Suppose I have the system:
$\dot{x} = -x^3 - y^2$
$\dot{y} = xy - y^3$
... and am asked to find the domain of stability of the system.
Is my attempt and reasoning below deemed a correct approach?
I first find an appropriate Lyapunov function as follows:
$V(x, y) = x^2 + y^2$
$\dot{V} = 2x \dot{x} + 2 y \dot{y}$
$\dot{V} = 2x (-x^3 - y^2) + 2y(xy - y^3)$
$\dot{V} = -2x^4 - 2 y^4$
$\dot{V} = -2(x^4 + y^4)$
$\leq 0$ for all $(x, y) \neq (0, 0)$
By definition, the above is a Lyapunov function and by the Lyapunov theorem, the system is stable and the nature of the stability is asymptotic.
To finally answer the question of "domain of stability", I note that $\dot{V} \leq 0$ holds true for all $(x, y) \neq (0, 0)$ and therefore I reason that the domain of stability is all values of $x$ and $y$, i.e. everywhere, or global.