Hi guys!(adsbygoogle = window.adsbygoogle || []).push({});

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?!

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

**Physics Forums | Science Articles, Homework Help, Discussion**