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Dynamical Systems
In simplest terms, a dynamical system
is a system that changes over time. Thus, the solar system is a dynamical
system; the United States economy is a dynamical system; the weather is
a dynamical system; the human heart is a dynamical system. Mathematically
speaking, a dynamical system is a system whose behavior at a given time
depends, in some sense, on its behavior at one or more previous times. The
words "in some sense" in the preceding sentence should be taken to mean
that we may or may not have a clue as to how a current state of a system
depends on a past state; but we have reason to believe that it does. Furthermore,
it is the task of the mathematical modeler to
come up with a mathematical construct, a model, that will describe this
relationship between current and past states of the system so that predictions
about the future course of events for the system may be made with some degree
of accuracy. This is what the great Isaac Newton did in his seventeenth
century development of the laws of motion and gravitational attraction.
Newton invented differential equations to mathematically model the way in
which the current position of a planet or moon depended upon its previous
positions. So successful was Newton's model, that in 1969, human beings
rode a giant Saturn V rocket and Newton's equations to the moon and back.
The spectacular predictive successes of Newton's continuous
model made it practically the only paradigm for modeling physical systems
for more than two centuries after it was first set forth. The influence
of Newtonian mechanics, and the calculus that was its foundation, was
so pervasive in the physical sciences that it became almost an assumption
that Newton's model and the Creator's model were one and the same. Implicit
in this acceptance of Newton's version of the universe was that, since
Newton's Laws yielded almost perfect predictions about the solar system,
the universe must be deterministic in the sense that knowledge of the
present state of the system was sufficient for determining all future
states of the system. (For a quick and easy overview of determinism, see
the UT Austin
chaos site.) Such a comforting and overpowering world view kept the
continuous models of the calculus at the foreground of science and any
potential discrete models in the background.
One of the attractive aspects of Newton's calculus is that
a properly schooled individual can practice it with nothing more than
pencil and paper. The differential equations are easy to write down and
purely mental techniques may be used to write down their solutions. If
one happens to write down a differential equation which does not have
a "closed form" solution, then such an "anomaly" is simply not part of
the Newtonian canon and, therefore, not part of God's divine plan. It
is not one of the Platonic "forms," so to speak. That the "anomalous"
differential equations far outnumber the "solvable" ones was a problem
that was pushed to the side amidst the euphoria created by the successes
of Newton's magnificent calculus and Gottfried Wilhelm von Leibniz's even
more "user friendly" version. It was not as if mathematicians and scientists
were unaware of the existence of "non-analytically solvable" differential
equations. Some, such as Leonhard Euler, even came up with methods for
numerically approximating the "solutions" of such bothersome equations
by discrete processes. But even given the conceptual simplicity of Euler's
method and others, the numerical difficulties in carrying out more than
a few steps of these discrete procedures with only pen and paper rendered
such procedures unattractive to people who were constitutionally more
inclined toward "the big picture" than the messy numerical details. This
would change dramatically in the second half of the twentieth century
when the advent of the digital computer would relegate the distasteful
"number crunching" to machines. In the 1970s and 80s, as digital computation
devices, such as personal computers and handheld calculators, became both
less expensive and more sophisticated, more and more scientists and mathematicians
began to experiment with discrete models for dynamical systems. The biologist
Robert May and the physicist Mitchell Feigenbaum made some startling discoveries
in the 1970s using hand calculators to explore what would happen if the
well-known phenomenon of logistic growth were treated as a discrete dynamical
system rather than a continuous one. Their discoveries (see one of them
here
and another here.
) helped to establish the importance of the strange new science which
would become known as Chaos Theory.
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