Institute for Christian Learning
Education Department of Seventh-day Adventists
CHAOS: A NEW MATHEMATICAL PARADIGM
by
Vernon Howe
Department of Mathematics and Computing
La Sierra University
Riverside, California 92515
USA
Prepared for the
International Faith and Learning Seminar
held at
Newbold College
Bracknell, Berkshire, England
June 1994
197-94 Institute for Christian Teaching
12501 Old Columbia Pike
Silver Spring, MD 20904 USA
"You believe in a God
who plays dice, and I in complete law and order."
Albert Einstein in a letter to Max Born
Introduction
Mathematics is traditionally done in obscurity and
mathematicians go about their business with the realization that hardly any
non-specialist is willing to invest the time to grasp the significance of the
important ideas and theorems of mathematics.
Recently, however, this situation changed when the new mathematical
theory referred to as chaos[1]
burst on the popular scene in an unprecedented fashion.[2]Articles
on chaos routinely appeared in the popular press. James Gleick's book, Chaos, became a New York Times best
seller. Non-specialists have jumped on
the bandwagon with almost a religious fervor.
The most passionate advocates of the new science go so far as to say thattwentieth-century science will be remembered for just three things: relativity, quantum mechanics, and chaos. . . . Of the three, the revolution in chaos applies to the universe we see and touch, to objects at human scale. Everyday experience and real pictures of the world have become legitimate targets for inquiry. (Gleick 5-6)
Because of all the hype and exaggerated claims that
have accompanied this phenomenon, it is important to have some idea of what the
mathematical theory of chaos is. The
main of this essay is to briefly explain what chaos is and to outline how this theory
can inform and modify our mathematical worldview. I hope this information will help take some of the mystery out of
what chaos theory is and will aid readers in applying the theory to their own
disciplines.[3]
Two Mathematical Cautions
All disciplines use jargon and technical words; for
example, biologists use Latin names and chemists use symbolic names such as
NaCl. In contrast, mathematicians use
everyday words but give them very different meanings.[4]
Mathematicians have a completely different meaning for the word chaos
and the reader should guard against thinking in terms of the way it is commonly
used. For example, mathematically, the
word chaos is not synonymous with disorder, clutter, pandemonium, or confusion. The mathematical meaning of the world chaos
is outlined in a later section entitled "What is Chaos?" In this essay the term chaos will
always be used in its technical, mathematical sense.
It is important to keep in mind that the mathematical
world is a very abstract and precise concept. Many times it is not at all clear how a mathematical idea or
theorem informs our real world and the tendency is overreach in the
application of mathematics when there is no real connection. An example of this "metamorphic"
mode of thought is to say that since mathematicians believe in the fourth
dimension (which they do), therefore God must live in the fourth
dimension. The point is not whether God
lives in the fourth dimension but whether the abstract mathematical concept has
anything directly to say about where God lives. However, if a new paradigm is discovered in one area it can
direct one's mind to the discovery of a paradigm in another area that uses
similar modes of thought. The
mathematical theory of chaos seems to be an important new paradigm of this
type.
The Old Paradigm[5]
In the rise of classical
science and mathematics, several "axioms" that still affect our
thinking today became part of the prevailing paradigm. Five of these are: 1)
determinism-complete current knowledge yields complete predictability; 2)
linearity– the standard linear mathematical models suffice; 3) reductionism–
the belief that a complex system can be analyzed in terms of its constituent
parts,[6] 4) complexity–complex problems
must have complex solutions;[7] 5) randomness–seemingly random phenomena do not have natural patterns.
In many ways classical
science can be viewed in terms of a search for certainty. De Jong describes this view as follows:
The Cartesian-Newtonian paradigm contends that the physical world is made up of basic entities with distinct properties distinguishing one element from another. Isolating and reducing the physical world to is most basic entities, its separate parts, provides us with completely knowable, predictable, and therefore controllable physical universe. . . .The Cartesian-Newtonian paradigm contends that the physical universe is governed by immutable laws and therefore is determined and predictable, like an enormous machine. In principle, knowledge of the world could be complete in all its details. (De Jong 100-101).
Certainly during the twentieth century this search for
certainty has been under siege (e.g., by relativity and quantum mechanics in
physics and Godel's Theorem in mathematics).[8]
Chaos can be viewed as the next "nail in the coffin" in the search
for certainty.
It
is ironic that the Cartesian-Newtonian paradigm was motivated by "the
Judeo-Christian conviction that God is a rational being and thus created a
rationally knowable world to be one of the inspirations for the emergence of
modern science" (Beck 154). Alfred
North Whitehead said it this way:
The greatest contribution of medievalism to the formation of the scientific movement...[was] the inexpugnable belief that every detailed occurrence can be correlated with its antecedents in a perfectly definite manner. . . . How has this conviction been so vividly implanted in the European mind?. . . It must come from the medieval insistence on the rationality of God. (Whitehead 12).
What is chaos?
Chaos (in a mathematical sense) is very difficult to
define[9]
and is hard to handle theoretically; however, it is often easy to
"recognize it when you see it."
For our purposes all we need is the idea of a dynamical system and the
concept of sensitive dependence on initial conditions.
To a scientist a "system" is a collection of
objects that are interrelated.[10]Examples
of systems are: the population of rabbits and foxes in Yosemite Valley; the
solar system; the Landers earthquake. A
mathematical system is one in which the salient features can be qualified
mathematically in terms of variables. A
particular set of values for the system variables is called a "state"
of the system (e.g., the number of rabbits and foxes on July 1 would be a state
of the system). A dynamical system is
one that changes in time and it is usually defined by an "initial"
state and rules for changing a given state into some future (or past) state. The rules for changing states are usually
given by a set of equations.
Historically, dynamical systems were defined in terms of differential
equations but difference equations work just as well and are much easier to
deal with.
An example of a dynamical system, due to Robert May,
for the population of some species of imaginary fish is given by the single
rule–the Logistic Equation: xt+1 = r(xt - xt2). One can think of x as the fraction of the
maximum allowable population of fish and xt as the population
fraction at time t. Here time
can only be discrete quantities[11]such
as 0, 1, 2; hence, this is a difference equation dynamical system.[12]
An initial value, xo, is chosen; it is substituted into the
rule and x1 is computed.
The process is repeated and values for x2,x3,x4,
. . . are computed.[13]
The rule or function is repeated or iterated to compute the population at any
future time.
The symbol r is called a parameter and can be viewed
as some type of growth or fitness factor for the fish. For a particular population of fish the
value of the parameter, r, remains constant but different populations of fish
could have different r values.
In contrast the variable xt changes over time for a
particular population of fish. One goal
of chaos theory is to study how the value of r affects the behavior or
dynamics of the population over time.
The logistic Equation is an example of a non-linear equation due to the xt2
term. Chaos can only be found in
nonlinear systems and this is essentially the simplest such system.[14]
It could be argued that a study of very simple
nonlinear difference equations . . . should be part of high schools or
elementary college mathematics courses.
They would enrich the intuition of students who are currently nurtured
on a diet of almost exclusively linear problems. (May and Oster 573).
Following May's and Oster's advice and studying the
Logistic Equation gives insight into the nature of chaos. Readers are encouraged to play with the
Logistic Equations[15]
and discover for themselves how complicated this simple dynamical system
is. I will outline some of the
features.
For example, if the parameter r is 2.1, then
the population soon settles down to a final, stable population of approximately
0.52 no matter what the starting population is (as long as it is strictly
between 0 and 1).[16]
If the parameter drips below 1 the population becomes extinct. At first, when
the parameter rises past 2.1, the final population also rises but soon settles
down to a fixed value. Everything seems to be progressing in a predictable
fashion until the parameter passes 3.
Suddenly the final population does not settle down to a single value,
rather it oscillates between two values every other year. This correlates to a "boom-bust"
2-cycle of the fish in alternating years.
As the parameter is increased the 2-year cycle continues but the high
and low values move further apart.
Increasing the parameter still more, all of a sudden[17]
the population moves out of a 2-cycle into a 4-cycle.[18]
These splits from a single value into a 2-cycle and then
from a 2-cycle into 4-cycle are called bifurcations. As the parameter is further increased these bifurcations come
faster and faster and one finds 4, 8, 16, . . .-cycles. At a certain point all periodicity seems to
disappear and the population appears to fluctuate in a very complicated and
seemingly random fashion that is called chaotic. However, as the parameter increases through the chaotic region,
the population will again become regular and settle down in some odd period,
such as a 5-cycle.[19]
Period-doubling bifurcations into 10, 20, 40, . . .-cycles now occur much like
before but they come at a much faster rate and then when the parameter is
raised just a little more the system returns to chaos. When the parameter becomes 4 the behavior of
the populations dynamics appears completely random.[20]
Sensitive Dependence on Initial Conditions
There are several features of chaos that are nicely
illustrated by the Logistic Equation.
For certain values of the parameter the population settles down to
periodic behavior. For other values the
population never approaches a periodic population. The values of the parameter that do not yield periodic
populations are called the chaotic region and we say that the system exhibits
chaos (in the mathematical sense) in this region.
The chaotic region of the Logistic Equation is
separated into subregions by non-chaotic regions called periodic windows. If one looks closely inside one of these
periodic windows there are period doubling bifurcations that lead back into a
smaller chaotic subregion. This smaller
chaotic subregion has still smaller windows of periodicity. When any portion of
this subregion is magnified, the small region turns out to behave similarly to
the whole region. This property of
subregions within regions happens at every possible magnification. The property of similar detail at every
level of scale is a hallmark of all chaotic systems. A chaotic system cannot be studied by breaking it down into
smaller pieces since the pieces will be as complicated in detail as the
complete system. For this reason chaos
is inherently anti-reductionist and thus a holistic theory.
For a parameter in the chaotic region of the Logistic
Equation, the population fluctuates in an unpredictable manner. However, if the population is plotted for
many generations patterns emerge.
Population points will cluster together in some places and might
completely miss other places.[21]
These patterns in the long-range dynamics of a chaotic system are another
hallmark of chaotic systems.
For a parameter in the periodic region of the Logistic
Equation, the population settles down to the same value independent of the
starting value. In the chaotic region
the situation is very different. Two
different starting values can yield completely different results, even if the
populations are run for an infinite length of time. This phenomenon happens with starting values that are arbitrarily
chosen together. For example we could
begin with two starting values that only differ in the millionth decimal place
and run them through the dynamical systems for many generations. Quickly the two runs will drift apart and
follow completely different population paths.
This is what is meant by sensitive dependence on initial
conditions and is the most important feature shared by all chaotic
dynamical systems.
An interesting historical note is that the first
dynamical system in which chaos was observed and clearly recognized was
probably Edward Lorenz's model of the weather.[22] Lorenz realized that his system was chaotic
and expressed its sensitivity to initial conditions by his famous
"butterfly effect."[23] A single butterfly flying by today would
alter the initial state of the weather a little bit. Over a period of time the weather could turn out to be
drastically different than it would have been if the butterfly had not flown
by.
To understand the concept of sensitive dependence on
initial conditions it is instructive to look at a system that does not exhibit
it. An example is: xt+1 =
xt + c. If two different
starting populations are used the two populations might drift apart, as far as
you like, but over a long period of time.
There are two key points when comparing this system with a chaotic
system such as the Logistic Equation.
When close starting values are used each system's population will
diverge; however, the rate of divergence is much different and faster for the
chaotic system. Also for a non-chaotic
system the dynamics tends to be qualitatively similar for nearby starting
values. This is not the case for a
chaotic system.
Chaos and a Linear Versus Nonlinear World View
An
over-simplified description of the Cartesian-Newtonian method is to first
reduce a physical system to a set of equations and then solve the
equations. The belief is that then one
has complete predictability. Ian
Stewart calls this a "clockwork world" view.
The revolution in scientific thought that culminated in Newton led to a vision of the universe as some gigantic mechanism, functioning "like clockwork", a phrase that we still use - however inappropriate it is in an age of digital watches - to represent the ultimate in reliability and mechanical perfection. In such a vision, a machine is above all predictable. Under identical conditions it will do identical things. An engineer, who knows the specifications of the machine and its state at any one moment, can in principle work out exactly what it will do for all time. (Stewart 9)
Certainly this has been a very productive approach for
science for the last three hundred years.[24]
In a clockwork universe it appears as if there is no
place for chaos and its sensitive dependence on social conditions. Why has this Newtonian "clockwork"
approach been so effective for so long and why has it taken so long for chaos
to be discovered? The answer is, of
course, complex but a large part of the answer is that until the mid-twentieth
century mathematicians and physicists concentrated almost exclusively on linear
dynamical systems and, as we have seen earlier, chaos can only occur in a
nonlinear system.
The solvable systems are the ones shown in textbooks. They behave. Confronted with a nonlinear system, scientists would have to substitute linear approximations or find some other uncertain backdoor approach. Textbooks showed students only the rare nonlinear systems that would give way to such techniques. They did not display sensitive dependence on initial conditions. Nonlinear systems with real chaos were rarely taught and rarely learned. When people stumbled across such things–and people did–all their training argued for dismissing them as aberrations. Only a few were able to remember that the solvable, orderly, linear systems were the aberrations. Only a few, that is, understood how nonlinear nature is in its soul. Enrico Fermi once exclaimed, "It does not say in the Bible that all laws of nature are expressible linearly!" The mathematician Stanislaw Ulam remarked that to call the study of chaos "nonlinear science" was like calling zoology "the study of non-elephant animals." (Gleick 68).
The questions, "What kind of equations can we solve?" was answered pragmatically: "Only linear equations." This answer became a pillar of the Cartesian-Newtonian paradigm and permeated the world view of the scientific community.
Classical mathematics concentrated on linear equations
for a sound pragmatic reason: it couldn't solve anything else. In comparison tot he unruly hooligan antics
of a typical differential equation, linear ones are a bunch of choirboys. . . .
So docile are linear equations that he classical mathematicians were willing to
compromise physics to get them. . . .
So ingrained became the linear habit that by the 1940s and 1950s many scientists and engineers knew little else. 'God would not be so unkind,' said a prominent engineer, 'as the make the equations of nature nonlinear.' Once more the Deity was carrying the can for humanity's obtuseness. The engineer meant he didn't know how to solve non-linear equations, but wasn't honest enough to admit it.
Linearity is a trap.
The behavior of linear equations - like that of choirboys - is far from
typical. But if you decide that only
linear equations are worth thinking about, self-censorship sets in. Your textbooks fill with triumphs of linear
analysis, its failures buried so deep that the graves go unmarked and the
existence of the graves goes unremarked. (Stewart 83).
What had started out by Newton as a belief
in a "clockwork world" had been replaced by the middle of the
twentieth century by a belief in a "linear world." "Linear worlds" do not allow for
chaos; thus, sensitive dependence on initial conditions was not even an issue.
The
"linear world" view came under attack during the last half of the
twentieth century. It became apparent that what Lorenz had noticed in his
weather model was not an odd exception but rather it was a common
occurrence. His butterfly effect
acquired a technical name: sensitive dependence on initial condition. Stewart points out that "sensitive
dependence on initial conditions was not an altogether new notion" by
quoting the old rhyme from the fast (Stewart 23):
For the want of a nail, the shoe was lost;
For the want of a shoe, the horse was lost;
For the want of a horse, the rider was lost;
For the want of a rider, the battle was lost;
For the want of a battle, the kingdom was lost!
Chaos and Determinism
The Cartesian-Newtonian plan has always been mathematically
ambitions–even for linear systems the equations are usually very hard or beyond
our abilities to solve. The dream of
actually achieving complete predictability for any complicated system has
always been more of an ideal rather than a practical reality. We now understand from the theory of chaos
and its sensitive dependence on initial conditions that total predictability
has always been a "false god."
Nonlinear systems have an inherent limitation since, in the chaotic
region, absolute accuracy is necessary for useful predictions. "This does not mean that all chaotic
systems require impossible accuracy for all useful prediction task; some
systems may manifest a mixture of predictable and unpredictable behaviors. But when we speak of chaotic phenomena as
being impossible to predict, I would maintain that this is a good way to spell
out that limitation" (Kellert 35).
At this point it is fair to ask why we are so
preoccupied by determinism. The answer
lies in what we accept as ultimate explanations. For the last three hundred years the "holy grail" of
science has been prediction. A litmus
test of good scientific research is repeatability. In the Cartesian-Newtonian paradigm to say that we can explain an
even usually means that we can predict the event. "If the only alternative to determinism are final causes
(e.g., divine intervention) and hazard (e.g., accident or chance), then
determinism is attractive as an a priori truth or a methodological
imperative of scientific inquiry" (Earman 23). Chaos has given another option to this dichotomy.
Stephen Kellert's In the Wake of Chaos is an
outstanding book that discusses in detail what we mean by
"predictable," the layers of meaning of the word
"determinism" and the impact of the theory of chaos. I will just touch on one aspect of
Kellert's argument and hope that the reader will be stimulated to read
Kellert's book, on which the rest of this section heavily relies.
Kellert points out that the term
"determinism" can be viewed in at least four ways and lists them in
the following "rough order, from the simpler and less restrictive to the
more robust and full-blown" (Kellert 50).
1. Differential
or Difference Equations from a sufficient model of the system.
2. Unique evolution, i.e., the evolution of the system is uniquely fixed once we pecified the state of the system at any one moment.
3. Value
determinateness. Do all properties of the system have well-defined real values?
4. Total
predictability ("whatever that means" (Davis 6)).
Each of the four shadings of the meanings of
"determinism" listed above includes the ones listed before it.
The
fourth meaning of total predictability is the goal of the Cartesian-Newtonian
paradigm. Karl Popper calls this
meaning "scientific determinism," namely,
the doctrine that the state of any closed physical system can be predicted, even from within the system, with any specified degree of precision, by deducing the prediction from theories, in conjunction with initial conditions whose required degree of precision can always be calculated (in accordance with the principle of accountability) if the prediction task is given. (Popper 36).
Nonlinear models and chaos have made determinism
unreachable if we employ the criteria of total predictability.
"Value determinateness, the their level of
determinism, means that physical quantities have exact values" (Kellert
60). Some systems could have parameter
values that are spread out or somehow indistinct. Some authors have argued that a system without the precision of
value determinateness could not qualify for being a deterministic system.[25]
Unique evolution[26]
says that the complete description of a system at a certain time completely
fixes the future (and past) of the system.
This means that each set of initial conditions dictates a unique
trajectory or history of the system.
Chaos brings a real tension to this meaning of determinism. While a system can exhibit unique evolution,[27]
sensitive dependence on initial conditions "means that we will never be
able to tell which unique trajectory a system is following, but that
does not mean such trajectories do not exist" (Kellert 63).
Chaos has forced us to reconsider three of the four
meanings Kellert has given to determinism.
At this point all we have left is the first meaning: Differential or
Difference equation models. The obvious
question is what do we have left and what type of information can we get from
such models?
The activity of building and using these models
[differential or difference dynamic systems] has three important
characteristics . . . the behavior of the system is not studied by reducing it
to its parts; the results are not presented in the form of deductive proofs;
and the systems are not treated as if instantaneous descriptions are complete.
(Keller 85)
These three characteristics point to important ways in which chaos is
changing the operating paradigm of mathematics; consequently a new paradigm for
how we view our world becomes possible.
Chaos Versus Reductionism, Complexity, and Randomness
Chaos is an intrinsically a holistic,
non-reductionistic paradigm. A
nonlinear system is not the sum of its parts. In fact, as we have seen in the
chaotic region a nonlinear system has similar detail at every level of
magnification. The Cartesian-Newtonian
paradigm focused on quantitative solutions.
Chaos focuses on qualitative aspects such as when does a system have
periodic behavior and, when the behavior is not periodic, what is the general
shape of its attractor. Determining the
general shape of the attractor in phase space becomes a central problem. The geometry rather than the algebra of the
dynamical system becomes the focus.
The Cartesian-Newtonian
paradigm assumes that complex outcomes must have complex explanations. This is not true of many chaotic
systems. The Logistic Equation is an
example of a very complex outcome with a very simple explanation–a simple quadratic
equation.
A delightful example of a
system that clearly demonstrates the nonreductionistic nature of chaos
illustrates that a complex outcome can have a very simple explanation, and
which shows a pattern in a seemingly random process is the Chaos Game. The game
proceeds by choosing, at random, one of the three vertices of the triangle and
moving half the distance from the current point to that vertex. Another vertex is randomly chosen, the point
is moved half the distance to that vertex ad so on. If the successive points
are marked in the triangle the pattern illustrated below emerges (5,000-10,000
points should suffice to get a good picture).
The pattern (called the Sierpinski Gasket) is independent of the
starting point or of the random sequence of vertices chosen. The Chaos Game is a picture of chaotic
system. This system clearly shows a
pattern in a seeming random process. If
an infinite number of points were plotted the pattern would look exactly the
same at every level of magnification and thus a reductionist approach to
studying the figure would be useless.[28]
Conclusion
It is certainly fair to ask questions such as: Is
chaos theory a fad? Will it go the direction of mathematical catastrophe theory
and become more of a belief/religion than a valid theory? Has chaos been overly hyped? Will chaos
revolutionize science? For questions of
this type only time will tell; however, I am firmly convinced that
mathematically the theory of chaos is valid, important, here to stay, and will
significantly change the way we do mathematics. Chaos theory helps us to focus on important features that have
been neglected. "Far better to
consider chaos theory as a search for order, a concept broader than law.
. . models, not laws, form the heart of a science" (Kellert 117). Chaos does not eliminate prediction; rather
it addresses the issues of what kind of prediction[29]
is possible and aids in predicting the limits of our predictions.
One view is that the Cartesian-Newtonian paradigm was
developed in order to answer questions about what appeared to be a disordered
world (chaotic in a non-mathematical chaos has brought us full circle. "There is a theory that history moves
in cycles. But, like a spiral
staircase, when the course of human events comes full circle it does so on a
new level" (Stewart 1). I believe
that chaos theory has moved us up to a new level.
Kellert proposes that chaos theory gives us a "dynamic understanding."
First, it calls to mind the connection with dynamical
systems theory, the qualitative study of the behavior of simple mathematical
systems. Second, it connotes change and
process, tying together the various uses of the word "how," . . .
Chaos theory lets us understand how patterns and unpredictability arise by
showing us how certain geometric mechanisms bring them forth (114).
Some may contend that this search for patterns
actually strives to discover new laws governing qualitative features of
systems. (112)
It is important to understand that the patterns referred to are at the heart of the qualitative not quantitative aspects of chaos; furthermore these patterns have little to do with the type of patterns that statistical techniques address.
Chaos theory is intrinsically holistic and
antireductionist. In my judgment this
is one of the most important aspects of chaos from a Christian point of
view. It changes the way we view and
analyze our world.
. . . you know the right equations but they're just not helpful. You add up all the microscopic pieces and you find that you cannot extend them to the long term. They're not what's important in the problem. It completely changes what it means to know something. (Gleick 175)
When Albert Einstein wrote to Max Born, "You
believe in a God who plays dice, and I in complete law and order," it was
in the context of quantum mechanics not chaos (after all, the theory of chaos
did not exist at that time). At the
time Einstein wrote this, the Cartesian-Newtonian paradigm was under
siege. The goal of complete predictability
was being replaced by the theory that nature has a fundamentally unpredictable
or even random aspect. Einstein was
hoping for a more satisfactory explanation than the hypothesis of a God who
acts in a random manner. I agree with
Stewart that "The question is not so much whether God plays dice,
but how God plays dice" (2).
Currently my answer is that God does not play dice in a traditional
random sense but, as Stewart says, "He's playing a much deeper game that
we have yet to fathom" (293). It
is my belief that chaos theory is one component of this deeper game of
"dice" that God is playing.
As a mathematician the new paradigm of chaos theory
has been important to my gaining a deeper understanding of mathematical
phenomena. I now study mathematics from
a more holistic, nonreductionist point of view. I view complexity and randomness in a completely different light.
Above all I have come to realize that I really do not
understand what mathematical determinism is and that it is a far deeper concept
than I had realized. I do not know how
these ideas will ultimately transfer to other areas of my non-mathematical life
but I do know that the mathematical paradigm of chaos has opened my eyes to
exciting possibilities about the nature of God.
References
Beck, W. David,
ed. Opening the American Mind: The
Integration of Biblical Truth in the Curriculum of the University. Grand
Rapids, MI: Baker Book House, 1991.
Davis, Philip J.
"Chaos and the New Understanding."
SIAM NEWS 27 (February 1994): 6.
De Jong, Arthur
J. Reclaiming a Mission: New
Direction for the Church-Related College.
Grand Rapids, MI: William B. Eerdmans, 1990.
Earman, J. A
Primer on Determinism. Dordrecht:
D. Reidel, 1986.
Gleick,
James. Chaotic: Making a New
Science. New York: Vicking, 1987.
Hofstadter,
Douglas R. Godel, Escher, Bach: An
Eternal Golden Braid. New York:
Basic Books, 1979.
Kellert, Stephen
H. In the Wake of Chaos. Chicago, IL: University of Chicago Press,
1993.
May, R. M. and
Oster, G. G. "Bifurcations and Dynamic Complexity in Simple Ecological Models." American Naturalist 110 (1976):
573-599.
Popper,
Karl. The Open Universe. Totowa, N.J.: Rowman & Littlefield, 1956.
Stewart,
Ian. Does God Play Dice? The
Mathematics of Chaos. Cambridge, MA: Basil Balckwell, 1989.
Whitehead, Alfred
North. Science and the Modern World. New York: Macmillan, 1967.
Yorke,
James. "Period Three Implies
Chaos." American Mathematical Monthly 82 (1975): 985-992.