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Chaos & Fractals in Financial Markets, index to articles, by J. Orlin Grabbe


Chaos & Fractals in Financial Markets
by J. Orlin Grabbe

The rolling of the golden apple. I meet chaos. Preliminary
pictures and poems. Dynamical systems. What is chaos? I'm
sensitive, don't perturb me. Why chaos? How fast do
Chaos and Fractals in
Financial Markets, Part 1 forecasts go wrong?--the Lyapunov exponent. Simple
calculation using a Lyapunov exponent. Enough for now.
Problems.
The French gambler and the pollen grains. The square root
of time. Normal versus lognormal. How big is it? History's
Chaos and Fractals in
Financial Markets, Part 2 first fractal. Fractal time. Probability is a one-pound jar of
jelly. Problems and answers.
Hazardous world. Coin flips and Brownian motion. A simple
stochastic fractal. Sierpinski and Cantor revisited. Blob
Chaos and Fractals in
Financial Markets, Part 3 measures are no good. Coastlines and Koch curves. Using a
Hausdorff measure. Jam session.
Gamblers, zero-sets, and fractal mountains. Futures trading
Chaos and Fractals in
and the gambler's ruin problem. An example. Gauss versus
Financial Markets, Part 4
Cauchy. Location and scale.
Louis Bachelier visits the New York Stock Exchange.
Chaos and Fractals in
Bachelier's scale for stock prices. Volatility. Fractal sums of
Financial Markets, Part 5
random variables. Some fun with logistic art. Julia sets.
Prechter's drum roll. Symmetric stable distributions and the
Chaos and Fractals in
Financial Markets, Part 6 gold mean law. The Fibonacci dynamical system.
Grow brain. Hurst, hydrology and the annual flooding of the
Chaos and Fractals in
Nile. Calculating the Hurst exponent. A misunderstanding to
Financial Markets, Part 7
avoid. Bull and bear markets.

These articles are parts of a work in progress. ©1999-2001 J. Orlin Grabbe. All Rights
Reserved.




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Chaos and Fractals in Financial Markets
Part 1
by J. Orlin Grabbe
Prologue: The Rolling of the Golden Apple
In 1776, a year in which political rebels in Philadelphia
were proclaiming their independence and freedom, a
physicist in Europe was proclaiming total dependence
and determinism. According to Pierre-Simon Laplace, if
you knew the initial conditions of any situation, you
could determine the future far in advance: "The present
state of the system of nature is evidently a consequence
of what it was in the preceding moment, and if we
conceive of an intelligence which at a given instant
comprehends all the relations of the entities of this
universe, it could state the respective positions, motions,
and general effects of all these entities at any time in the
past or future."
The Laplacian universe is just a giant pool table. If you
know where the balls were, and you hit and bank them
correctly, the right ball will always go into the intended
pocket.
Laplace's hubris in his ability (or that of his
"intelligence") to forecast the future was completely
consistent with the equations and point of view of
classical mechanics. Laplace had not encountered
nonequilibrium thermodynamics, quantum physics, or
chaos. Today some people are frightened by the very
notion of chaos. (I have explored this at length in an
essay devoted to chaos from a philosophical perspective.
But the same is also true with respect to the somewhat
related mathematical notion of chaos.) Today there is no
justification for a Laplacian point of view.
At the beginning of this century, the mathematician
Henri Poincar©, who was studying planetary motion,
began to get an inkling of the basic problem:
"It may happen that small differences in the
initial conditions produce very great ones in
the final phenomena. A small error in the
former will produce an enormous error in
the latter. Prediction becomes impossible"
(1903).


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In other words, he began to realize "deterministic" isn™t
what it™s often cracked up to be, even leaving aside the
possibility of other, nondeterministic systems. An
engineer might say to himself: "I know where a system is
now. I know the location of this (planet, spaceship,
automobile, fulcrum, molecule) almost precisely.
Therefore I can predict its position X days in the future
with a margin of error precisely related to the error in my
initial observations."
Yeah. Well, that™s not saying much. The prediction error
may explode off to infinity at an exponential rate (read
the discussion of Lyapunov exponents later). Even God
couldn™t deal with the margin of error, if the system is
chaotic. (There is no omniscience. Sorry.) And it gets
even worse, if the system is nondeterministic.
The distant future? You™ll know it when you see it, and
that™s the first time you™ll have a clue. (This statement
will be slightly modified when we discuss a system™s
global properties.)
I Meet Chaos
I first came across something called "dynamical
systems" while I was at the University of California at
Berkeley. But I hadn't paid much attention to them. I
went through Berkeley very fast, and didn't have time to
screw around. But when I got to Harvard for grad school,
I bought Ren© Thom's book Structural Stability and
Morphogenesis, which had just come out in English. The
best part of the book was the photos.
Consider a crown worn by a king or a princess, in fairy
tales or sometimes in real life. Why does a crown look
the way it does? Well, a crown is kind of round, so it will
fit on the head, and it has spires on the rim, like little
triangular hats”but who knows why”and sometimes
on the end of the spires are little round balls, jewels or
globs of gold. Other than the requirement that it fit on
the head, the form of a crown seems kind of arbitrary.
But right there in Thom's book was a photo of a steel ball
that had been dropped into molten lead, along with the
reactive splash of the molten liquid. The lead splash was
a perfect crown--a round vertical column rising upward,
then branching into triangular spires that get thinner and
thinner (and spread out away from the center of the
crown) as you approached the tips, but instead of ending
in a point, each spire was capped with a spherical blob of
lead. In other words, the shape of a crown isn't arbitrary
at all: under certain conditions its form occurs

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spontaneously whenever a sphere is dropped into liquid.
So the king™s crown wasn™t created to "symbolize" this
or that. The form came first, a natural occurrence, and
the interpretation came later.
The word "morphogenesis" refers to the forms things
take when they grow: bugs grow into a particular shape,
as do human organs. I had read a number of books on
general systems theory by Ervin Laszlo and Ludwig von
Bertalanffy, which discuss the concepts of
morphogenesis, so I was familiar with the basic ideas.
Frequent references were made to biologist D™Arcy
Thompson™s book On Growth and Form. But it was only
much later, when I began doing computer art, and
chaotically created a more or less perfectly formed ant
by iterating a fifth-degree complex equation (that is, an
equation containing a variable z raised to the fifth power,
z5, where z is a complex number, such as z = .5 + 1.2
sqrt(-1) ), that I really understood the power of the idea.
If the shape of ants is arbitrary, then why in the hell do
they look like fifth-degree complex equations?
Anyway, moving along, in grad school I was looking at
the forms taken by asset prices, foreign exchange rates in
particular. A foreign exchange rate is the price that one
fiat currency trades for another. But I could have been
looking at stock prices, interest rates, or commodity
prices”the principles are the same. Here the assumption
is that the systems generating the prices are
nondeterministic (stochastic, random)”but that doesn™t
prevent there being hidden form, hidden order, in the
shape of probability distributions.
Reading up on price distributions, I came across some
references to Benoit Mandelbrot. Mandelbrot, an applied
mathematician, had made a splash in economics in the
early-1960s with some heretical notions of the
probabilities involved in price distributions, and had
acquired as a disciple Eugene Fama [1] at the University
of Chicago. But then Fama abandoned this heresy (for
alleged empirical reasons that I find manifestly absurd),
and everyone breathed a sigh of relief and returned to the
familiar world of least squares, and price distributions
that were normal (as they believed) in the social sense as
well as the probability sense of a "normal" or Gaussian
distribution.
In economics, when you deal with prices, you first take
logs, and then look at the changes between the logs of
prices [2]. The changes between these log prices are
what are often referred to as the price distribution. They


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may, for example, form a Bell-shaped curve around a
mean of zero. In that case, the changes between logs
would have a normal (Gaussian) distribution, with a
mean of zero, and a standard deviation of whatever. (The
actual prices themselves would have a lognormal
distribution. But that™s not what is meant by
"non-normal" in most economic contexts, because the
usual reference is to changes in the logs of prices, and
not to the actual prices themselves.)
At the time I first looked at non-normal distributions,
they were very much out of vogue in economics. There
was even active hostility to the idea there could be such
things in real markets. Many people had their nice set of
tools and results that would be threatened (or at least
they thought would be threatened) if you changed their
probability assumptions. Most people had heard of
Mandelbrot, but curiously no one seemed to have the
slightest clue as to what the actual details of the issue
were. It was like option pricing theory in many ways: it
wasn™t taught in economic departments at the time,
because none of the professors understood it.
I went over to the Harvard Business School library to
read Mandelbrot™s early articles. The business school
library was better organized than the library at the
Economics Department, and it had a better collection of
books and journals, and it was extremely close to where I
lived on the Charles River in Cambridge. In one of the
articles, Mandelbrot said that the ideas therein were first
presented to an economic audience in Hendrik
Houthakker™s international economics seminar at
Harvard. Bingo. I had taken international finance from
Houthakker and went to talk to him about Mandelbrot.
Houthakker had been a member of Richard Nixon™s
Council of Economic Advisors, and was famous for the
remark: "[Nixon] had no strong interest in international
economic affairs, as shown by an incident recorded on
the Watergate tapes where Haldeman comes in and
wants to talk about the Italian lira. His response was
˜[expletive deleted] the Italian lira!™"
Houthakker told me he had studied the distribution of
cotton futures prices and didn™t believe they had a
normal distribution. He had given the same data to
Mandelbrot. He told me Mandelbrot was back in the
U.S. from a sojourn in France, and that he had seen him
a few weeks previously, and Mandelbrot had a new book
he was showing around. I went over to the Harvard Coop
(that™s pronounced "coupe" as in "a two-door coupe", no
French accent) and found a copy of Mandelbrot™s book.

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Great photos! That™s when I learned what a fractal was,
and ended up writing two of the three essays in my PhD
thesis on fractal price distributions [3].
Fractals led me back into chaos, because maps (graphics)
of chaos equations create fractal patterns.
Preliminary Pictures and Poems
The easiest way to begin to explain an elephant is to first show someone a picture. You point and say, "Look.
Elephant." So here™s a picture of a fractal, something called a Sierpenski carpet [4]:




Notice that it has a solid blue square in the center, with 8 additional smaller squares around the center one.


1 2 3




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8 center square 4




7 6 5




Each of the 8 smaller squares looks just like the original square. Multiply each side of a smaller square by 3
(increasing the area by 3 x 3 = 9), and you get the original square. Or, doing the reverse, divide each side of the
original large square by 3, and you end up with one of the 8 smaller squares. At a scale factor of 3, all the squares
look the same (leaving aside the disgarded center square).
You get 8 copies of the original square at a scale factor of 3. Later we will see that this defines a fractal dimension
of log 8 / log 3 = 1.8927. (I said later. Don™t worry about it now. Just notice that the dimension is not a nice round
number like 2 or 3.)
Each of the smaller squares can also be divided up the same way: a center blue square surrounded by 8 even
smaller squares. So the original 8 small squares can be divided into a total of 64 even smaller squares”each of
which will look like the original big square if you multiply its sides by 9. So the fractal dimension is log 64 / log 9
= 1.8927. (You didn™t expect the dimension to change, did you?) In a factal, this process goes on forever.
Meanwhile, without realizing it, we have just defined a fractal (or Hausdorff ) dimension. If the number of small
squares is N at a scale factor of r, then these two numbers are related by the fractal dimension D:
N = rD .
Or, taking logs, we have D = log N / log r.
The same things keep appearing when we scale by r, because the object we are dealing with has a fractal
dimension of D.
Here is a poem about fractal fleas:
Great fleas have little fleas, upon their backs to bite 'em
And little fleas have lesser fleas, and so ad infinitum,
And the great fleas themselves, in turn, have greater fleas to go on,
While these again have greater still, and greater still, and so on.
Okay. So much for a preliminary look at fractals. Let™s take a preliminary look at chaos, by asking what a
dynamical system is.
Dynamical Systems
What is a dynamical system? Here™s one: Johnny grows 2 inches a year. This system explains how Johnny™s
height changes over time. Let x(n) be Johnny™s height this year. Let his height next year be written as x(n+1).
Then we can write the dynamical system in the form of an equation as:



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x(n+1) = x(n) + 2.
See? Isn™t math simple? If we plug Johnny™s current height of x(n) = 38 inches in the right side of the equation,
we get Johnny™s height next year, x(n+1) = 40 inches:
x(n+1) = x(n) + 2 = 38 + 2 = 40.
Going from the right side of the equation to the left is called an iteration. We can iterate the equation again by
plugging Johnny™s new height of 40 inches into the right side of the equation (that is, let x(n)=40), and we get
x(n+1) = 42. If we iterate the equation 3 times, we get Johnny™s height in 3 years, namely 44 inches, starting from
a height of 38 inches).
This is a deterministic dynamical system. If we wanted to make it nondeterministic (stochastic), we could let the
model be: Johnny grows 2 inches a year, more or less, and write the equation as:
x(n+1) = x(n) + 2 + e
where e is a small error term (small relative to 2), and represents a drawing from some probability distribution.
Let's return to the original deterministic equation. The original equation, x(n+1) = x(n) + 2, is linear. Linear
means you either add variables or constants or multiply variables by constants. The equation
z(n+1) = z(n) + 5 y(n) “2 x(n)
is linear, for example. But if you multiply variables together, or raise them to a power other than one, the
equation (system) is nonlinear. For example, the equation
x(n+1) = x(n)2
is nonlinear because x(n) is squared. The equation
z = xy
is nonlinear because two variables, x and y, are multiplied together.
Okay. Enough of this. What is chaos? Here is a picture of chaos. The lines show how a dynamical system (in
particular, a Lorenz system) changes over time in three-dimensional space. Notice how the line (path, trajectory)
loops around and around, never intersecting itself.




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Notice also that the system keeps looping around two general areas, as though it were drawn to them. The points
from where a system feels compelled to go in a certain direction are called the basin of attraction. The place it
goes to is called the attractor.
Here™s an equation whose attractor is a single point, zero:
x(n+1) = .9 x(n) .
No matter what value you start with for x(n), the next value, x(n+1), is only 90 percent of that. If you keep
iterating the equation, the value of x(n+1) approaches zero. Since the attractor in this case is only a single point, it
is called a one-point attractor.
Some attractors are simple circles or odd-shaped closed loops”like a piece of string with the ends connected.
These are called limit cycles.
Other attractors, like the Lorenz attractor above, are really weird. Strange. They are called strange attractors.
Okay. Now let™s define chaos.
What is Chaos?
What are the characteristics of chaos? First, chaotic systems are nonlinear and follow trajectories (paths,

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highways) that end up on non-intersecting loops called strange attractors. Let's begin by understanding what
these two terms mean.
I am going to repeat some things I said in the previous section. D©jà vu. But, as in the movie The Matrix, d©jà vu
can communicate useful information. All over again.
Classical systems of equations from physics were linear. Linear simply means that outputs are proportional to
inputs. Proportional means you either multiply the inputs by constants to get the output, or add a constant to the
inputs to get the output, or both. For example, here is a simple linear equation from the capital-asset pricing
model used in corporate finance:
E(R) = ± + β E(Rm).
It says the expected return on a stock, E(R), is proportional to the return on the market, E(Rm). The input is
E(Rm). You multiply it by β ("beta"), then add ± ("alpha") to the result”to get the output E(R). This defines a
linear equation.
Equations which cannot be obtained by multiplying isolated variables (not raised to any power except the first) by
constants, and adding them together, are nonlinear. The equation y = x2 is nonlinear because it uses a power of
two: namely, x squared. The equation z = 4xy-10 is nonlinear because a variable x is multipled by a variable y.
The equation z = 5+ 3x-4y-10z is linear, because each variable is multiplied only by a constant, and the terms are
added together. If we multiply this last equation by 7, it is still linear: 7z = 35 + 21x “ 28y “ 70z. If we multiply it
by the variable y, however, it becomes nonlinear: zy = 5y + 3xy-4y2-10zy.
The science of chaos looks for characteristic patterns that appear in complex systems. Unless these patterns were
exceedingly simple, like a single equilibrium point ("the equilibrium price of gold is $300"), or a simple closed or
oscillatory curve (a circle or a sine wave, for example), the patterns are referred to as strange attractors.
Such patterns are traced out by self-organizing systems. Names other than strange attractor may be used in
different areas of science. In biology (or sociobiology) one refers to collective patterns of animal (or social)
behavior. In Jungian psychology, such patterns may be called archetypes [5].
The main feature of chaos is that simple deterministic systems can generate what appears to be random behavior.
Think of what this means. On the good side, if we observe what appears to be complicated, random behavior,
perhaps it is being generated by a few deterministic rules. And maybe we can discover what these are. Maybe life
isn't so complicated after all. On the bad side, suppose we have a simple deterministic system. We may think we
understand it§ it looks so simple. But it may turn out to have exceedingly complex properties. In any case, chaos
tells us that whether a given random-appearing behavior is at basis random or deterministic may be undecidable.
Most of us already know this. We may have used random number generators (really pseudo-random number
generators) on the computer. The "random" numbers in this case were produced by simple deterministic
equations.
I™m Sensitive”Don™t Perturb Me
Chaotic systems are very sensitive to initial conditions. Suppose we have the following simple system (called a
logistic equation) with a single variable, appearing as input, x(n), and output, x(n+1):
x(n+1) = 4 x(n) [1-x(n)].
The input is x(n). The output is x(n+1). The system is nonlinear, because if you multiply out the right hand side of
the equation, there is an x(n)2 term. So the output is not proportional to the input. Let's play with this system. Let
x(n) = .75. The output is



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4 (.75) [1- .75] = .75.
That is, x(n+1) = .75. If this were an equation describing the price behavior of a market, the market would be in
equilibrium, because today™s price (.75) would generate the same price tomorrow. If x(n) and x(n+1) were
expectations, they would be self-fulfilling. Given today's price of x(n) = .75, tomorrow's price will be x(n+1) =
.75. The value .75 is called a fixed point of the equation, because using it as an input returns it as an output. It
stays fixed, and doesn't get transformed into a new number.
But, suppose the market starts out at x(0) = .7499. The output is
4 (.7499) [1-.7499] = .7502 = x(1).
Now using the previous day's output x(1) = .7502 as the next input, we get as the new output:
4 (.7502) [1-.7502] = .7496 = x(2).
And so on. Going from one set of inputs to an output is called an iteration. Then, in the next iteration, the new
output value is used as the input value, to get another output value. The first 100 iterations of the logistic
equation, starting with x(0) = .7499, are shown in Table 1.
Finally, we repeat the entire process, using as our first input x(0) = .74999. These results are also shown in Table
1. Each set of solution paths”x(n), x(n+1), x(n+2), etc.”are called trajectories. Table 1 shows three different
trajectories for three different starting values of x(0).
Look at iteration number 20. If you started with x(0) = .75, you have x(20) = .75. But if you started with
x(0) = .7499, you get x(20) = .359844. Finally, if you started with x(0) = .74999, you get x(20) = .995773. Clearly
a small change in the intitial starting value causes a large change in the outcome after a few steps. The equation is
very sensitive to initial conditions.
A meteorologist name Lorenz discovered this phenomena in 1963 at MIT [6]. He was rounding off his weather
prediction equations at certain intervals from six to three decimals, because his printed output only had three
decimals. Suddenly he realized that the entire sequence of later numbers he was getting were different. Starting
from two nearby points, the trajectories diverged from each other rapidly. This implied that long-term weather
prediction was impossible. He was dealing with chaotic equations.

Table 1: First One Hundred Iterations of the Equation
x(n+1) = 4 x(n) [1- x(n)] with Different Values of x(0).

x(0): .75000 .74990 .74999

Iteration

1 .7500000 .750200 .750020

2 .7500000 .749600 .749960

3 .7500000 .750800 .750080

4 .7500000 .748398 .749840

5 .7500000 .753193 .750320

6 .7500000 .743573 .749360


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7 .7500000 .762688 .751279

8 .7500000 .723980 .747436

9 .7500000 .799332 .755102

10 .7500000 .641601 .739691

11 .7500000 .919796 .770193

12 .7500000 .295084 .707984

13 .7500000 .832038 .826971

14 .7500000 .559002 .572360

15 .7500000 .986075 .979056

16 .7500000 .054924 .082020

17 .7500000 .207628 .301170

18 .7500000 .658075 .841867

19 .7500000 .900049 .532507

20 .7500000 .359844 .995773

21 .7500000 .921426 .016836

22 .7500000 .289602 .066210

23 .7500000 .822930 .247305

24 .7500000 .582864 .744581

25 .7500000 .972534 .760720

26 .7500000 .106845 .728099

27 .7500000 .381716 .791883

28 .7500000 .944036 .659218

29 .7500000 .211328 .898598

30 .7500000 .666675 .364478

31 .7500000 .888878 .926535




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32 .7500000 .395096 .272271

33 .7500000 .955981 .792558

34 .7500000 .168326 .657640

35 .7500000 .559969 .900599

36 .7500000 .985615 .358082

37 .7500000 .056712 .919437

38 .7500000 .213985 .296289

39 .7500000 .672781 .834008

40 .7500000 .880587 .553754

41 .7500000 .420613 .988442

42 .7500000 .974791 .045698

43 .7500000 .098295 .174440

44 .7500000 .354534 .576042

45 .7500000 .915358 .976870

46 .7500000 .309910 .090379

47 .7500000 .855464 .328843

48 .7500000 .494582 .882822

49 .7500000 .999883 .413790

50 .7500000 .000470 .970272

51 .7500000 .001877 .115378

52 .7500000 .007495 .408264

53 .7500000 .029756 .966338

54 .7500000 .115484 .130115

55 .7500000 .408589 .452740

56 .7500000 .966576 .991066

57 .7500000 .129226 .035417


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58 .7500000 .450106 .136649

59 .7500000 .990042 .471905

60 .7500000 .039434 .996843

61 .7500000 .151515 .012589

62 .7500000 .514232 .049723

63 .7500000 .999190 .189001

64 .7500000 .003238 .613120

65 .7500000 .012911 .948816

66 .7500000 .050976 .194258

67 .7500000 .193508 .626087

68 .7500000 .624252 .936409

69 .7500000 .938246 .238190

70 .7500000 .231761 .725821

71 .7500000 .712191 .796019

72 .7500000 .819899 .649491

73 .7500000 .590658 .910609

74 .7500000 .967125 .325600

75 .7500000 .127178 .878338

76 .7500000 .444014 .427440

77 .7500000 .987462 .978940

78 .7500000 .049522 .082465

79 .7500000 .188278 .302657

80 .7500000 .611319 .844223

81 .7500000 .950432 .526042

82 .7500000 .188442 .997287




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83 .7500000 .611727 .010822

84 .7500000 .950068 .042818

85 .7500000 .189755 .163938

86 .7500000 .614991 .548250

87 .7500000 .947108 .990688

88 .7500000 .200378 .036901

89 .7500000 .640906 .142159

90 .7500000 .920582 .487798

91 .7500000 .292444 .999404

92 .7500000 .827682 .002381

93 .7500000 .570498 .009500

94 .7500000 .980120 .037638

95 .7500000 .077939 .144886

96 .7500000 .287457 .495576

97 .7500000 .819301 .999922

98 .7500000 .592186 .000313

99 .7500000 .966007 .001252

100 .7500000 .131350 .005003



The different solution trajectories of chaotic equations form patterns called strange attractors. If similar patterns
appear in the strange attractor at different scales (larger or smaller, governed by some multiplier or scale factor r,
as we saw previously), they are said to be fractal. They have a fractal dimension D, governed by the relationship
N = rD. Chaos equations like the one here (namely, the logistic equation) generate fractal patterns.
Why Chaos?
Why chaos? Does it have a physical or biological function? The answer is yes.
One role of chaos is the prevention of entrainment. In the old days, marching soldiers used to break step when
marching over bridges, because the natural vibratory rate of the bridge might become entrained with the soldiers'
steps, and the bridge would become increasingly unstable and collapse. (That is, the bridge would be destroyed
due to bad vibes.) Chaos, by contrast, allows individual components to function somewhat independently.



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A chaotic world economic system is desirable in itself. It prevents the development of an international business
cycle, whereby many national economies enter downturns simultaneously. Otherwise national business cycles
may become harmonized so that many economies go into recession at the same time. Macroeconomic policy
co-ordination through G7 (G8, whatever) meetings, for example, risks the creation of economic entrainment,
thereby making the world economy less robust to the absorption of shocks.
"A chaotic system with a strange attractor can actually dissipate disturbance much more rapidly. Such systems are
highly initial-condition sensitive, so it might seem that they cannot dissipate disturbance at all. But if the system
possesses a strange attractor which makes all the trajectories acceptable from the functional point of view, the
initial-condition sensitivity provides the most effective mechanism for dissipating disturbance" [7].
In other words, because the system is so sensitive to initial conditions, the initial conditions quickly become
unimportant, provided it is the strange attractor itself that delivers the benefits. Ary Goldberger of the Harvard
Medical School has argued that a healthy heart is chaotic [8]. This comes from comparing electrocardiograms of
normal individuals with heart-attack patients. The ECG™s of healthy patients have complex irregularities, while
those about to have a heart attack show much simpler rhythms.
How Fast Do Forecasts Go Wrong?”The Lyapunov Exponent
The Lyapunov exponent » is a measure of the exponential rate of divergence of neighboring trajectories.
We saw that a small change in the initial conditions of the logistic equation (Table 1) resulted in widely divergent
trajectories after a few iterations. How fast these trajectories diverge is a measure of our ability to forecast.
For a few iterations, the three trajectories of Table 1 look pretty much the same. This suggests that short-term
prediction may be possible. A prediction of "x(n+1) = .75", based solely on the first trajectory, starting at x(0) =
.75, will serve reasonably well for the other two trajectories also, at least for the first few iterations. But, by
iteration 20, the values of x(n+1) are quite different among the three trajectories. This suggests that long-term
prediction is impossible.
So let's think about the short term. How short is it? How fast do trajectories diverge due to small observational
errors, small shocks, or other small differences? That™s what the Lyapunov exponent tells us.
Let µ denote the error in our initial observation, or the difference in two initial conditions. In Table 1, it could
represent the difference between .75 and .7499, or between .75 and .74999.
Let R be a distance (plus or minus) around a reference trajectory, and suppose we ask the question: how quickly
does a second trajectory§ which includes the error µ § get outside the range R? The answer is a function of the
number of steps n, and the Lyapunov exponent » , according to the following equation (where "exp" means the
exponential e = 2.7182818¦, the basis of the natural logarithms):
R = µ • exp(» n).
For example, it can be shown that the Lyapunov exponent of the logistic equation is » = log 2 = .693147 [9]. So
in this instance, we have R = µ • exp(.693147 n ).
So, let™s do a sample calculation, and compare with the results we got in Table 1.
Sample Calculation Using a Lyapunov Exponent
In Table 1 we used starting values of .75, .7499, and .74999. Suppose we ask the question, how long (at what
value of n) does it take us to get out of the range of +.01 or -.01 from our first (constant) trajectory of x(n) = .75?
That is, with a slightly different starting value, how many steps does it take before the system departs from the
interval (.74, .76)?


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In this case the distance R = .01. For the second trajectory, with a starting value of .7499, the change in the initial
condition is µ = .0001 (that is, µ = 75-.7499). Hence, applying the equation R = µ • exp(» n), we have
.01 = .0001 exp (.693147 n).
Solving for n, we get n = 6.64. Looking at Table 1, we see that that for n = 7 (the 7th iteration), the value is x(7) =
.762688, and that this is the first value that has gone outside the interval (.74, .76).
Similarly, for the third trajectory, with a starting value of .74999, the change in the initial condition is µ = .00001
(i.e., . µ = 75-.74999). Applying the equation R = µ • exp(» n) yields
.01 = .00001 exp (.693147 n).
Which solves to n = 9.96. Looking at Table 1, we see that for n = 10 (the 10th iteration), we have x(10) =
.739691, and this is the first value outside the interval (.74, .76) for this trajectory.
In this sample calculation, the system diverges because the Lyapunov exponent is positive. If it were the case the
Lyapunov exponent were negative, » < 0, then exp(» n) would get smaller with each step. So it must be the case
that » > 0 for the system to be chaotic.
Note also that the particular logistic equation, x(n+1) = 4 x(n) [1-x(n)], which we used in Table 1, is a simple
equation with only one variable, namely x(n). So it has only one Lyapunov exponent. In general, a system with M
variables may have as many as M Lyapunov exponents. In that case, an attractor is chaotic if at least one of its
Lyapunov exponents is positive.
The Lyapunov exponent for an equation f (x(n)) is the average absolute value of the natural logarithm (log) of its
derivative:



Σ
»= (1/n) log |df /dx(n)|
n ’∞

For example, the derivative of the right-hand side of the logistic equation
x(n+1) = 4 x(n)[1-x(n)] = 4 x(n) “ 4 x(n)2
is
4 - 8 x(n) .
Thus for the first iteration of the second trajectory in Table 1, where x(n) = .7502, we have | df /dx(n)| =
| 4[1-2 (.7502)] | = 2.0016, and log (2.0016) = .6939. If we sum over this and subsequent values, and take the
average, we have the Lyapunov exponent. In this case the first term is already close to the true value. But it
doesn't matter. We can start with x(0) = .1, and obtain the Lyapunov exponent. This is done in Table 2, below,
where after only ten iterations the empirically calculated Lyapunov exponent is .697226, near its true value of
.693147.

Table 2: Empirical Calculation of Lyapunov Exponent from
the Logistic Equation with x(0) = .1
x(n) log|df/dx(n)|


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Iteration:

1 .360000 .113329

2 .921600 1.215743

3 .289014 .523479

4 .821939 .946049

5 .585421 -.380727

6 .970813 1.326148

7 .113339 1.129234

8 .401974 -.243079

9 .961563 1.306306

10 .147837 1.035782



Average .697226


Enough for Now
In the next part of this series, we will discuss fractals some more, which will lead directly into economics and
finance. In the meantime, here are some exercises for eager students.
Exercise 1: Iterate the following system: x(n+1) = 2 x(n) mod 1. [By "mod 1" is meant that only the fractional
part of the result is kept. For example, 3.1416 mod 1 = .1416.] Is this system chaotic?
Exercise 2: Calculate the Lyapunov exponent for the system in Exercise 1. Suppose you change the initial
starting point x(0) by .0001. Calculate, using the Lyapunov exponent, how many steps it takes for the new
trajectory to diverge from the previous trajectory by an amount greater than .002.
Finally, here is a nice fractal graphic for you to enjoy:




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Notes
[1] Eugene F. Fama, "Mandelbrot and the Stable Paretian Hypothesis," Journal of Business, 36, 420-429, 1963.
[2] If you really want to know why, read J. Aitchison and J.A.C. Brown, The Lognormal Distribution, Cambridge
University Press, Cambridge, 1957.
[3] J. Orlin Grabbe, Three Essays in International Finance, Department of Economics, Harvard University, 1981.
[4] The Sierpinski Carpet graphic and the following one, the Lorentz attractor graphic, were taken from the web
site of Clint Sprott: http://sprott.physics.wisc.edu/ .

[5] Ernest Lawrence Rossi, "Archetypes as Strange Attractors," Psychological Perspectives, 20(1), The C.G. Jung
Institute of Los Angeles, Spring-Summer 1989.
[6] E. N. Lorenz, "Deterministic Non-periodic Flow," J. Atmos. Sci., 20, 130-141, 1963.
[7] M. Conrad, "What is the Use of Chaos?", in Arun V. Holden, ed., Chaos, Princeton University Press,
Princeton, NJ, 1986.
[8] Ary L. Goldberger, "Fractal Variability Versus Pathologic Periodicity: Complexity Loss and Stereotypy In
Disease," Perspectives in Biology and Medicine, 40, 543-561, Summer 1997.
[9] Hans A. Lauwerier, "One-dimensional Iterative Maps," in Arun V. Holden, ed., Chaos, Princeton University
Press, Princeton, NJ, 1986.

J. Orlin Grabbe is the author of International Financial Markets, and is an internationally recognized derivatives
expert. He has recently branched out into cryptology, banking security, and digital cash. His home page is located
at http://www.aci.net/kalliste/homepage.html .
-30-



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Chaos and Fractals in Financial Markets, Part 1, by J. Orlin Grabbe

from The Laissez Faire City Times, Vol 3, No 22, May 31, 1999




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In Praise of Chaos

[Email Reply]


In Praise of Chaos
by J. Orlin Grabbe
Speech Presented to Eris Society, August 12, 1993


Introduction: The Intrusion of Eris
Chaos has a bad name in some parts. It was chaos that brought us the Trojan War (Robert Graves, The
Greek Myths, chapter 159). Eris, goddess of chaos, upset at not being invited to the wedding of Peleus
and Thetis, showed up anyway and rolled a golden apple marked "kalliste" ("for the prettiest one")
among the guests. Each of the goddesses Hera, Athena, and Aphrodite claimed the golden apple as her
own. Zeus, no fool, appointed Paris, son of Priam, king of Troy, judge of the beauty contest. Hermes
brought the goddesses to the mountain Ida, where Paris first tried to divide the apple among the
goddesses, then made them swear they wouldn't hold the decision against them. Hermes asked Paris if he
needed the goddesses to undress to make his judgment, and he replied, Of course. Athena insisted
Aphrodite remove her magic girdle, the sexy underwear that made everyone fall in love with her, and
Aphrodite retorted Athena would have to remove her battle helmet, since she would look hideous without
it.
As Paris examined the goddesses individually, Hera promised to make Paris the lord of Asia and the
richest man alive, if she got the apple. Paris said he couldn't be bribed. Athena promised to make Paris
victorious in all his battles, and the wisest man alive. Paris said there was peace in these parts. Aphrodite
stood so close to Paris he blushed, and not only urged him not to miss a detail of her lovely body, but
said also that he was the handsomest man she had seen lately, and he deserved a woman as beautiful as
she was. Had he heard about Helen, the wife of the king of Sparta? The goddess promised Paris she
would make Helen fall in love with him. Naturally Paris gave the apple to Aphrodite, and Hera and
Athena went off fuming to plot the destruction of Troy. That is, Aphrodite got the apple, and Paris got
screwed.
While the Greeks had a specific goddess dedicated to Chaos, early religions gave chaos an even more
fundamental role. In the Babylonian New Year festival, Marduk separated Tiamat, the dragon of chaos,
from the forces of law and order. This primal division is seen in all early religions. Yearly homage was
paid to the threat of chaos's return. Traditional New Year festivals returned symbolically to primordial
chaos through a deliberate disruption of civilized life. One shut down the temples, extinguished fires, had
orgies and otherwise broke social norms. The dead mingled with the living; Afterward you purified
yourself, reenacted the creation myth whereby the dragon of chaos was overthrown, and went back to
normal. Everyone had fun, but afterward order was restored, and the implication was it was a good thing
we had civilization, because otherwise people would always be putting out the fires and having orgies.
Around us in the world today we see the age-old battle between order and chaos. In the international
sphere, the old order of communism has collapsed. In its place is a chaotic matrix of competing,

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breakaway states, wanting not only political freedom and at least a semi-market economy, but also their
own money supplies and nuclear weapons, and in some cases a society with a single race, religion, or
culture. Is this alarming or reassuring? We also have proclamations of a New World Order, on one hand,
accompanied by the outbreak of sporadic wars and US bombing raids in Africa, Europe, and Asia, on the
other.
In the domestic sphere we have grass roots political movements, such as the populist followers of H.
Ross Perot challenging the old order imposed by the single-party Democratic- Republican monolith. We
have a President who is making a mockery out of the office, and a Vice President who tells us we should
not listen to any dissenting opinions with respect to global warming. Is this reassuring or alarming?
In the corporate-statist world of Japan we see the current demolition of the mythic pillars of Japanese
society: the myth of high-growth, the myth of endless trust between the US and Japan, the myth of full
employment, the myth that land and stock prices will always rise, and the myth that the Liberal
Democratic Party will always remain in power. Is the shattering of these myths reassuring or alarming?
In fact, wherever we look, central command is losing control. Even in the sphere of the human mind we
have increasing attention paid to cases of multiple personality. The most recent theories see human
identity and the human ego as a network of cooperative subsystems, rather than a single entity.
(Examples of viewpoint are found in Robert Ornstein, Multimind, and Michael Gazzanaga, The Social
Brain.) If, as Carl Jung claimed, "our true religion is a monotheism of consciousness, a possession by it,
coupled with a fanatical denial of the existence of fragmentary autonomous systems," then it can be said
that psychological polytheism is on the rise. Or, as some would say, mental chaos. Is this reassuring or
alarming?

Myth of Causality Denies Role of Eris
The average person, educated or not, is not comfortable with chaos. Faced with chaos, people begin
talking about the fall of Rome, the end of time. Faced with chaos people begin to deny its existence, and
present the alternative explanation that what appears as chaos is a hidden agenda of historical or
prophetic forces that lie behind the apparent disorder. They begin talking about the "laws of history" or
proclaiming that "God has a hidden plan". The creation, Genesis, was preceded by chaos (tohu-va- bohu),
and the New World Order (the millennium), it is claimed, will be preceded by pre-ordained apocalyptic
chaos. In this view of things, chaos is just part of a master agenda. Well, is it really the case there is a
hidden plan, or does the goddess Eris have a non- hidden non-plan? Will there be a Thousand Year Reign
of the Messiah, or the Thousand Year Reich of Adolph Hitler, or are these one and the same?
People are so uncomfortable with chaos, in fact, that Newtonian science as interpreted by Laplace and
others saw the underlying reality of the world as deterministic. If you knew the initial conditions you
could predict the future far in advance. With a steady hand and the right cue tip, you could run the table
in pool. Then came quantum mechanics, with uncertainty and indeterminism, which even Einstein
refused to accept, saying "God doesn't play dice." Philosophically, Einstein couldn't believe in a universe
with a sense of whimsy. He was afraid of the threatened return of chaos, preferring to believe for every
effect there was a cause. A consequence of this was the notion that if you could control the cause, you
could control the effect.
The modern proponents of law and order don't stop with the assertion that for every effect, there is a

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cause. And they also assert they "know" the cause. We see this attitude reflected by social problem
solvers, who proclaimed: "The cause of famine in Ethiopia is lack of food in Ethiopia." So we had rock
crusades to feed the starving Ethiopians and ignored the role of the Ethiopian government. Other
asserted: "The case of drug abuse is the presence of drugs," so they enacted a war on certain drugs which
drove up their price, drove up the profit margins available to those who dealt in prohibited drugs, and
created a criminal subclass who benefited from the prohibition. Psychologists assert: "The reason this
person is this way is because such-and-such happened in childhood, with parents, or siblings, or
whatever." So any evidence of abuse, trauma, or childhood molestation--which over time should assume
a trivial role in one's life--are given infinite power by the financial needs of the psychotherapy business.
You may respond: "Well, but these were just misidentified causes; there really is a cause." Maybe so, and
maybe not. Whatever story you tell yourself, you can't escape the fact that to you personally "the future is
a blinding mirage" (Stephen Vizinczey, The Rules of Chaos). You can't see the future precisely because
you don't really know what's causing it. The myth of causality denies the role of Eris. Science eventually
had to acknowledge the demon of serendipity, but not everyone is happy with that fact. The political
world, in the cause-and-effect marketing and sales profession, has a vested interest in denying its
existence.

Approaches to Chaos
In philosophy or religion there are three principal schools of thought (in a classification I'll use here).
Each school is distinguished by its basic philosophical outlook on life. The First School sees the universe
as indifferent to humanity's joys or sufferings, and accepts chaos as a principle of restoring balance. The
Second School sees humanity as burdened down with suffering, guilt, desire, and sin, and equates chaos
with punishment or broken law. The Third School considers chaos an integral part of creativity, freedom,
and growth.

I. First School Approach: Attempts to Impose Order Lead to Greater
Disorder
Too much law and order brings its opposite. Attempts to create World Government will lead to total
anarchy. What are some possible examples?
q The Branch Davidians at Waco. David Koresh's principal problem was, according to one FBI
spokesman, that he was "thumbing his nose at the law". So, to preserve order, the forces of law and
order brought chaos and destruction, and destroyed everything and everyone. To prevent the
misuse of firearms by cult members, firearms were marshaled to randomly kill them. To prevent
alleged child abuse, the forces of law and order burned the children to death.
q Handing out free food in "refugee" camps in Somalia leads to greater number of starving refugees,
because the existence of free food attracts a greater number of nomads to the camps, who then
become dependent on free food, and starve when they are not fed.
q States in the US. are concerned about wealth distribution. But, to finance themselves, more and
more states have turned to the lottery. These states thereby create inequality of wealth distribution
by giving away to a few, vast sums of cash extracted from the many.
The precepts of the first school find expression in a number of Oriental philosophies. In the view of this

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school, what happens in the universe is a fact, and does not merit the labels of "good" or "bad", or human
reactions of sympathy or hatred. Effort to control or alter the course of macro events (as opposed to
events in ones personal life) is wasted. One should cultivate detachment and contemplation, and learn
elasticity, learn to go with the universal flow of events. This flow tends toward a balance. This view finds
expression in the Tao Teh King:
The more prohibitions you have,
the less virtuous people will be
The more weapons you have,
the less secure people will be.
The more subsidies you have,
the less self-reliant people will be.
Therefore the Master says:
I let go of the law,
and people become honest.
I let go of economics,
and people become prosperous.
I let go of religion,
and people become serene.
I let go of all desire for the common good,
and the good becomes common as grass.
(Chapter 57, Stephen Mitchell translation.)
You don't fight chaos any more than you fight evil. "Give evil nothing to oppose, and it will disappear by
itself" (Tao Teh King, Chapter 60). Or as Jack Kerouac said in Dr. Sax: "The universe disposes of its
own evil." Again the reason is a principal of balance: You are controlled by what you love and what you
hate. But hate is the stronger emotion. Those who fight evil necessarily take on the characteristics of the
enemy and become evil themselves. Organized sin and organized sin-fighting are two sides of the same
corporate coin.

II. Second School Approach: Chaos is a Result of Breaking Laws
In the broadest sense, this approach a) asserts society is defective, and then b) tells us the reason it's
bad is because we've done wrong by our lawless actions. This is the view often presented by the front
page of any major newspaper. It's a fundamental belief in Western civilization.
In early Judaism and fundamentalist Christianity, evil is everywhere and it must be resisted. There is no
joy or pleasure without its hidden bad side. God is usually angry and has to be propitiated by sacrifice
and blood. The days of Noah ended in a flood. Sodom and Gomorra got atomized. Now, today, it's the
End Time and the wickedness of the earth will be smitten with the sword of Jesus or some other Messiah
whose return is imminent.
In this context, chaos is punishment from heaven. Or chaos is a natural degenerate tendency which must
be alertly resisted. In the Old Testament Book of Judges, a work of propaganda for the monarchy, it is
stated more than once: "In those days there was no king in Israel: every man did that which was right in
his own eyes" (Judg. 17:6; 21:25). Doing what appeals to you was not considered a good idea, because,


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as Jeremiah reminds us "The heart [of man] is deceitful above all things and desperately wicked" (Jer.
17:9).
And in the New Testament, the rabbinical lawyer Paul says "by the law is the knowledge of sin" (Rom.
3:20), and elsewhere is written, "Whososever committeth sin transgresseth also the law: for sin is the
transgression of the law." (1 John 3:4). And, naturally, "the wages of sin is death" (Rom. 6:23).
New age views of karma are similar. If you are bad, as somehow defined, you built up bad karma (New
Age view), or else God later burns you with fire (fundamentalist Christian view). For good deeds, you
get good karma or treasures in heaven. It's basically an accountant's view of the world. Someone's
keeping a balance sheet of all your actions, and toting up debits or credits. Of course, some religions
allow you to wipe the slate clean in one fell swoop, say by baptism, or an act of contrition, which is sort
of like declaring bankruptcy and getting relief from all your creditors. But that's only allowed because
there has already been a blood sacrifice in your place. Jesus or Mithra or one of the other Saviors has
already paid the price. But even so, old Santa Claus is up there somewhere checking who's naughty or
nice.
What is fundamental about this approach is not the specific solution to sin, or approach to salvation, but
the general pessimistic outlook on the ordinary flow of life. The first Noble Truth of Buddha was that
"Life is Sorrow". In the view of Schopenhauer, Life is Evil, and he says "Every great pain, whether
physical or spiritual, declares what we deserve; for it could not come to us if we did not deserve it" (The
World as Will and Representation). Also in the Second School bin of philosophy can be added Freud,
with his Death Wish and the image of the unconscious as a murky swamp of monsters. Psychiatry in
some interpretations sees the fearful dragon of chaos, Tiamat, lurking down below the civilized veneer of
the human cortex.
The liberal's preoccupation with social "problems" and the Club of Rome's obsession with entropy are
essentially expressions of the Second School view. Change, the fundamental motion of the universe,
is bad. If a business goes broke, it's never viewed as a source of creativity, freeing up resources and
bringing about necessary changes. It's just more unemployment. The unemployment-inflation tradeoff as
seen by Sixties Keynesian macroeconomics is in the Second School spirit. These endemic evils must be
propitiated by the watchful Priests of Fiscal Policy and the Federal Reserve, and you can only reduce one
by increasing the other. This view refuses to acknowledge that one of the positive roles of the Market is
as a job destroyer as well as a job creator.
More generally, the second school has generated whole industries of "problem solvers"-- politicians,
bureaucrats, demagogues, counselors, and charity workers who have found the way to power, fame, and
wealth lies in championing causes and mucking about in other people's lives. Whatever their motivations,
they operate as parasites and vampires who are healthy only when others are sick, whose well-being
increases in direct proportion to other people's misery, and whose method of operation is to give the
appearance of working on the problems of others. Of course if the problems they champion were actually
solved, they would be out of a job. Hence they are really interested in the process of "solving"
problems--not in actual solutions. They create chaos and destruction under the pretense of chaos control
and elimination.




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III. Third School Approach: Chaos is Necessary for Creativity,
Freedom, and Growth
You find this view in a few of the ancient Greek writers, and more recently in Nietzsche. Nietzsche says:
"One must still have chaos in one to give birth to a dancing star." The first fundamental point of view
here is: Existence is pure joy. If you don't see that, your perception is wrong. And we are not talking
about Mary Baker Eddy Christian Science denial of the facts. In this approach you are supposed to learn
to alchemically transmute sorrow into joy, chaos into art. You exult in the random give and take of the
hard knocks of life. It's a daily feast. Every phenomenon is an Act of Love. Every experience, however
serendipitous, is necessary, is a sacrament, is a means of growth.
"Saying Yes to life even in its strangest and hardest problems, the will to life rejoicing over
its own inexhaustibility even in the very sacrifice of its highest types--that is what I called
Dionysian, that is what I guessed to be the bridge to the psychology of the tragic poet. Not
in order to be liberated from terror and pity, not in order to purge oneself of a dangerous
affect by its vehement discharge-- Aristotle understood it that way [as do the Freudians who
think one deals with ones neuroses through one's art, a point of view which Nietzsche is
here explicitly rejecting]--but in order to be oneself the eternal order of becoming, beyond
all terror and pity--that joy which included even joy in destroying." (Twilight of the Idols).
It is an approach centered in the here and now. You cannot foresee the future, so you must look at the
present. But because "nothing is certain, nothing is impossible" (Rules of Chaos). You are free and
nobody belongs to you. In the opening paragraphs of Tropic of Cancer, Henry Miller says: "It is now the
fall of my second year in Paris. I was sent here for a reason I have not yet been able to fathom. I have no
money, no resources, no hopes. I am the happiest man alive."
Your first responsibility is to take care of yourself, so you won't be a burden to other people. If you don't
do at least that, how can you be so arrogant as to think you can help others? You make progress by
adapting to your own nature. In Rabelais' Gargantua the Abbey of Theleme had the motto: Fay ce que
vouldras, or "Do as you will." Rabelais (unlike the Book of Judges) treats this in a very positive light.
The implication is: Don't go seeking after some ideal far removed from your own needs. Don't get
involved in some crusade to save the human race--because you falsely think that is the noble thing to
do--when what you may really want to do, if you are honest with yourself, is to stay home, grow
vegetables, and sell them in a roadside market. (Growing vegetables is, after all, real growth--more so
than some New Age conceptions.) You have no obligation under the sun other than to discover your real
needs, to fulfill them, and to rejoice in doing so.
In this approach you give other people the right to make their own choices, but you also hold them
responsible for the consequences. Most social "problems", after all, are a function of the choices
people make, and are therefore insolvable in principle, except by coercion. One is not under any
obligation to make up for the effects of other people's decisions. If, for example, people (poor or rich,
educated or not) have children they can't care for or feed, one has no responsibility to make up for their
negligence or to take on one's own shoulders responsibility for the consequent suffering. You can, if you
wish, if you want to become a martyr. If you are looking to become a martyr, the world will gladly
oblige, and then calmly carry on as before, the "problems" unaltered.
One may, of course, choose to help the rest of the world to the extent that one is able, assuming one
knows how. But it is a choice, not an obligation. Modern political correctness and prostituted religion

http://www.aci.net/kalliste/chaos.htm (6 of 7) [12/04/2001 1:29:06]
In Praise of Chaos

have tried to turn all of what used to be considered virtues into social obligations. Not that anyone is
expected to really practice what they preach; rather it is intended they feel guilty for not doing so, and
once the guilt trip is underway, their behavior can be manipulated for political purposes.
What would, after all, be left for social workers to do if all social problems were solved? One would still
need challenges, so presumably people would devote themselves to creative and artistic tasks. One would
still need chaos. One would still need Eris rolling golden apples.

Conclusion
In the revelation given to Greg Hill and Kerry Thornley, authors of Principia Discordia, or How I
Found Goddess and What I Did to Her When I Found Her, the goddess Eris (Roman Discordia) says: "I
am chaos. I am the substance from which your artists and scientists build rhythms. I am the spirit with
which your children and clowns laugh in happy anarchy. I am chaos. I am alive, and I tell you that you
are free."
Today, in Aspen, Eris says: I am chaos. I am alive, and I tell you that you are free.
Copyright 1993
J. Orlin Grabbe
1475 Terminal Way, Suite E
Reno, NV 89502
Homepage: http://www.aci.net/kalliste/homepage.html




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