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)