Stability of Critical Points of Second Order Equation

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

let

therefore

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,

Len

Re: Stability of Critical Points of Second Order Equation

Re: Stability of Critical Points of Second Order Equation

Quote:

Originally Posted by

**Danny** You are.

Thank you,

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?

Re: Stability of Critical Points of Second Order Equation

Len, are you in MATH3090 at York University?