Direction of trajectory, system of DE's and portrait phase in plane phase

[fluidistic]

Hi guys!
I had the following system of DE's to solve:
$\alpha '=-2i \alpha$
$\beta ' =2i \beta$.
Where alpha and beta depend on t.
I solved it by writing the system under matricial form, found the eigenvalues and corresponding eigenvectors.
The solution is (and I've checked it, it works): $\alpha (t)=c_1e^{-2it}$, $\beta (t) =c_2 e^{2it}$.
Since the eigenvalues are purely complex the trajectories in the phase plane are either circles or ellipses around (0,0).
Now I've been reading http://tutorial.math.lamar.edu/Classes/DE/PhasePlane.aspx to check out how to determine the direction of rotation.
When I pick $(\alpha , \beta ) =(1,0)$, I get that $(\alpha ' , \beta ' )=(-2i ,0)$. However on the website I've just linked, there's no example of what happens when you get complex values. I don't know how to sketch the direction of the trajectory at the point (1,0) because of that complex number.
Is the direction counter clockwise, clockwise, none?!

 

[haruspex]

You appear to have a 4 dimensional phase space. How do you propose sketching trajectories in that in general?
OTOH, with beta initially zero, it will always be zero, so that makes it manageable.

 

[chiro]

Hey fluidistic.

Have you checked the orientability of the Jacobian (determinant)?