Find the critical value or values of alpha where the qualitative nature of the phase portrait for x'=(2, alpha, -5, -2)x changes. (this is 2x2 matrix, 2 and alpha on the left, -5 and -2 on the right.)

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# 1 Answer

Hello Sun,

Just for your reference (and since I don't have too much time right now), first find the eigenvalues of the matrix. Do this by solving

(2 - r)(-2 - r) + 5α = 0,

the solutions will be your eigenvalues (r = ±√(4-5α) for your reference). To find the critical values, simply locate where the solutions will change from complex to real---in this case, α = 4/5. If you need help on how to interpret the qualitative nature
of he phase portrait, I'm sure some of the other hyperactive tutors will come by and complete the picture for you, so to speak, maybe Kirill or Andre or Robert (just a friendly poke since I see their names often on the boards).

Regards,

Hassan H.

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