Chapter 10
Applications of Systems of Equations
In this chapter we initially present techniques for determining the behavior
of solutions to systems of first-order differential equations without first finding
the solutions. To this end, we define and discuss equilibrium points (critical
points), various types of stability and instability, and phase-plane graphs.
Next, we show how to use computer software to solve systems of first-order
differential equations numerically, how to graph the solution components, and
how to produce phase-plane graphs. We also state stability theorems for sys-
tems of first-order differential equations. Throughout this chapter we develop
and discuss a wide variety of models and appli cations which can be written
as vector initial value problems and then solved numerically.10.1 Richardson's Arms Race Model
Throughout recorded history, there have been many discussions regarding
the causes of war. In his account of the Peloponnesian war, written over
two thousand years ago, Thucydides asserted that armaments cause war. He
wrote:
"The real though unavowed cause I believe to have been the growth
of Athenian power, which terrified the Lacedaemonians and forced them
into war."The mathematical model of an arms race which we will study in this sec-
tion was developed by Lewis Fry Richardson. He was born at Newcastle-
upon-Tyne, England, on October 11, 1881. Richardson attended Cambridge
University where he was a pupil of Sir J. J. Thomson, the discoverer of the
electron. In 1923, Richardson published his book, Weather Prediction by
Numerical Process. This text is considered a classic work in the field of me-
teorology. In 1926, Richardson was awarded a Doctor of Science degree from
the University of London for his contributions in the fields of meteorology
and physics. The following year he was elected to the Fellowship of the Royal
Society.
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