Bifurcations (Next Section)
Bifurcations and Phase Lines (Cover Page)
The phase line and the (Previous Section)

# 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.

Bifurcations (Next Section)
Bifurcations and Phase Lines (Cover Page)
The phase line and the (Previous Section)

Robert L. Devaney
May 6, 1995