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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.

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    $\begingroup$ You are absolutely right. $\endgroup$ – GReyes May 22 at 21:46
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    $\begingroup$ An answer is an answer in my books so happy to upvote and accept your answer if you wish to post so :-) $\endgroup$ – Dean P May 22 at 21:50
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    $\begingroup$ Thanks, I appreciate that. I am just glad to help and reserve "answers" for longer/more detailed ones. $\endgroup$ – GReyes May 22 at 22:01

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