# Determining the domain of stability of a dynamic system

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.

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