d​xd​t=α x+β y&IndentingNewLine;d​yd​t=-β x+α y.

The eigenvalues of this simple canonical system are . The extremum, , is shown as a green dot.

If , the extremum is an unstable focus.

If , the extremum is a stable focus.

If , the dynamic behavior is that of a limit cycle and the critical point is a center.

If , the trajectories spiral clockwise around the origin.

If , the trajectories spiral counterclockwise around the origin.

The red curve is the parametric plot of the solution of the system of ODEs with an initial condition (shown as a cyan dot).