I am a little bit unsure about the following problem:
Given the system:
$\dot{x} = y - x^3$
$\dot{y} = -x + 2x^3 -2y$
Using the Liapunov function $V(x,y) = x^2 + y^2$, show that the origin in the $(x,y)$-plane is asymptotically stable.
ATTEMPT AT SOLUTION:
OK, so the family of curves:
$V(x,y) = x^2 + y^2 = \alpha$, $0 < \alpha < \infty$
is a topographic system. We thus have:
$\dot{V}(x,y) = 2x\dot{x} + 2y\dot{y}$
$= 2x(y - x^3) + 2y(-x + 2x^3 -2y) = 2xy - 2x^4 - 2xy +4x^{3}y -4y^2 = -2x^4 + 4x^{3}y - 4y^2$
In order for us to achieve asymptotic stability, we must have that $\dot{V}(x,y) < 0$ for all points except the origin. However, I can not see this being the case here. Say we take $x = -5$, $y = -5$, then $\dot{V}(x,y) > 0$. So I must do something wrong here.
Any help/tips would be greatly appreciated!