# Dynamic Behavior of a Simple Canonical System

Dynamic Behavior of a Simple Canonical System

Consider the following system of ODEs:

dx

dt

dy

dt

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

λ=α±βi

M=(0,0)

If , the extremum is an unstable focus.

α>0

If , the extremum is a stable focus.

α<0

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

α=0

If , the trajectories spiral clockwise around the origin.

β>0

If , the trajectories spiral counterclockwise around the origin.

β<0

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

P=(1,1)