| 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 | similar to center or spiral point | ? | ||
``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 looks more and more like the phase portrait of the corresponding linear system.
``?'' means that this cannot be determined on basis of the corresponding linear problem.
Note that the table only considers the case of nonzero eigenvalues. In this case we always have an isolated critical point.