Determine stability of the equilibrium state

Question

Please help me.

I am struggling to determine the stability of the equilibrium state $x = 0$ of the system $$x_1' = x_1(x_1^2 + x_2^2 - \beta^2) + x_2 \\ x_2' = x_2(x_1^2 + x_2^2 - \beta^2) - x_1$$

with a Lyapunov function $$V(t) = \dfrac{1}{2}\left(x_1^2(t)+x_2^2(t)\right)$$

Answer 1

The time derivative of the Lyapunov function is

$$ \dot{V}(x) = (x_1^2 + x_2^2) (x_1^2 + x_2^2 - \beta^2) $$

If $x_1^2 + x_2^2 > \beta^2$ then $\dot{V}(x) > 0$ so global stability can't be concluded. But we can use it for local stability.

If $x_1^2 + x_2^2 < \beta^2$ then $\dot{V}(x) < 0$ for $x \neq 0$ so the system is locally stable assuming $\beta \neq 0$.