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Classification of equilibrium points.

The phase line and graph of f also provide an easy method to classify equilibrium points for autonomous differential equations. There are only three basic types: sinks (nearby solutions converge to the equilibrium point), sources (nearby solutions diverge), and nodes (all other behavior). With an eye toward the classification of equilibria in systems, we discuss the ``first derivative test'' for autonomous equations: Suppose p is an equilibrium point for the equation

dy / dt = f(y)

If f'(p)> 0 then p is a source.
If f'(p) < 0 then p is a sink.
If f'(p) = 0 then we get no information.

Figure 5: Sinks, sources, and nodes

See Figure 5. As above, students need to understand the relationship between the sign of f'(p), the rise or fall of the graph of f near p, and the behavior of solutions to appreciate this result.

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Robert L. Devaney
May 6, 1995