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 |
? |

** ^{*}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:

- 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**.

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) |

Remaining cases:

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