In the previous problem, the fixed point at the origin (*R*=0, *J*=0)
formed a *neutral center* where trajectories simply orbit around
the fixed point. In 2-D there are lots of different ways for system to behave near
fixed points. In general, there are four different qualitative
behaviors (plus one more that's not a fixed point). They are

- Stable nodes and spirals
- Unstable nodes and spirals
- neutral centers
- Saddle points

In a linear model, there can only be one fixed point (the proof is actually graphical). In a non-linear model there can be lots of fixed points scattered around the plane. When you know where they are and whether they are attractors or repellers, you often can guess most of the behavior of your model.

Oh one last thing, in 2-D there is one more animal that can live on
the phase plane and that is a **limit cycle**, i.e. an isolated
closed orbit (sort of like a neutral center) that either attracts or
repels nearby trajectories. We won't deal with them here but they are
common in non-linear oscillators, chemical reactions and certain more
realistic population models.

Mon Sep 22 21:30:22 EDT 1997