Critical Points of Autonomous Systems

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
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.