| eigenvalues | linear system | nonlinear system | ||||
|---|---|---|---|---|---|---|
| real | both pos. | equal | proper or improper node | unstable | similar to node or spiral point | unstable |
| different | node | unstable | same | |||
| both neg. | equal | proper or improper node | as. stable | similar to node or spiral point | as. stable | |
| different | node | as. stable | same | |||
| pos. and neg. | saddle point | unstable | same | |||
| complex not real |
real part pos. | spiral point | unstable | same | ||
| real part neg. | spiral point | as. stable | same | |||
| real part zero | center | stable | ? | ? | ||
``same'' means that type and stability for the nonlinear problem are the same as for the corresponding linear problem. If we look at at smaller and smaller neighborhoods of the critical point, the phase portrait gets closer and closer to the phase portrait of the corresponding linear system.
``?'' means that this cannot be determined on basis of the corresponding linear problem.
``similar to node or spiral point'' means that the behavior resembles that of a node or a spiral point in the sense that the trajectories go into the critical point either from specific directions, or rotate infinitely often around the critical point. But even for arbitrarily small neighborhoods the trajectories can look very different from those of the corresponding linear equation, see examples.
Note that the table only considers the case of nonzero eigenvalues. In this case we always have an isolated critical point.