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**.