Upon hearing the word chaos, one's mind usually conjectures a place of total disorder and confusion. This is the usual meaning of the word in normal usage. However, there has been a literal explosion of scientific interest in chaos and how to control it or at least understand it. If the term chaos really implied total disorder or randomness, there would probably be no point in studying the phenomenon. However, in technical literature, the term chaos means something that appears to be random and disordered but is actually deterministic in nature, meaning that it is precisely controlled by natural laws. The apparent disorder arises from an extreme sensitivity to initial conditions, much like the path of the ball in a pinball machine seeming to defy human control. This paper discusses the scientific meaning of the word chaos and how understanding chaos may be of great benefit to mankind.
Most people like to have a sense of order and predictability in their
lives; they like to plan for the future and know that there is a reasonable
probability of seeing their dreams fulfilled. However, the natural world
around us, in spite of its outward beauty and longevity, seems to defy
all efforts at predicting its future. Mankind has not yet learned the secrets
to predicting the weather more than a few days in advance-and with questionable
accuracy at that. Some years ago the multiflora rose, native to Asia, was
advocated to farmers as a natural fence for their cattle. It did stop cattle
in their tracks, however, it is now illegal to sell multiflora rose in
Ohio because it's difficult to stop the multiflora rose-nature has a mind
of its own.
Can we improve our ability to foresee the consequences
of our seemingly little actions, or is it hopelessly difficult as Poincaré
seemed to imply in 1903:
A very small cause which escapes our notice determines a considerable effect that we cannot fail to see...even if the case that the natural laws had no longer secret for us...we could only know the initial situation approximately...It may happen that small differences in initial conditions produce very great ones in the final phenomena.
Understanding chaos would undoubtedly be of great benefit to mankind.
Chaos in itself has been of great benefit, for as Henry Brooks Adams put
it: ``Chaos often breeds life, when order breeds habit.'' It may be critically
important to understand chaos in order to sustain our own existence as
John Fiske eloquently stated in
Modern discussions of chaos are almost always
based on the work of Edward N. Lorenz. In his book
It's not to important to be able to understand
and control the chaos in a pinball machine, but how about chaos in the
atmosphere, or chaos in the human brain and heart, or chaos in important
industrial processes? Obviously the answer to this is yes. As a matter
of fact, then senator, now vice president, Al Gore thought it important
enough to devote several pages to chaos theory in his book:
It is quite natural that atmospheric chaos would be one of the first targets of modern chaos theory, after all, Edward Lorenz was a meteorologist. His paper on deterministic non-periodic flow [2] is considered by many to be the birth of chaos theory. In studying the behavior of a gaseous system, Lorenz simplified the Navior-Stokes equations and produce a set of three ordinary looking nonlinear differential equations:
The constants a, r, and b determine the behavior
of the system. These three equations look innocent enough, however they
exhibit chaotic behavior-they are extremely sensitive to initial conditions.
The classic values used to demonstrate chaos are a=10, r=28, and b=8/3.
Choosing a suitable initial value and solving with a numerical procedure
leads to the well known butterfly shown in Figure 2.
Other views-all with isoview scaling-are provided in Figures 3
and 4. However, it is difficult to see the strange
behavior of the Lorenz Attractor in these two dimensional views. Figure
1 shows the development of the Lorenz attractor
in a box in steps of delta t=5 so that the reader may see how the trajectory
weaves back and forth between the two lobes of the butterfly, never repeating
the same path twice. Also, the behavior in one time segment doesn't really
give a clue as to how it will behave in the next time segment. Sometimes
it spends the majority of its time whirling around on just one side, while
at other times it weaves back and forth, sharing time on each side. Yet
at other times, it seems to find a nearly stable orbit and stays so close
for awhile that it looks periodic and stable-but not quite.
I found out just how chaotic-sensitive to initial
conditions-these equations are when I tried to find initial values for
x, y and z that would produce a nice "balanced" butterfly with
equally large wings. I have provided the Scilab source code in the appendix
where you will see that I specified the initial conditions to fourteen
significant digits; when I tried the same case to eight significant digits-which
was still very accurate-I got a lopsided butterfly!
The whole point of the aforegoing discussion of
the Lorenz Attractor was to demonstrate how difficult even a simplified
gaseous system in a box can be. Imagine now that we try to simulate the
weather by breaking the atmosphere into many millions of interactive little
boxes and attempt to predict conditions several weeks into the future,
keeping in mind that we don't know the initial conditions at time t=0 very
accurately. Now you know why your local weather reporter isn't using chaos
theory on the evening weather report. However, progress is being made in
this area, and some day large scale simulations will be possible.
Since chaotic systems are systems that only appear to be random, but
are really deterministic in nature, they possess an underlying order. This
underlying order leads to the possibility of controlling chaotic systems.
William Ditto and Louis Pecora have been pioneering methods of controlling
chaotic mechanical, electrical and biological systems [3]. For example,
Pecora and Thomas Carroll of the U.S. Naval Research Laboratory realized
that synchronized chaos might be used for encoding private electronic communications.
An explanation of synchronized chaos is necessary here to understand what
they did.
Chaotic systems are described as having an infinite
number of unstable periodic motions. This instability means that no two
chaotic systems can be built that provide the same output. However, it
has been shown that if two identical stable systems are driven by the same
chaotic signal, they will generate a chaotic output, but their inherent
stability causes them to suppress differences between them. The result
is two identical chaotic outputs. This doesn't appear too significant in
itself, but consider the following possibility.
At the sending location, generate a chaotic signal
and use it as the input to a stable system to generate another chaotic
output signal. Mix the chaotic output signal with a message; the result
is another signal that looks chaotic but has a message embedded in it.
Now transmit two signals to the receiving location: the original chaotic
signal and the message-carrying chaotic output. At the receiving location,
feed the original chaotic signal into a stable system identical to that
at the sending location; the result is the same chaotic signal that was
mixed with the message at the sending station. Subtract this from the message-encoded
chaotic signal and you have the original message. While it has been shown
that it is not too difficult to intercept and extract the message from
the chaos encoded message, if this method is layered on top of other encrypting
methods, it enhances the security of private transmissions.
While the technique just discussed is a method
of using chaos, others are working on methods of controlling chaos. Since
a chaotic system is composed of an infinite number of unstable periodic
orbits, the key to controlling the system is to wait until the system comes
near the desired periodic orbit and then perturb the input parameters just
enough to encourage the system to stay on that orbit. This method-referred
to as OGY after Edward Ott, Celso Grebogi and James A. Yorke who developed
the system-has been used successfully in stabilizing lasers and other industrial
systems. But the techniques employed in the OGY method are not limited
to industry; they show promise in controlling chaotic behavior in the human
body as well.
It has been argued that some cardiac arrhythmias are instances of chaos.
This opens the doors to new strategies of control. The traditional method
of controlling a system is to model it mathematically in sufficient detail
to be able to control critical parameters. However, this method fails in
chaotic systems since no model can be developed for a system with an infinite
number of unstable orbits. The OGY method mentioned above was able to exploit
the properties of chaotic mechanical and electrical systems, however, system-wide
parameters in the human body can not be manipulated quickly enough to control
cardiac chaos. Therefore, Garfinkel, Spano, Ditto and Weiss [5] developed
a similar method which they called proportional perturbation feedback (PPF).
In their words: ``Both methods use a linear approximation of the dynamics
in the neighborhood of the desired fixed point. OGY then varies a system-wide
parameter to move the stable manifold to the system state point; our method
perturbs the system state point to move it toward the stable manifold."
Without trying to figure out what that means, the important point is that
it worked.
In eleven separate experimental runs, the technique
was successful at controlling induced arrhythmia in eight cases. The good
thing is that the stimuli did not simply over drive the heart; stimuli
did not even have to be delivered on every beat. This contrasted well with
the periodic method which was never successful in restoring a periodic
rhythm, and even showed a tendency to make the rhythm more aperiodic. Therefore,
besides providing a successful method of control,the method would be a
less dramatic intrusion into the patient's system.
Similar efforts are being made to control epileptic
brain seizures which exhibit chaotic behavior. This technique, controls
by waiting for the system to make a close approach to an unstable fixed
point along the stable direction. It then makes a minimal intervention
to bring the system back on the stable manifold [4]. Again, an important
benefit is the minimal amount of intervention required to control the chaotic
event.
To get an idea of how many groups are studying the control of chaotic
systems, one has only to do a search on the Internet for ``chaos'' sites.
For example, the University of Texas at San Antonio is studying four types
of chaotic dynamics in cellular flames. Other interesting sites in the
United States are at the University of Maryland and Georgia Institute of
Technology. Ohio State University is doing studies on chaotic dynamics
of ferromagnetic resonance under the directions of Professor P.E. Wigen.
And the interest in chaos is not limited to the United States; interesting
papers regarding intelligent control systems are available online at a
site in France, and a long list of available publications is available
at another site in Russia. These and other interesting sites with their
Internet addresses are listed in the appendix.
With so much worldwide interest and active research,
it can only be expected that advancements will continue at a rapid pace
in the interesting field of chaos theory-hopefully to the benefit of all
mankind.
http://www.etca.fr/English/Projects/Chaos/
DCE/CTME/GIP Papers about Fractals, Grammars, Splines and their application
to Commuter Science. Dr Jacques Blanc-Talon - Scientific Consultant Fractals,
Formal Languages, Image Processing, Topology
http://www-physics.mps.ohio-state.edu/ dwp/nonlin.html
Nonlinear Dynamics in Ferromagnetic Resonance - Describes research
of the chaotic dynamics of ferromagnetic resonance under the directions
of Professor P.E. Wigen at The Ohio State University.
http://www-chaos.umd.edu/
University of Maryland - Chaos Group
http://acl2.physics.gatech.edu/
Georgia Institute of Technology - Applied Chaos Lab
http://vip.cs.utsa.edu/flames/overview.html
U Houston and UT San Antonio - Combustion Chaos Group
http://www.ipme.ru/ccs.html
Russian Academy of Sciences - Control of Complex Systems Lab - Russian
Academy of Sciences
http://nonlinear.ucsc.edu/
Santa Cruz Institute of Nonlinear Science
http://www.physik.th-darmstadt.de/nlp/
TH Darmstadt-Institut für Angewandte Physik-Nonlinear Physics
Group
http://inls.ucsd.edu/
UC San Diego - Institute for Nonlinear Science
http://www.iddr.ie.psu.edu/People/satish/chaos_research.html
Chaos Research - Dr. Kumara's group works on applied chaos theory for
developing monitoring and control schemes for machining processes
http://www.nonlin.tu-muenchen.de/
Chaos Group founded 1990 by Werner Eberl - including the dynamical
Who-Is-Who-Handbook of Nonlinear Dynamics
http://galahad.elte.hu/
Eötvös Loránd University - Solid State
and Chaos Group
http://www-chaos.engr.utk.edu
University of Tennessee - research in chaos applied to engineering
systems.
http://www.vanderbilt.edu/AnS/psychology/cogsci/chaos/PaperCall96.html
Sixth Annual International Conference of the Society for Chaos Theory
in Psychology and the Life Sciences (SCTPLS) - June 25-28 at UC Berkeley;
the study of chaos theory, fractals, nonlinear dynamics, self-organizational
processes and related principles applied to the various psychological sub-disciplines.
http://www.students.uiuc.edu/ ag-ho/chaos/chaos.html
Chaos Theory - Overview with emphasis on thought processes over actual
mathematical theory.
http://tqd.advanced.org/3120/
The Chaos Experience - to explain, in simple terms, the basic principles
behind chaos theory and to demonstrate their use in everyday life.
http://www.perfstrat.com/rfb/chaostoc.htm
Application of Chaos Theory to Psychological Models - chaos theory
offers a viable basis for improved understanding of human behavior, and
provides achievable frameworks for potential identification, assessment
and adjustment of human behavior patterns.
http://www.cbc.net/ steve/R1.HTM
Review of Chaos Oriented Books
http://t13.lanl.gov/ronnie/
Cycle Expansions in Chaos - Cycle expansions are used to compute the
properties of chaotic systems. There's an introduction and a searchable
database.
http://www.geocities.com/CapeCanaveral/6211/
Ian Malcolm's Chaos Theory Page
http://www.ipme.ru/coc97.html
Control of Oscillations and Chaos '97 - call for papers for conference
on controlling oscillatory dynamical systems (August 27-29, 1997).
http://lenti.med.umn.edu/ mwd/robot.html
Robotics, Learning, Chaos, Complexity, Systems
http://www.virtualchaos.org/
Virtual Chaos - A theory of everything; educates the public about common
questions about science and the humanities.
http://www.geocities.com/CapeCanaveral/3001/
Black Holes and Mysteries of the Cosmos - collection of ideas and opinions
about black holes and the mysteries of the cosmos including: chaos theory,
quantum theory, the universe, the cosmos, and the Big Bang.
http://linac.ikp.physik.th-darmstadt.de/heiko/chaosmain.html
Institute for Nuclear Physics Darmstadt - informations about our work
in the field of quantum chaos. For that we investigate the spectrum of
the eigenfrequencies of superconducting microwave cavities in a range up
to 20 GHz.
http://members.aol.com/ingallsd/private/logistic.htm
Generalized Logistic Equation - y = a*x + b*x*(1-x), one of the simpler
equations in chaos, is considered. Results and graphs are shown. Links
to other references are provided.
http://sun158.dees.unict.it
Catania Systems and Control Group - describes the staff and the research
activity of the Systems and Control Group of the University of Catania.
Infos on Neural Networks, Chaos, Robotics, Electronics and so on are included.
http://www.phys.boun.edu.tr/
Bogazici University - research interests: High Energy Physics, Chaos
and Dynamical Systems, Solid State, Astrophysics, Mathematical Physics.
http://chaos.chm.nau.edu/
Northern Arizona University
http://chaos.fullerton.edu/motf.html
Molecule of the Fortnight
http://chaos.fullerton.edu/mhslinks.html
Index - Particle-Surface Resources on the Internet - an actively maintained
and commented list of sites that are of interest to researchers in the
area of ion-beam and cluster-beam interactions with surfaces.
http://chaos.fccc.edu/research/labs/roder/
Heinrich Roder Lab - researches protein folding, dynamics, and function.
http://morpheus.hartford.edu/cs/student/srosenbl/chaos/chaos/index.htm
Mr Sierpinski's Triangle
The Lorenz Attractor was produced with Scilab 2.2 running on a Linux operating system. Scilab and Linux are both freely available. Scilab may be downloaded for Linux and other Unix operating systems from:
ftp://ftp.inria.fr:/INRIA/Scilab
For Linux specifically, it can be obtained from mirror sites such as:
ftp://sunsite.unc.edu/pub/Linux/apps/math/matrix/
Linux is now very popular and available from many mirror sites and on
many inexpensive CD distributions.
The program used to produce Figure 1
was as follows:
deff('[dydt]=lorenz(t,y)',...
"a=[-10,10,0;28,-1,-y(1);0,y(1),-8/3];...
dydt=a*y")
comp(lorenz);
format("e",20)
y0=[-3.2917495672888E+00;-6.3058819810691E+00;8.1821792963329E+00];
t0=0;dt=0.01;t1=5;
t=t0:dt:t1;
for i=1:1:20;
y=ode(y0,t0,t,lorenz);
xbasc(0);param3d(y(1,:),y(2,:),y(3,:),45,30,"X@Y@Z",[1,4],...
[-20,20,-28,28,0,50])
halt();
y0=y(:,501);
end
The program to produce the two-dimensional views was:
deff('[dydt]=lorenz(t,y)',...
"xt=y(1);yt=y(2);zt=y(3);...
dydt=[10*(yt-xt); 28*xt-yt-xt*zt; xt*yt-8*zt/3]")
comp(lorenz);
format("e",20)
y0=[-3.2917495672888E+00;-6.3058819810691E+00;8.1821792963329E+00];
t0=0;dt=0.01;t1=50;
t=t0:dt:t1;
y=ode(y0,t0,t,lorenz);
xbasc(0);plot2d([y(1,:)]',[y(3,:)]',-1,'030',"X@Y@Z",[-18,0,20,49])
halt();
xbasc(0);plot2d([y(2,:)]',[y(3,:)]',-1,'030',"X@Y@Z",[-24,0,28,49])
halt();
xbasc(0);plot2d([y(1,:)]',[y(2,:)]',-1,'030',"X@Y@Z",[-18,-24,20,28])
halt();
[1] Edward N. Lorenz,
[2] Edward N. Lorenz, ``Deterministic non-periodic flow'' in
[3] W. Ditto and Lou Pecora, ``Mastering Chaos'' in
[4] S. J. Schiff, K. Jerger, D. H. Duong, T. Chang, M. L. Spano and
W. L. Ditto, ``Controlling Chaos in the Brain'' in
[5] A. Garfinkel, M. L. Spano, W. L. Ditto and J. Weiss, ``Controlling
Cardiac Chaos'' in
[6] Senator Al Gore,