.net in
/usenet/comp.sources.x/volume8/xmntns.
xfroot: generates a fractal root window.
xmartin: generates a Martin hopalong root window.
xmandel: generates Mandelbrot/Julia sets.
xfroot, xmartin, xmandel are part of the X11 distribution.
For Mac:
fractal, L-System, 3DL-System, IFS, FracHill are available from
ftphost.aukuni.ac.nz [130.216.1.5] in the architec directory.
Distributed X systems:
MandelSpawn: computes Mandelbrot/Julia sets on a network of machines.
Ftp from: funic.funet.fi:/pub/X11/contrib/mandelspawn-0.06.tar.Z
Please let me know of any other programs you know of.
Q3a: What if I can't use ftp?
A3a: There is allegedly a ftp mail server: send mail to
"ftpmail@decwrl.dec.com" with "help" in the body. Let me know if this
works.
Q4: Where is alt.fractals.pictures archived?
A4: The pictures are available via anonymous FTP from csus.edu
[130.86.90.1] in /pub/alt.fractals.pictures.
Q5: I want to learn about fractals. What should I read first?
A5: There is a book list at the end. (Please send me suggestions on
which books are best to start with. I'd also like brief descriptions
of the books.) _Chaos_ is a good book to get a general overview and
history. _Fractals Everywhere_ is a textbook on fractals that describes
what fractals are and how to generate them. _Chaos, Fractals, and
Dynamics_ is also a good start.
Q6: What is the Mandelbrot set?
A6: The Mandelbrot set is the set of all complex c such that iterating
z -> z^2+c does not go to infinity (starting with z=0).
Q7: How is the Mandelbrot set actually computed?
A7: The basic algorithm is:
For each c, start with z=0. Repeat z=z^2+c up to N times, exiting if
the magnitude of z gets large.
If you finish the loop, the point is probably inside the Mandelbrot set.
If you exit, the point is outside and can be colored according to how
many iterating were completed. You can exit if |z|>2, since if z gets
this big it will go to infinity. The maximum number of iterations, N,
can be selected as desired, for instance 100. Larger N will give sharper
detail but takes longer.
Q8: How can I speed up Mandelbrot set generation?
A8: Look at R. Rojas: "A tutorial on efficient computer graphic representations
of the Mandelbrot set", Computers and Graphics, 1991, V15 N1 p91-100.
There are a bunch of text files on fast integer math available by
ftp to cs.dal.ca [129.173.4.5] in the pub directory.
Q9: What is the difference between the Mandelbrot set and a Julia set?
A9: 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.
Q10: What is the connection between the Mandelbrot set and Julia sets?
A10: 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.
Q11: How is a Julia set actually computed?
A11: The Julia set can be computed by iteration similar to the Mandelbrot
computation.
Alternatively, points on the boundary of the Julia set can be computed
quickly by using inverse iterations.
Q12: How does complex arithmetic work?
A12: It works mostly like regular algebra with a couple additional formulas:
(note: a,b are reals, x,y are complex, i is the root of -1)
i^2 = -1
exp(a+i*b) = exp(a)(cos(b)+i*sin(b))
From these:
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)
Sine: sin(x) = (exp(i*x)-exp(-i*x))/(2*i)
Cos: 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)
Complex powers: x^y = exp(y*log(x))
More details can be found in any complex analysis book.
Q13: What is an iterated function system?
A13: If a fractal is self-similar, you can specify various mappings that map
the whole onto the parts. By taking a point and repeatedly applying
these mappings you end up with a collection of points on the fractal.
Iterated function systems can be used to make fractal ferns and trees
and are also used in fractal image compression.
_Fractals Everywhere_ by Barnsley is mostly about iterated function
systems.
Q14: How are fractal mountains generated?
A14: Usually by a method such as taking a triangle, dividing it into 3
subtriangles, and perturbing the center point. This process is then
repeated on the subtriangles. This results in a 2-d table of heights,
which can then be rendered as a 3-d image.
(Does anyone have references on this, to save me digging them up?)
Q15: What are plasma clouds?
A15: 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.
Q16: Where are the popular Lyapunov fractals described?
A16: Scientific American September 1991, and Computers and Graphics, Vol 13,
Number 4.
Q17: What is the logistic equation?
A17: 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.
For larger c, the population will oscillate between two values, then four
values, eight, sixteen, etc.
For still larger c (between 3.57 and 4), the population behavior is
chaotic (for most c values).
Q18: I want to know what is meant by fractal, chaos, etc.
A18: Matthew Bernhardt did a good job of collecting definitions of these terms.
The rest of this file is from his collected definitions. Note that
many of these definitions are very technical.
Q19: What is chaos?
A19: -An attractor is chaotic if at least one of its Lyapunov exponents
is positive. Chaos results from the existence of a chaotic attractor.
-Chaos is the recurrent behavior of a deterministic dynamical system
in which the phase-space divergence of nearby trajectories at an
exponential rate results in a limited predictability horizon.
-In chaotic iterated systems of the form x_{i+1}=f(x_i), the result
after iteration is extremely sensitive to the initial value such that
f^n(x_0+(epsilon)) is nowhere near f^n(x_0).
-Chaos results from our inability to predict the future behavior of a
deterministic system from initial conditions because of its great
sensitivity to initial conditions.
-Apparently unpredictable behavior arising in a deterministic system.
-Unpredictable yet nonrandom behavior of a system.
-Stochastic behavior occurring in a deterministic system.
(Ian Stewart's _Does God Play Dice?_)
Chaos References:
1. Devaney - An Introduction to Chaotic Dynamical Systems
(second edition) Addison-Wesley: Redwood City, CA, 1989
2. _Chaos, Fractals, and Dynamics: Computer Experiments in Mathematics_,
Devaney. in Mathematics, Addison-Wesley: Menlo Park, CA, 1990
This is elementary and readable, yet rigorous.
Continued in the next message...
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Q20: What is nonlinearity? What are nonlinear equations?
A20: -Nonlinear maps fail to satisfy the condition that
f(ax+by)=af(x)+bf(y)
where x and y are vectors, and a and b are scalars.
e.g. f(x)=ax is linear.
f(x)=x^2 is nonlinear.
Nonlinearity is a map or term that is not linear.
Nonlinear equations are equations containing nonlinearities.
e.g. f(x)=ax+x^2 and f(x)=ax^2+bx+c are nonlinear.
-A nonlinear system gives an output which is not proportional to the
corresponding input. Nonlinear dynamical systems possess nonlinear
dynamical laws, which are functions of the system's state variables.
-In linear systems, dy/dx is a constant, while in nonlinear systems
dy/dx=some nonconstant function of x.
-Nonlinear equations fail to exhibit linear superimposability.
Nonlinear equations can be categorized by differentiability,
discontinuity, and "memory" (e.g. hysteresis in an electric circuit),
etc. This can be important to some types of nonlinear analysis such
as the Popov hyperstability criterion.
Nonlinearity References:
1. Zelinsky, D. "A First Course in Linear Algebra" (1973) p. 65. Academic
Press.
2. W.A. Brock and E.G. Baek "Some theory of statistical inference for
nonlinear science." (1991) Review of Economic Studies 58(4), 697-716.
3. Guckenheimer, John and Philip Holmes. _Nonlinear Oscillations,
Dynamical Systems, and Bifurcations of Vector Fields_. Springer-
Verlag, New York, 1983.
Q21: What is a fractal? What are some examples of fractals?
A21: -A set of points whose fractal dimension exceeds its topological
dimension.
-Examples of fractals:
Sierpinski triangle, Koch snowflake, Peano curve, Mandlebrot set.
Fractal References:
1. Rudy Rucker's _Mind Tools_
Plus many of the books mentioned later.
Q22: What is fractal dimension? How is it calculated?
-Fractal dimension is the Hausdorff-Besikovich Dimension.
-Roughly, fractal dimension can be calculated by taking the limit of the
quotient of the log change in object size and the log change in
measurement scale, as the measurement scale approaches zero.
e.g.
Consider a straight line. Now blow up the line by a factor of two.
The line is now twice as long as before. Log 2 / Log 2 = 1,
corresponding to dimension 1.
Consider a square. Now blow up the square by a factor of two. The
square is now 4 times as large as before (i.e. 4 original squares
can be placed on the original square). Log 4 / log 2 = 2,
corresponding to dimension 2 for the square.
Consider a snowflake curve formed by repeatedly replacing ___ with _/\_,
where each of the 4 new lines is 1/3 the length of the old line.
Blowing up the snowflake curve by a factor of 3 results in a snowflake
curve 4 times as large (one of the old snowflake curves can be placed
on each of the 4 segments _/\_). Log 4 / log 3 = 1.261... Since the
dimension 1.261 is larger than the dimension 1 of the lines making up
the curve, the snowflake curve is a fractal.
Fractal Dimension References:
1. Barnsley et. al., "Science of Fractal Images", Springer Verlag, 1985.
2. Falconer, K.J. (1985) "The Geometry of Fractal Sets" Cambridge Univ.
Press.
3. T.S. Parker and L.O. Chua (1989) "Practical Numerical Algorithms for
Chaotic Systems." Springer Verlag, Ch. 7.
4. J.P. Eckmann and D. Ruelle (1985) Reviews of Modern Physics 57(3)
617-656
5. J. Theiler (1990) Physical Review A 41, 3038-3051.
6. Grassberger and Procaccia (1983) Physica D 9, 189-208.
Q23: What is a strange attractor?
A23: -An indecomposable closed invariant set that "attracts" the points about
it which contains a transversal homoclinic orbit. (This orbit accounts
for the strangeness.)
-The limit set of a chaotic trajectory.
-A phase space locus of a bounded long-term dynamical behavior which has
a nonzero probability of being observed - its basin of attraction has
positive measure - and contains not a smooth manifold structure, but
rather a self-similar or fractal structure.
Note: While all chaotic attractors are strange, not all strange
attractors are chaotic. (Grebogi: et. al. (1984) "Strange Attractors
that are not chaotic." Physica D13, 261-268.)
-Consider a volume in phase space defined by all the initial conditions
a system may have. For a dissipative system, this volume will shrink as
the system evolves in time (Liouville's Theorem). If the system is
sensitive to initial conditions, the trajectories of the points defining
initial conditions will move apart in some directions, closer in others,
but there will be a net shrinkage in volume. Ultimately, all points will
lie along a fine line of zero volume. This is the strange attractor.
All initial points in phase space which ultimately land on the attractor
form a Basin of Attraction.
Note: A strange attractor results if a system is sensitive to initial
conditions and is not conservative.
-The surfaces which the state of a chaotic system will be confined to,
given time for transients to die out.
Q24: What are Lyapunov exponents? How are they calculated?
A24: -A generalization of an eigenvalue.
-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
in one iteration. Lambda is the Lyapunov exponent. At least one
Lyapunov exponent must be positive in a strange attractor. A simple
derivation is available in "Deterministic Chaos" by Heinz Georg Schuster,
Physics Verlag, 1984, p. 18.
Lyapunov Exponent References:
1. Frank, M. and T. Stenges (1988) Journal of Economic Surveys 2, 103-133.
2. T.S. Parker and L.O. Chua (1989) "Practical Numerical Algorithms for
Chaotic Systems." Springer Verlag, Ch. 3
3. Scientific American, Sept. 1991
References on self-affine fractals: (Anyone know what these are?)
Bedford, Math. Proc. Cambridge Phil. Soc. 106 (1989) 325
Falconer, Math. Proc. Cambridge Phil. Soc. 103 (1988) 339
McMullen, Nagoya Math. J. 96 (1984) 1
Przytycki & Urbanski, Studia Math. 93 (1989) 155
Miscellaneous References:
_Chaos_, James Gleick (1987)
_Does God Play Dice?_, Ian Stewart (1990)
_The Beauty of Fractals_, Peitgen and Saupe.
_The Fractal Geometry of Nature_, Benoit Mandlebrot
_Fractals Everywhere_, Michael Barnsley
Scientific American
Algorithm
Mandala
Proceedings of the Santa Fe Institute's Complex Systems Summer School (1988)
Official Title: Volume I "Lectures in the Sciences of Complexity"
ed. Dan Stein
see esp. the first article by David Campbell
"Introduction to nonlinear phenomena."
_Chaos and Fractals: The Mathematics Behind the Computer Graphics_, R.
Devaney and L. Keen. This book contains detailed mathematical
descriptions of chaos, the Mandelbrot set, etc.
_Computers, pattern, chaos, and beauty_, C. Pickover. This book
contains a bunch of interesting explorations of different fractals.
"Chaos and Graphics" section in the quarterly journal _Computers and
Graphics_. This contains recent work in fractals from the graphics
perspective.
_Fractal Creations_. This is the book on the fractint program.
_Fractals for the Classroom_, Peitgen, Jurgens, Saupe. This book
is aimed at advanced secondary school students, has lots of examples,
and BASIC programs.
Acknowledgements -- thanks to:
Michael A. Rolenz, Murray Frank, Bruce Stewart, Kai-Mikael J, Leon Katz,
Miriam Nadel, Erik Boman, Abhijit V. Deshmukh, Tom Parker, Matt Perry,
Guy Metcalfe, Robert Hood, Gordon Erlebacher, Gerald Edgar, Duncan Foster,
Tim Scavio, and others.
Special thanks to Matthew J. Bernhardt (mjb@acsu.buffalo.edu) for collecting
the chaos definitions.
Copyright 1992 Ken Shirriff (shirriff@sprite.Berkeley.EDU)
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