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I had the following system of DE's to solve:

[itex]\alpha '=-2i \alpha[/itex]

[itex]\beta ' =2i \beta[/itex].

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): [itex]\alpha (t)=c_1e^{-2it}[/itex], [itex]\beta (t) =c_2 e^{2it}[/itex].

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 [itex](\alpha , \beta ) =(1,0)[/itex], I get that [itex](\alpha ' , \beta ' )=(-2i ,0)[/itex]. 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?!

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

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