Stationary Points of Autonomous ODE System

We want to find functions \(x(t),\;y(t)\) such that \[ \left[ \matrix{x'(t)\vphantom{\bigl(} \cr y'(t) \vphantom{\bigl(}} \right] = \left[ \matrix{f\bigl(x(t),y(t)\bigr) \cr g\bigl(x(t),y(t)\bigr)} \right] \] where \(f(x,y),\;g(x,y)\) are given functions.

We say \( (x_*,y_*) \) is a stationary point if \( \left[ \matrix{f(x_*,y_*) \cr g(x_*,y_*)} \right] = \left[\matrix{0\cr0}\right]\), corresponding to a constant solution of the ODE.

The type and stability of the stationary point depends on the eigenvalues of the Jacobian matrix \[ A = \left[\matrix{\partial_xf(x_*,y_*) & \partial_yf(x_*,y_*) \cr \partial_xg(x_*,y_*) & \partial_y g(x_*,y_*)} \right] \]

eigenvalues linear ODE system nonlinear ODE system

real both pos. different nodal source unstable, repelling same

equal radial source or twist source^*

both neg. different nodal sink stable, attracting

equal radial sink or twist sink^*

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 ?

eigenvalues	linear ODE system	nonlinear ODE system
real	both pos.	different	nodal source	unstable, repelling	same
equal	radial source or twist source^*
both neg.	different	nodal sink	stable, attracting
equal	radial sink or twist sink^*
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.

For twist sinks/sources, spiral sinks/sources and centers you should find out whether they are clockwise/counterclockwise. You can decide this by looking at the arrow at (1,0) (1st column of A), or the arrow at (0,1) (2nd column of A).

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

If the linearized problem has a center: We cannot draw conclusions about type and stability just based on the linearized problem.
The nonlinear problem may have a stable or unstable stationary point. We may have a center, but we could also could get unstable spirals, stable spirals, or other cases.
We need to use integral methods (orbital equation) to draw further conclusions.

Note: This page only considers the case of nonzero eigenvalues (i.e., matrix \(A\) is nonsingular). In this case both the linear and nonlinear ODE system have an isolated stationary point.

If all eigenvalues have

negative real part: sink (stable, attracting)
positive real part: source (unstable, repelling)

nodal sink	radial sink	twist sink (ccw)	spiral sink (cw)
nodal source	radial source	twist source (cw)	spiral source (ccw)

(cw=clockwise, ccw=counterclockwise)

Remaining cases:

saddle (unstable, not repelling)
center (stable, not attracting, cw)