I'll tell you what we normally did in our ODE course.

We looked at systems of the form X'=AX+K (*)

(where X' is the vector (x',y') , A is 2x2 matrix, X is the vector (x,y) and K vector).

K is called the perturbation, or small disturbance, vector.

In your case, A is the matrix (-1,1;0,-1)

Meaning, A is the linear part of the system. The other terms go into the vector K.

One would hope that A is diagonalizable, so you could write A=EDE^-1, multiply the (*) equation from the left with E^-1, and you'd get X1' = DX1 + (E^-1)K

where X1 = (E^-1) X

You would then define a Lyapnunov function similar to what you've written, but in terms of the new variables (x1,y1)=X1. (i.e V=x1^2 + w * y1^2)

(To avoid confusion with the k in your original post, I've changed it to w)

You could in this fashion, find that domain (if it exists) where the system is asymptotically stable near the origin.

BUT, in your case A is not diagonalizable, in fact it's in a Jordan form, with -1 as an eigenvalue.

In our course we have not dealt with Jordan forms. But you could try it (I will definitely try it tomorrow out of curiosity ).

You'd write A=EJE^-1 and hope for the best :-)