Undergrad Phase Plane Diagram w/ Complex eigenvalues

[e101101]

Is the spiral I drew here clockwise or counterclockwise ? What’s a trick to know whether it’s going CCW or CW. Thanks!

 

[member 428835]

The ODE system looks something like this $y' = A y$. Let's pick $y_2 = 0, y_1 = 1$, which implies $y' = \langle-3,2\rangle$. This implies at the point $(1,0)$ in the phase plane there will be a vector pointing in the direction $\langle -3,2\rangle$, and hence moving in the direction along the spiral away from the center.

 

[e101101]

The ODE system looks something like this $y' = A y$. Let's pick $y_2 = 0, y_1 = 1$, which implies $y' = \langle-3,2\rangle$. This implies at the point $(1,0)$ in the phase plane there will be a vector pointing in the direction $\langle -3,2\rangle$, and hence moving in the direction along the spiral away from the center.

Im not quite sure if that means all solns would be approaching the center? Clockwise or Counterclockwise

 

[member 428835]

Im not quite sure if that means all solns would be approaching the center? Clockwise or Counterclockwise

Plot several points and you'll see the behavior, such as the point $(y_1 = a > 0, y_2 = 0)$. Alternatively, look at the sign of the real component of the eigenvalue to determine whether or not solutions converge to the origin or not.