Dynamic Behavior of a Simple Canonical System

α

0.1

β

0.1

Consider the following system of ODEs:

dx

dt

=αx+βy

dy

dt

=-βx+αy.

The eigenvalues of this simple canonical system are

λ=α±βi

. The extremum,

M=(0,0)

, is shown as a green dot.

If

α>0

, the extremum is an unstable focus.

If

α<0

, the extremum is a stable focus.

If

α=0

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

If

β>0

, the trajectories spiral clockwise around the origin.

If

β<0

, 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

P=(1,1)

(shown as a cyan dot).