Computer Algebra Recipes

Chapter 4
Nonlinear ODE Models
The elegant body of mathematical theory pertaining to linear
systems ... tends to dominate even moderately advanced university
courses. The mathematical intuition so developed ill equips the
student to confront the bizarre behavior exhibited by the simplest of
...nonlinear systems. Yet such nonlinear systems are surely the rule,
not the exception .... Not only in research, but also in the everyday
world of politics and economics, we would all be better o® if more
people realized that simple nonlinear systems do not necessarily
possess simple dynamic properties.
Robert M. May, mathematical biologist, Nature, Vol. 261, 459 (1976)
In Chapter 1, phase-plane portraits were used to explore some simple nonlinear
ODE models whose temporal evolution could not have been predicted, even
qualitatively, before the portraits were numerically constructed. An example
was the period-doubling route to chaos exhibited by the Du±ng equation
Äx +2° _x + ®x+ ¯x 3 = F cos(!t) (4.1)
when the amplitude F of the driving force was increased, the other parameters
being held ¯xed. If the nonlinear term, ¯x 3 , were not present, this \bizarre"
period-doubling behavior would not even be possible. If we were to change the
various coe±cient values in (4.1), the response of the nonlinear system would
in general be entirely di®erent and not easily predicted on the basis of mathematical
or physical intuition alone. To aid in the qualitative understanding of
the behavior of nonlinear ODE systems such as this one, the concepts of ¯xed
points and phase-plane analysis were discussed in Chapter 2.
One might well ask whether there are mathematical techniques for obtaining
the exact analytic solutions to nonlinear ODEs, and if so, whether Maple
can be used to ¯nd these solutions. The answer is that the vast majority of
nonlinear ODEs of interest to physicists and engineers do not possess exact
analytic solutions. Only a handful of ODEs of physical interest can be solved
exactly, and for those equations Maple can be used to ¯nd the solutions. Some
examples of ¯rst-order nonlinear ODEs for which this is the case are illustrated
in the following section.
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