Chaos in the 21st Century (A lesson)

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Ever since I found out about Y2K the concept of Chaos ran through my mind. But it is not the Chaos of disorder,  it is the Chaos of natural order. This is a subject that few know about yet the ramifications will influence mankind's understanding of life itself.

Y2K can be looked at as a interaction of dynamic systems. Chaos theory try's to explain these systems. I have posted this as a bit of information for you to digest as we move towards the millennium. It is real, influences all aspects of life yet remains elusive regarding its true meaning. This is not just the domain of engineers and mathematics, it is studied by health professionals, biologists, chemists, philosophers, artists and Y2Kers (I hope).

Below is a good rundown on the subject by Manus J. Donahue III. I have taken the liberties to highlight in bold important points about Chaos theory. It is important to remember that Chaos Theory describes dynamic systems moving in time. This of course has a direct relationship to the events that may transpire during the near future. Chaos is the turbulence of a system, and if Y2K has even a moderate effect on the systems we rely on then the disturbances due to initial conditions will be impossible to map accurately due to the increase in uncertainty.

Society has demonstrated with Y2K an absolute disregard to the realities of time. This is ignorance and should be cured. No time like the present to start so enjoy folks. I would welcome comments from Chaologists if there are any out there.

 Chaos Theory and Fractal Phenomena

Chaos Theory and Fractal Phenomena

The following essay was compiled by Manus J. Donahue III. Please cite this page as a reference if you use any of the material on this page in essays, documents, or compositions. Also, you may e-mail me at mjd@duke.edu. Thank you and enjoy.

"Physicists like to think that all you have to do is say, these are the conditions, now what happens next?"

-Richard P. Feynman

The world of mathematics has been confined to the linear world for centuries. That is to say, mathematicians and physicists have overlooked dynamical systems as random and unpredictable. The only systems that could be understood in the past were those that were believed to be linear, that is to say, systems that follow predictable patterns and arrangements. Linear equations, linear functions, linear algebra, linear programming, and linear accelerators are all areas that have been understood and mastered by the human race. However, the problem arisesthat we humans do not live in an even remotely linear world;in fact, our world should indeed be categorized as nonlinear; hence, proportion and linearity is scarce.

How may one go about pursuing and understanding a nonlinear system in a world that is confined to the easy, logical linearity of everything? This is the question that scientists and mathematicians became burdened with in the 19th Century; hence, a new science and mathematics was derived: chaos theory.

The very name "chaos theory" seems to contradict reason, in fact it seems somewhat of an oxymoron. The name "chaos theory" leads the reader to believe that mathematicians have discovered some new and definitive knowledge about utterly random and incomprehensible phenomena; however, this is not entirely the case. The acceptable definition of chaost heory states, chaos theory is the qualitative study of unstable aperiodic behavior in deterministic nonlinear dynamical systems. A dynamical system may be defined to be a simplified model for the time-varying behavior of an actual system, and aperiodic behavior is simply the behavior that occurs when no variable describing the state of the system undergoes a regular repetition of values. Aperiodic behavior never repeats and it continues to manifest the effects of any small perturbation; hence, any prediction of afuture state in a given system that is aperiodic is impossible. Assessing the idea of aperiodic behavior to a relevant example, one may look at human history. History is indeed aperiodic since broad patterns in the rise and fall of civilizations may be sketched; however, no events ever repeat exactly.

What is so incredible about chaos theory is that unstable aperiodic behavior can be found in mathematically simply systems. These very simple mathematical systems display behavior so complex and unpredictable that it is acceptable to merit their descriptions as random.

An interesting question arises from many skeptics concerning why chaos has just recently been noticed. If chaotic systems are so mandatory to our every day life, how come mathematicians have not studied chaos theory earlier? The answer can be given in one word: computers. The calculations involved in studying chaos are repetitive, boring and number in the millions. No human is stupid enough to endure the boredom; however, a computer is always up to the challenge. Computers have always been known for their excellence at mindless repetition; hence, the computer is our telescope when studying chaos. For, without a doubt, one cannot really explore chaos without a computer.

Before advancing into the more precocious and advanced areas of chaos, it is necessary to touch on the basic principle that adequately describes chaos theory, the Butterfly Effect. The Butterfly Effect was vaguely understood centuries ago and is still satisfactorily portrayed in folklore:

"For want of a nail, the shoe was lost;

For want of a shoe, the horse was lost;

For want of a horse, the rider was lost;

For want of a rider, the battle was lost;

For want of a battle, the kingdom was lost!"

Small variations in initial conditions result in huge, dynamic transformations in concluding events. That is to say that there was no nail, and, therefore, the kingdom was lost. The graphs of what seem to be identical, dynamic systems appear to diverge as time goes on until all resemblance disappears.

Perhaps the most identifiable symbol linked with the Butterfly Effect is the famed Lorenz Attractor. Edward Lorenz, a curious meteorologist, was looking for a way to model the action of the chaotic behavior of a gaseous system. Hence, he took a few equations from the physics field of fluid dynamics, simplified them, and got the following three-dimensional system:

dx/dt=delta*(y-x)

dy/dt=r*x-y-x*z

dz/dt=x*y-b*z

Delta represents the "Prandtl number," the ratio of the fluid viscosity of a substance to its thermal conductivity; however, one does not have to know the exact value of this constant; hence, Lorenz simply used 10. The variable "r" represents the difference in temperature between the top and bottom of the gaseous system. The variable "b" is the width to height ratio of the box which is being used to hold the gas in the gaseous system. Lorenz used 8/3 for this variable. The resultant x of the equation represents the rate of rotation of the cylinder, "y" represents the difference in temperature at opposite sides of the cylinder, and the variable "z" represents the deviation of the system from a linear, vertical graphed line representing temperature. If one were to plot the three differential equations on a three-dimensional plane, using the help of a computer of course, no geometric structure or even complex curve would appear; instead, a weaving object known as the Lorenz Attractor appears. Because the system never exactly repeats itself, the trajectory never intersects itself. Instead it loops around forever. I have included a computer animated Lorenz Attractor which is quite similar to the production of Lorenz himself. The following Lorenz Attractor was generated by running data through a 4th-order Runge-Kutta fixed-timestep integrator with a step of .0001, printing every 100th data point. It ran for 100 seconds, and only took the last 4096 points. The original parameters were a =16, r =45, and b = 4 for the following equations (similar to the original Lorenz equations):

x'=a(y-x)

y'=rx-y-xz

z'=xy-bz

The initial position of the projectory was (8,8,14). When the points were generated and graphed, the Lorenz Attractor was produced in 3-D:

(Sniped Graphic)

The attractor will continue weaving back and forth between the two wings, its motion seemingly random, its very action mirroring the chaos which drives the process.

Lorenz had obviously made an immense breakthrough in not only chaos theory, but life. Lorenz had proved that complex, dynamical systems show order, but they never repeat. Since our world is classified as a dynamical, complex system, our lives, our weather, and our experiences will never repeat; however, they should form patterns.

Lorenz, not quite convinced with his results, did a follow-up experiment in order to support his previous conclusions. Lorenz established an experiment that was quite simple; it is known today as the Lorenzian Waterwheel. Lorenz took a waterwheel; it had about eight buckets spaced evenly around its rim with a small hole at the bottom of each . The buckets were mounted on swivels, similar to Ferris-wheel seats, so that the buckets would always point upwards. The entire system was placed under a waterspout. A slow, constant stream of water was propelled from the waterspout; hence, the waterwheel began to spin at a fairly constant rate.

Lorenz decided to increase the flow of water, and, as predicted in his Lorenz Attractor, an interesting phenomena arose. The increased velocity of the water resulted in a chaotic motion for the waterwheel. The waterwheel would revolve in one direction as before, but then it would suddenly jerk about and revolve in the opposite direction.The filling and emptying of the buckets was no longer synchronized; the system was now chaotic.

Lorenz observed his mysterious waterwheel for hours, and, no matter how long he recorded the positions and contents of the buckets, there was never and instance where the waterwheel was in the same position twice. The waterwheel would continue on in chaotic behavior without ever repeating any of its previous conditions. A graph of the waterwheel would resemble the Lorenz Attractor.

Now it may be accepted from Lorenz and his comrades that our world is indeed linked with an eery form of chaos. Chaos and randomness are no longer ideas of a hypothetical world; they are quite realistic here in the status quo. A basis for chaos is established in the Butterfly Effect, the Lorenz Attractor, and the Lorenz Waterwheel; therefore, there must be an immense world of chaos beyond the rudimentary fundamentals. This new form mentioned is highly complex,repetitive, and replete with intrigue.

Big Snip

It is now established that fractals are quite real and incredible; however, what do these newly discovered objects have to do with real life? Is there a purpose behind these fascinating images? The answer is a somewhat surprising yes. Homer Smith, a computer engineer of Art Matrix, once said, "If you like fractals, it is because you are made of them. If you can't stand fractals, it's because you can't stand yourself." Fractals make up a large part of the biological world. Clouds, arteries, veins, nerves, parotid gland ducts, and the bronchial tree all show some type of fractal organization. In addition, fractals can be found in regional distribution of pulmonary blood flow, pulmonary alveolar structure, regional myocardial blood flow heterogeneity, surfaces of proteins, mammographic parenchymal pattern as a risk for breast cancer, and in the distribution of arthropod body lengths. Understanding and mastering the concepts that govern fractals will undoubtedly lead to breakthroughs in the area of biological understanding. Fractals are one of the most interesting branches of chaos theory, and they are beginning to become ever more key in the world of biology and medicine.

(Snip)

The stock markets are said to be nonlinear, dynamic systems. Chaos theory is the mathematics of studying such nonlinear, dynamic systems. Does this mean that chaoticians can predict when stocks will rise and fall? Not quite; however, chaoticians have determined that the market prices are highly random, but with a trend. The stock market is accepted as a self-similar system in the sense that the individual parts are related to the whole. Another self-similar system in the area of mathematics are fractals. Could the stock market be associated with a fractal? Why not? In the market price action, if one looks at the market monthly, weekly, daily, and intra day bar charts, the structure has a similar appearance.
 
However, just like a fractal, the stockmarket has sensitive dependence on initial conditions. This factor is what makes dynamic market systems so difficult to predict. Because we cannot accurately describe the current situation with the detail necessary, we cannot accurately predict the state of the system at a future time. Stock market success can be predicted by chaoticians.

Short-term investing, such as intra day exchanges are a waste of time. Short-term traders will fail over time due to nothing more than the cost of trading. However, over time, long-term price action is not random. Traders can succeed trading from daily or weekly charts if they follow the trends. A system can be random in the short-term and deterministic in the long term.

Perhaps even more important than stock market chaos and predictability is solar system chaos. Astronomers and cosmologists have known for quite some time that the solar system does not "run with the precision of a Swiss watch." Inabilities occur in the motions of Saturn's moon Hyperion, gaps in the asteroid belt between Mars and Jupiter, and in the orbit of the planets themselves. For centuries astronomers tried to compare the solar system to a gigantic clock around the sun; however, they found that their equations never actually predicted the real planets' movement. It is easy to understand how two bodies will revolve around a common center of gravity. However, what happens when a third, fourth, fifth or infinite number of gravitational attractions are introduced? The vectors become infinite and the system becomes chaotic. This prevents a definitive analytical solution to the equations of motion. Even with the advanced computers that we have today, the long term calculations are far too lengthy. Stephen Hawking once said, "If we find the answer to that (the universe), it would be the ultimate triumph of human reason-for then we would know the mind of God.

The applications of chaos theory are infinite; seemingly random systems produce patterns of spooky understandable irregularity. From the Mandelbrot set to turbulence to feedback and strange attractors; chaos appears to be everywhere. Breakthroughs have been made in the past in the area chaos theory, and, in order to achieve any more colossal accomplishments in the future, they must continue to be made. Understanding chaos is understanding life as we know it.

However, if we do discover a complete theory, it should in time be understandable in broad principle by everyone, not just a few scientists. Then we shall all, philosophers, scientists, and just ordinary people, be able to take part in the discussion of the question of why it is that we and the universe exist.

Stephen Hawking

-- Brian (imager@home.com), November 01, 1999

Answers

I should also mention that the site this came from has graphics to help understand Chaos and Fractals. Well worth checking out.

-- Brian (imager@home.com), November 01, 1999.

Thank you, brian. it's always nice to read Stephen. One of the few treatments that can help people understand the theory, but you shoulda left the animation. It's too cool.

Chuck

-- Chuck, a night driver (rienzoo@en.com), November 01, 1999.


Chuck

Yes maybe next time I will post something on Quantum physics or Cosmology dealing with time from Steven.

The graphics can be a bit bandwidth heavy so they were left out. Maybe another thread I will :o)

-- Brian (imager@home.com), November 01, 1999.


Brian, you might want to check out Ilya Prigogen's work on Dissipative Structures, for which he won a Nobel Prize. Chaos and dissipative structures are intimately connected imho, perhaps analogous to magnetism and electricity.

-- Mitchell Barnes (spanda@inreach.com), November 01, 1999.

Mitch

I have Order out of Chaos. Been at least 12 yrs since I read it though.

How time flies.

 Amazon.com: A Glance: Order Out of Chaos
Order Out of Chaos  by Ilya Prigogine
Paperback (November 1986)

And here is the difinitive learners guide to Chaos Theory

 Amazon.com: A Glance: Chaos : Making a New Science
Chaos : Making a New Science by James Gleick
Paperback - 352 pages Reprint edition (December 1988) Penguin USA

-- Brian (imager@home.com), November 01, 1999.



And so what do we conclude from this---that we might experience a discontinuous leap to a new dynamic equilibrium (limit cycle) because of a spontaneous perturbation (y2k) to our nonlinear, complexly ineractive world, ie, we can't do jack shit?

-- (kaos@ivegota.headache), November 01, 1999.

Kaos

In twenty words or less, possibly. How about a quantum leap.

We don't know, its never been done before :o)

After the exponential growth in IT without the same growth in IT management, it could get messy. I like to think of it as a flow of information that could get restricted then there would be turbulance and the flow would have to be reduced or things would get real screwed up.

Unfortunately the Y2K mantra is full bore ahead. I think that would be a problem.

-- Brian (imager@home.com), November 02, 1999.


Brian, Chuck and Mitchell:

Prigogine's newest work (for the semi-layman) is called "The End of Certainty."

It updates the work pioneered in "Order Out of Chaos", and in many ways is more readable. I highly recommend it.

As your post suggests, Brian, as we approach a phase-transition in a dynamical complex system, the instabilities increase, and the importance of the random factors is exagerated. Just before the transition (or bifurcation, if you prefer), with the instabilities at maximum values, the random factors have an inordinate ability to influence the future-history of the system. Or, put another way; very small 'pushes' or 'nudges' at the critical time can constitute the 'initial conditions' to which the system is so susceptible. Very tiny, seemingly insignificant, actions, events and happenings can , at the right moment, determine to a large extent the future of the system.

We are in a time of transition. The instabilities are rapidly mounting. WE are the random factors in this equation. We are the Agents of Fortune. We cannot know the future with certainty, but our small efforts CAN affect the outcome of the phase transition, the future history of the system.

This transition will almost certainly take us to a lower energy state. The question we are all seeking an answer to is: How low will it go?

Try and do the best you can, everyday.

Godspeed,

-- Pinkrock (aphotonboy@aol.com), November 02, 1999.


Pinkrock

A good run down of the situation. The chance that any individual can make a change is an important one. So not only is there a potential of disruption from the CDC but if the time and contitions are right an individual or group could influence the future in a positive way.

But timing is everything. And know one knows the time. That is what makes life so interesting.

-- Brian (imager@home.com), November 02, 1999.


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