
Stability of a solution
Dear all,
I have a question on the stability of a solution.
Let a(t), b(t) and c(t) be continuous functions of t over the interval . Assume (x,y) = is a solution of the system
Show that this solution is stable.
I rearranged the system to get
I've only dealt with constant coefficient linear systems before, so I'm having trouble with this question.
Let .
I'm not sure if I can treat A like a constant matrix, i.e. take the determinant of A to be and trace(A) = 0 by treating t as constant. Also, what role does the play here?
Can someone please help me?
Thank you.
Regards,
Rayne

You are attempting to use Lyapunov's theorem. Let's find the stationary solution. Solve to get . Goes without saying that , or we get a trivial system.
Now let's rename: . By Lyapunov's theorem, if all eigenvalues of the Jacobian matrix evaluated at have nonpositive real parts, the solution is stable. But this is just as you say , so the characteristic polynomial is .