A menagerie of fixed points

 

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

Figure 5 shows them schematically.

  
Figure 5: The taxonomy of fixed points in 2-D.

Stable fixed points are attractors in that any trajectory that starts near them will eventually end up at the steady state fixed point after some time. Whether they spiral in or come in as nodes depends on the details of the problem. Likewise, unstable fixed points are repellers and everything flies away from them. The saddle node is essentially unstable but has a stable manifold, that is parts of phase space are attracted to the fixed point until they get close enough and then fly away along the unstable manifold. There are a few more kinds of fixed points that occur when you change from one kind of fixed point to another (for example as the parameters change in your problem) but we don't need to worry about those.

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.