Hi Everybody.
Our second unit is on dynamics and chaos.
Dynamics is how things change, as I
described earlier, a fundamental aspect
of understanding complex systems is
characterizing their dynamics. How their
complex behavior unfolds and how it
changes over time. First, I will give a
brief history of the science of dynamics.
Then we will look at the notion of
iteration, and how iteration of simple
behaviors can give rise to complex
patterns. We will spend some time
discussing the crucial notion of
non-linearity and how non-linear
interactions form the crux of complex
systems. We will then look in depth at
a simple model of population growth that
gives rise to unexpected behavior. I will
explain what is meant by "chaos",
otherwise known as sensitive dependence on
initial conditions.
Dynamics is the general study of how
systems change over time. I will show
you some examples of the kinds of topics
people study under the rubrick of
dynamics. Planetary dynamics studies the
movement of planets under the force of
gravity and characterizes their orbits,
deviations from their orbits, eclipses
and so on. Fluid dynamics studies flow of
fluids and includes things like study of
ocean flows, hurricanes, gas clouds in
space and turbulent air flow like the kind
we might experience when we are flying in
an airplane. Electrical dynamics studies
the flow of electricity in circuits,
Climate dynamics looks at how climate
changes over time in terms of temperature
pressure and so on. Crowd dynamics look
at how crowds of people act. That can
either be in an ordered way or in a
disordered way, for example, when someone
calls, "Fire!" in a crowded room, people
might stampede. Population dynamics
looks at how populations vary over time.
We will be talking about that quite a bit
in this unit. Financial dynamics looks
at phenomena related to stock prices or
other financial activity.
Group dynamics looks at how groups of
animals or humans form and how they work
together to accomplish tasks. There is
also work on social dynamics. That
includes the dynamics of conflicts and of
cooperation; for example, among nations.
Dynamics is a very general field.
It has been a huge triumph of mathematics
and science to develop quantitative
tools, such as differential equations
that can be applied to explain so many
different phenomena and dynamical systems
theory is the general area of mathematics
concerned with dynamical systems, in short
it's the branch of mathematics which
describes how systems change over time
and it includes many sub-branches
including calculus, differential equations
iterated maps and so on. We'll talk
about some of this during this unit.
The dynamics of a system refers to the
manner in which the system changes.
Dynamical systems theory gives us a
vocabulary and set of mathematical tools
for describing dynamics. Let me briefly
give some history of dynamical systems
theory and mention some of the historical
big names. In the West, the study of
dynamical systems really started with
Aristotle. Aristotle believed that there
are two sets of laws. One set for earth
where objects move in straight lines,
according to Aristotle, and only under
force. Things fall to the ground at a
rate depending on how heavy they are. He
believed that there are a separate set of
physical laws for the heavens.
For example, other planets and the sun
orbit in perfect circles around the earth.
Aristotle based his views on logic and
common sense and some naive observation.
He didn't really see the need to do
systematic experiments. Only after a
couple of thousands of years did people
start questioning his views.
Nicholas Copernicus for example proposed
a new set of laws for the heavens. In his
theory the sun is stationary and the
planets orbit around it. Galileo was a
pioneer of the experimental method,
as far as studying motion was concerned.
He proved experimentally that most of
Aristotle's laws of motion were false.
Isaac Newton was the founder of the modern
science of dynamics. Newton discovered
much of what we use today to understand
the physics of motion under the force of
gravity. He proposed the then radical
view that the laws of motion are the same
on earth and in the heavens. That is,
gravity is a universal force that acts
the same no matter where in the universe.
Newton, along with Leibnitz, also invented
the branch of mathematics that we call
calculus which has been the primary tool
used ever since to study how systems
change over time and space.
Pierre-Simon Laplace was a big proponent
of Newtonian reductionism and determinism.
I will read a very famous quotation from
him which sums up his view of a
deterministic universe in which everything
is knowable in principle. Laplace said,
We may regard the present state of the
universe as the effect of its past and the
cause of its future, an intellect which at
a certain moment would know all forces
that set nature in motion and all
positions of items of which nature is
composed. If this intellect were also
vast enough to submit these data to
analysis it would embrace in a single
formula the movements of the greatest
bodies of the universe and those of the
tiniest atoms. For such an intellect
nothing would be uncertain, and the future
just like the past would be present before
its eyes. While Laplace wrote this in
the early 1800's, we could imagine today
that the intellect might be a computer,
a supercomputer that would be able to
model all of the particles in the universe
for example and the forces that act on
them and would be able to predict anything
at all. This view of the possibility of
complete prediction was widely accepted
until the late 19th or early 20th century.
Although before that Henri Poincare,
French mathematician, started to speculate
on possible reasons why such perfect
prediction might not be possible. He was
a pioneer of modern dynamical systems
theory and the notion of chaos. Let me
give his most famous quotation,
Poincare said, If we knew exactly the laws
of nature and the situation of the
universe at the initial moment, we could
predict exactly the situation of that same
universe at a succeeding moment. Now that
goes along with what Laplace said. But,
then he goes on, But even if it were the
case that the natural laws no longer had
any secret for us, we could still only
know the initial situation approximately.
If that enabled us to predict the
succeeding situation with the same
approximation, that's all we require and
we should say that the phenomenon has
been predicted. That is it governed by
laws. But it is not always so. 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. This quotation introduces the
notion called sensitive dependence on
initial conditions. Consider Laplace's
idea. If at a precise time, you know
the exact position and velocity of every
atom in the universe, you could use
Newton's laws to predict exactly what the
positions and velocities of the atoms will
be at some precise time in the future.
But, suppose you don't quite know the
positions exactly. Suppose you know them
only to a finite number of decimal places.
What Poincare is saying is that there are
some systems, not all, but some, in which
if you got some decimal place wrong for
position or velocity, your calculation
will eventually end up being way off.
This could be the 10th decimal place, the
hundredth, the thousandth, or however
far out you want to go. The system, if
it is sensitive to initial conditions,
where the initial conditions, for
example, would be the positions and
velocities of every atom, at some
particular time, if it is sensitive in
that way, then if you don't know exact
values, for the initial conditions,
prediction becomes impossible. Poincare's
notion of sensitive dependence on initial
conditions, is illustrated by the famous
so called butterfly effect. In this
hypothetical example, a small butterfly
flaps its wings in Tokyo. This causes a
change in the position and velocity of a
few air molecules. If the whole weather
system is sensitive to initial conditions
and the weather forecasters don't take the
butterfly into account, after some time,
their predictions will be way off, and the
butterfly might in fact create a hurricane
This is not to say that this actually
happens, or the weather is sensitive
to initial conditions, Poincare is just
saying that there are some systems with
this property, and we don't know what
they all are. We will look at a simple
one a bit later in this unit. Now we can
define the notion of chaos. Chaos is used
in colloquial every day language to mean
roughly disorder, but in dynamical
systems theory, it means something
specific. It is one particular type of
the dynamics of a system. It is one way
in which systems change. It is defined
as sensitive dependence on initial
conditions. We will make this quite a
bit more precise later on. Now you might
be familiar with this fellow,
Dr. Ian Malcolm, who asked, "You've never
heard of chaos theory? Non-linear
equations? Strange attractors?"
If this is sounding familiar, you might
have actually seen him. He was a
character in a book and then a movie,
called Jurassic Park, back in the 90's.
You may or may not know, that the sequel
to Jurassic Park, also written by Michael
Creighton, and made into a movie was
called The Lost World.
Part of The Lost World took place at the
Santa Fe Institute. In the Prologue, we
have "Life at the Edge of Chaos",
Creighton writes that the Santa Fe
Institute was housed in a series of
buildings on Canyon Road which had
formerly been a convent. That is actually
true. The Institute seminars were held
in a room which had served as a chapel
which is also true. Now standing at the
podium with a shaft of sunlight shining
down on him, Ian Malcolm, paused
dramatically before continuing his lecture
dressed entirely in black, leaning on a
cane, Malcolm gave the impression of
severity. He was known in the Institute
for his unconventional analysis and his
tendency to pessimism. His talk that
August entitled, Life at the Edge of Chaos
was typical of his thinking. In it,
Malcolm presented his analysis of chaos
theory as it applied to evolution.
Once this book came out, and the movie
came out, a lot of people noticed that
there was a place called the Santa Fe
Institute. I happened to be there in the
mid-90's, as a resident faculty member,
and one day the Santa Fe Institute
librarian came to lunch and was sitting
with a group of faculty and post-docs at
the Institute and mentioned humorously
that someone had written her a letter
requesting Professor Ian Malcolm's
papers. So, of course, post-docs, being
post-docs, decided to do the obvious thing
which was make Ian Malcolm a web site.
Here is his web site at the Santa Fe
Institute, which had some of his papers
in it, and his research interests and so
on, and only after the Board of Trustees
of the Santa Fe Institute decided that
this was unprofessional did Ian Malcolm's
web page come down. Chaos is a very
important area in dynamical systems
theory and shows up in many different
contexts. You can see all these different
areas in which chaos has been seen such as
brain activity, population growth,
financial data, and so on.
We're going to look at the phenomenon of
chaos in population growth, in a very
simple model of population growth.
We're going to address the question,
What is the difference between chaos and
randomness? Which turns out to be a more
subtle question than one might think.
We're going to explore it through the
notion of "deterministic chaos".