Find the critical points of the following second-order equation (by transferring into a system of first order differential equations) and determine the stability of these critical points:
My start at a solution: transferring into a system of first order differential equations
So to get the critical points I can set This implies
so , where
so , implies
Then my critical points are
(1,0) and (-1,0)
At this point I would take the Jacobian at the two critical points and solve for the eigenvalues. The eigenvalues would then tell me about the stability? I will do this, I just want to confirm I am on the right track first.
Thanks so much,
So for the point (1,0) I got eigenvalues -2 and 1. This would mean the point is unstable and a saddle.
and for (-1,0) I got eigenvalues 1/2 + (sqrt(7)/2)i and 1/2 - (sqrt(7)/2)i
So it has two positive real parts. therefore it is unstable.
Is there anything else I can say about the stability and the critical points?