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I need to find the values of $k$ for which $x^2+ky^2$ is a Liapunov function for the system $$\dot{x}=-x+y-x^2-y^2+xy^2, \dot{y}=-y+xy-y^2-x^2y$$

**My attempt:** $$\dot{V} = \frac{\partial V}{\partial x} \times \frac{dx}{dt} + \frac{\partial V}{\partial y} \times \frac{dy}{dt} = 2x(-x+y-x^2-y^2+xy^2)+2ky(-y+xy-y^2-x^2y)=-2x^2+2xy-2x^3-2xy^2+2x^2y^2-2ky^2+2kxy^2-2ky^3-2kx^2y^2 \leq 0$$

The thing that throws me off the most is the plural **values**.

Now this turns into a minimization problem.

I keep running into terms like $xy, y^3, x^3$, which are not "sign" friendly.

Should I look for terms that will "knockout" each other?

Should I use the AM-GM-HM inequality to help?