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)?