Stationary Points of Autonomous System

eigenvalues			linear system		nonlinear system
real	both pos.	different	nodal source	unstable, repelling	same
	both pos.	equal	radial source or twist source^*	unstable, repelling
	both neg.	different	nodal sink	stable, attracting
	both neg.	equal	radial sink or twist sink^*	stable, attracting
	pos. and neg.		saddle	unstable, not repelling
nonreal	real part positive		spiral source	unstable, repelling
	real part negative		spiral sink	stable, attracting
	real part zero		center	stable, not attracting	?

^*equal eigenvalues: If there are two eigenvectors we get a radial sink/source. If there is only one eigenvector (deficient case) we obtain a twist sink/source.

"same" means: type and stability for the nonlinear system are the same as for the corresponding linear system:

If we look at at smaller and smaller neighborhoods of the stationary point, the phase portrait looks more and more like the phase portrait of the corresponding linear system.
Case of real eigenvalues (nodal sink/source, radial sink/source, twist sink/source, saddle):
For the linear problem we have trajectories which are straight lines, given by the eigenvectors.
For the nonlinear problem we will have curved trajectories in general. But the tangents of the trajectories at the stationary point are the same as for the linear problem, given by the eigenvectors.
If the linear problem has a spiral sink/source:
For the linear problem we have trajectories which make infinitely many revolutions around the stationary point (clockwise or counterclockwise). The same is true for the nonlinear problem, with the same clockwise/counterclockwise sense of rotation.

Note: This page only considers the case of nonzero eigenvalues. In this case both the linear and nonlinear ODE system have an isolated stationary point.