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Mathematical Modeling II
with Calculus BC
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 helped to establish the importance of the strange new science
which would become known as Chaos Theory.
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