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Consider the following system of ODEs:

The eigenvalues of this simple canonical system are

. The extremum,

, is shown as a green dot.

, the extremum is an unstable focus.

, the extremum is a stable focus.

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

, the trajectories spiral clockwise around the origin.

, 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).

S. Lynch, Dynamical Systems with Applications using Mathematica, Boston: Birkhäuser, 2007.

Contributed by: Housam Binous

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Dynamic Behavior of a Simple Canonical System