Chaos Theory & Fractals,

8/22/2019 Chaos Theory & Fractals,

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EE454Applied Signal Processing

July 18, 1999

Research Project

Chaos Theory & Fractals,Their Applications in Real Life

Adeel Aslam Bhutta960001, EE


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Chaos, Fractals & their Real Life

Applications

Introduction:

Chaos has been called, by some, the third revolution of the twentieth century in

Physics, after Relativity and Quantum Physics. Chaos, contrary to popular belief, does

not involve the absence of patterns rather, it is the study of complex nonlinear

dynamic systems. Thus, Chaos Theory is the study of forever changing complex

systems based on mathematical concepts of recursion, whether in form of a recursive

process or a set of differential equations modeling a physical system. One of thecentral concepts of Chaos Theory is that while it is important to exactly predict the

state of a system, it is usually a universal rule that it is possible to model the overall

behavior of a system. Thus, Chaos Theory lays emphasis not on the disorder but on

the order inherent in the system.

In the next section, we present definition of Chaos Theory, characteristics of Chaotic

systems and some real life examples. While looking at the real world examples of

Chaos Theory, we shall define Fractal and look at its importance followed by some

examples. In the end, we shall look at the uses of Chaos Theory in modern age

specifically in the areas of Signal Processing. Implementation issues and problems

shall be discussed in the end.

What is Chaos Theory?

Chaos Theory is among the youngest of the sciences, and has rocketed from its

obscure roots in the seventies to become one of the most fascinating fields in

existence. At the forefront of much research on physical systems, and already being

implemented in fields covering as diverse matter as image compression, fluiddynamics, chaos science promises to continue to yield absorbing scientific

information which may shape the face of science in the future. The acceptable

definition of chaos theory states, Chaos Theory is the qualitative study of unstable

aperiodic behavior in deterministic nonlinear systems. Complex implies just that,

aperiodic is simply the behavior that never repeats, nonlinear implies recursion &

higher mathematical algorithms, and dynamic implies non-constant and non-periodic

(time variables). Thus, Chaos Theory is the study of forever changing complex

systems based on mathematical concepts of recursion, whether in form of a recursive

process or a set of differential equations modeling a physical system.

Newhouse's definition states,A bounded deterministic dynamical system with at leastone positive Liaponov exponent is a chaotic system; a chaotic signal is an observation


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of a chaotic system. The presence of a positive Liaponov exponent causes trajectories

that are initially close to each other to separate exponentially. This, in turn, implies

sensitive dependence of the dynamics on initial conditions, which is one of the most

important characteristics of chaotic systems. What is so incredible about Chaos

Theory is that unstable aperiodic behavior can be found in mathematically simple

systems. Lorenz proved that the complex, dynamical systems show order, but theynever repeat. Since our world is classified as a dynamical, complex system, our lives,

our weather, and our experiences never repeat; however, they should form patterns.

History of Chaos Theory:

Chaos theory is the result of the studies begun by Edward Lorenz in 1960's. Lorenz

was a meteorologist who used computers to simulate weather systems using nonlinear

equations. He discovered inadvertently that the small changes (as little as 1/1000) in

the initial conditions produced dramatic changes in the overall system.

Lorenz also discovered that there was orderliness to this chaos. If computer

simulations were run long enough, spiral patterns emerged.

Benoit Mandelbrot began to investigate the images that arose from nonlinear

equations. He based his work on previous investigations by Gaston Julia from the

1920's. Julia theorized that iterations of a rational function stayed within confines

even as the number of iterations increased to infinity. Mandelbrot found that plotting

those iterations resulted in images called Fractals. Like strange attractors, theseimages were unpredictable. Fractal geometry, unlike Euclidean geometry, can

describe chaotic systems.

Nowadays, the world is looking at the Chaos Theory for some real life applications.

Some are using Fuzzy-Logic Circuits to model the chaotic behavior of systems. Some

companies have even come up even with consumer products.

Chaotic Systems are real:Dynamic systems are variables of time. Examples include the stock market,

ecosystems, weather, human body, solar systemthe list is endless. Even a simple

pendulum can be chaotic under certain conditions. Traditional mathematics based on

Newtonian principles has only been able to understand and model these systems by

taking them apart and looking at the individual pieces. We can use linear equations to

model the pieces, however, this gives us an incomplete picture of the behavior of

these systems. Eventually, we run up against the need to model these systems using

non-linear equations, most of them are unsolvable. But many of the pioneers in chaos

discovered that graphing these equations using feedback loops allowed them to look


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at pictures of these systems, and we are beginning to understand much more about

them using these graphs.

Chaotic Systems:Chaos theory can be studies in systems that are complex as well as those that are

relatively simple. The calculations involved in chaos are repetitive, boring and

number in millions that's why computers are used for such calculations. Therefore, the

computer is our telescope while studying chaos.

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 as Small

variations in initial condition result in huge, dynamic transformations in concludingevents. 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*zdz/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 that 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.

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 theoriginal Lorenz equations):


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x'=a(y-x)

y'=rx-y-xz

z'=xy-bzThe initial position of the projectory was (8,8,14). When the points were generated

and graphed, the Lorenz Attractor was produced in 3-D:

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 in

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 did a follow-up experiment in order to support his previous conclusions. He

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 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. Lorenzdecided to increase the flow of water, and, as predicted in his Lorenz Attractor, an


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interesting phenomenon 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 andinstance 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.

The extending and folding of chaotic systems give strange attractors, such as the

Lorenz Attractor, the distinguishing characteristic of a non-integral dimension. This

non-integral dimension is most commonly referred to as a fractal dimension.

Fractals:

Mathematically, fractals are pictures that result from iterations of nonlinear equations,

usually in a feedback loop. Using the output value for the next input value, a set of

points is produced. Graphing these points produces images. Again, by creating a vast

number of points using computers to generate those points, mathematicians

discovered these wonderfully complex images which were called fractals, a term

coined by Benoit Mandelbrot, one of the first to discover and examine these images.

Two important properties of fractals are: self-similarity

fractional dimensions

Self-similarity means that at every level, the fractal image repeats itself. Sierpinski's

Triangle demonstrates this quite well: a triangle within smaller triangles within

smaller triangles within ever smaller triangles, on and on. Many shapes in nature

display this same quality of self-similarity. Clouds, ferns, coastlines, mountains, etc.

all possess this feature.


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Fractional dimension means that a shape is neither 1, nor 2, nor 3 dimensional, but

actually may fall between these numbers, being composed of fractions. Mandelbrot

calculated that fractals have a fractional dimension between 1 and 2. By studying

fractals, mathematicians have a whole new geometry for describing the universe,

beyond the boundaries of Euclidean geometry.

In addition to the famous Sierpenski Triangle, the Koch Snowflake is also a wellnoted, simple fractal image. To construct a Koch Snowflake, begin with a triangle

with sides of length 1. At the middle of each side, add a new triangle one-third the

size; and repeat this process for an infinite amount of iterations. The length of the

boundary is 3 X 4/3 X 4/3 X 4/3...-infinity. However, the area remains less than the

area of a circle drawn around the original triangle. What this means is that an

infinitely long line surrounds a finite area. The end construction of a Koch Snowflake

resembles the coastline of a shore.

Some other fractals are shown below:

The above are called Julia set of fractals while the followings are Mandelbrot

fractals:


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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 aremade 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.

Applications of Chaos Theory:

Everyone always wants to know one thing about new discoveries----what good are

those?

First and foremost, chaos theory is a theory. As such, much of it is of use more as

scientific background than as direct applicable knowledge. Chaos theory is great as a

way of looking at events which happen in the world differently from the more

traditional strictly deterministic view which has dominated science from Newtonian

times. Instead of a traditional X-Y plot, scientists can now interpret phase-spacediagrams which--rather than describing the exact position of some variable with

respect to time--represents the overall behavior of a system. Instead of looking for

strict equations conforming to statistical data, we can now look for dynamic systems

with behavior similar in nature to the statistical data--systems, that is, with similar

Attractors. Chaos theory provides a sound framework with which to develop scientific

knowledge.

However, this is not to say that chaos theory has no applications in real life. Chaos

theory techniques have been used to model biological systems, which are of course

some of the most chaotic systems imaginable. Systems of dynamic equations have

been used to model everything from population growth to epidemics to arrhythmicheart palpitations.

In fact, almost any chaotic system can be readily modeled--the stock market provides

trends that can be analyzed with strange Attractors more readily than with

conventional explicit equations.

Nowadays, Chaotic phenomenon is used formilitary operations too. The world is

working on the transmission of military signals using chaotic models that would make

it very difficult to be captured.

Fractal image compression techniques are still under research, but promise suchamazing results as 600:1 graphic compression ratios. The movie special effects


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industry would have much less realistic clouds, rocks, and shadows without fractal

graphic technology.

The first consumer product to exploit chaos theory was produced in 1993 by Goldstar

Co. in the form of a revolutionary washing machine. A chaotic washing machine!

The washing machine is based on the principle that there are identifiable andpredictable movements in nonlinear systems. The new washing machine was designed

to produce cleaner and less tangled clothes. The key to the chaotic cleaning process

can be found in a small pulsator that rises and falls randomly as the main pulsator

rotates. The new machine was surprisingly successful. However, Daewoo, a

competitor of Goldstar claims that they first started commercializing chaos theory in

their "bubble machine" which was released in 1990. The "bubble machine" was the

first to use the revolutionary "fuzzy logic circuits." These circuits are capable of

making choices between zero and one, and between true and false. Hence, the "fuzzy

logic circuits" are responsible for controlling the amount of bubbles, the turbulence of

the machine, and even the wobble of the machine. Indeed, chaos theory is very much

a factor in today's consumer world market.

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.

From a signal processing perspective, an issue of paramount importance is the

reconstruction of dynamics from measurements made on a single coordinate of the

system. We can use chaotic models for the prediction of next samples in any model.

For chaos, such prediction can only be short-term. If we try to predict long-termvalues in any chaotic system then it would not be of any use because the error

associated with that would be very high. The practical example of such prediction is

weather prediction.

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. Understanding

chaos is 'understanding life' as we know it.


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Conclusion:

The natural world has always had a chaotic way about it. We can find chaos theory

everywhere around us: in simple pendulum, stock market, solar system, weather

forecasting, image processing, biological systems, human bodyand so on. Chaoticsystems are not random. They may appear to be. They have some simple defining

features that they are deterministic that means they have something determining their

behavior. Chaotic systems are very sensitive to the initial conditions which means that

a very slight change in the starting point can lead to enormously different outcomes.

This makes the system fairly unpredictable. Chaos systems never repeat but they

always have some order. Most of the systems we find in the world predicted by

classical physics are the exceptions. In this world of order, chaos rules!

There is a strong link between chaos and fractals. Fractal geometry is the geometry

that describes the chaotic systems we find in nature. Fractals are a language, a way to

describe geometry. Euclidean geometry is a description of lines, circles, triangles, and

so on. Fractal geometry is described in algorithms- a set of instructions on how to

create the fractal. Computers translate the instructions into the magnificent patterns,

we see as fractal images. Fractals are self-similar and possess fractional dimensions.

Clouds, ferns, coastlines, mountains, veins, nerves, parotid gland ducts, etc. all

possess these features. "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."

Chaos theory and fractal are everywhere. We find them around us, with us and even

in us. We study these chaotic systems using basic principles of recursion or a set of

differential equations modeling a physical system. The applications of Chaos theorycan be found in biological systems, solar system, stock market, weather predictions,

signal processing, and many more.

In signal processing, chaotic systems can be modeled using fractals. We can apply

concepts of fractal dimensions that means that we could use 2.5 th past value for our

analysis. To calculate a precise model and hence predict a value, we need to have a

good approximation of the number of delays or lags needed which can be fractional in

fractal theory. In terms of the computations, the time series must be large enough for

estimating the Attractor dimensions and the Liaponov exponents and computing the

delay coordinate map. That is the implementation problem with chaos systems /

models that we need to have a very large time series that in turn increases the numberof computations required.

Chaos theory is a new way of thinking about what we have. It gives us a new concept

of measurements and scales. It looks at the universe in an entirely different way not in

Newtonian way but in chaotic way, not using Euclidean geometry but using fractal

geometry. Benoit Mandelbrot said, "Clouds are not spheres, mountains are not cones,

coastlines are not circles, and bark is not smooth, nor does lightning travel in straight

line." 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,


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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

References: Edward Ott, Chaos in Dynamical Systems, Cambridge University Press, 1993.

Michael Tabor, Chaos and Integrability in Nonlinera Dynamics (An Introduction),

John-Willy & Sons, Inc. 1989.

____, Characteristics of a Chaotic Process, IEEE Signal Processing Magazine,

vol. 1, pp. 30-31, March 1996.

Getteys, Keller & Skove, Physics, McGraw-Hill, Inc. 1994.

Manus J. Donahua III, Chaos Theory and Fractal Phenomena, Internet.

Some other documents from Internet.

************ The End ***************

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