Introduction to Chaos and It's Real World Applications

George T. Yurkon

May 28, 1997

Abstract

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.

1.  Introduction to Chaos

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 Bounding the United States: ``The United States-bounded on the north by the Aurora Borealis, on the south by the precession, on the east by primeval chaos, and on the west by the Day of Judgement.''
     Modern discussions of chaos are almost always based on the work of Edward N. Lorenz. In his book The Essence of Chaos [1], he expends considerable effort in explaining deterministic chaos to the layman. He defines a deterministic sequence as one in which only one thing can happen next. He then defines randomness as being identical to the absence of determinism. Deterministic chaos is then something that looks random, but is really deterministic. This can happen because, in Poincairé's words: ``...we could only know the initial situation approximately...''. Lorenz used the once popular pinball machine as an example. Anyone who has ever tried his hand at releasing a pinball with just the right velocity to make it repeat the desired trajectory knows that it cannot be done except on apparently random occasions. It's not that the pinball's path is random, it's just that the resultant trajectory is extremely sensitive to slight variations in initial conditions-striking a pin just a few thousandths of an inch to the left or right of the previous occurrence, can cause a deflection in a completely different direction.
     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: Earth in the Balance [6]. He points out that: ``Many are now convinced that in a similar way [similar to the way Newtonian physics led to a revolution in understanding cause and effect], the insights of Chaos Theory will soon be absorbed into political science and social analysis.'' Let's briefly explore some of the areas of active research.
    

2.  Chaos in the Atmosphere

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:

dx/dt=-ax+ay

dy/dt=-xz+rx-y

dz/dt=xy-bz

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.
    

Figure 1: Lorenz Attractor in Intervals of delta t=5

Figure 2: X-Z View of Lorenz Attractor

Figure 3: Y-Z View of Lorenz Attractor

Figure 4: X-Y View of Lorenz Attractor

3.  Chaos in Industry

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.
    

4.  Chaos in the Human Body

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.
    

5.  Concluding Remarks

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.
    

Appendix

A.  Chaos Sites on the Internet


     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

    

B.  Scilab Programs for the Lorenz Attractor

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();

References

[1] Edward N. Lorenz, The Essence of Chaos, University of Washington Press, 7-11 (1993)

[2] Edward N. Lorenz, ``Deterministic non-periodic flow'' in Journal of Atmospheric Sciences, 20, 130-41 (1963)

[3] W. Ditto and Lou Pecora, ``Mastering Chaos'' in Scientific American, 78-84 (August 1993)

[4] S. J. Schiff, K. Jerger, D. H. Duong, T. Chang, M. L. Spano and W. L. Ditto, ``Controlling Chaos in the Brain'' in Nature, 370, 615-20 (1994)

[5] A. Garfinkel, M. L. Spano, W. L. Ditto and J. Weiss, ``Controlling Cardiac Chaos'' in Science 257, 1230-5 (1992)

[6] Senator Al Gore, Earth in the Balance, Houghton Mifflin (1992)
    


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