Convergence of dynamical system to stable equilibrium $(xy-1)^2$ to set $\{xy = 1\}$.

Question

Consider the following dynamical system:

$\dot{x} = -(xy-1)y$ and $\dot{y} = -(xy-1)x.$

I would like to show that $x,y$ converge to set $xy = 1.$

1. Using Lyapunov function $V = (xy-1)^2,$ we can show that $\dot{V} = -(xy-1)^2(x^2+y^2)\le 0$ with equality only when $S = \{x=y=0 or \{xy = 1\}\}.$ Hence by Lasalle invariance principle we can show that it asymptotically converges to $S.$

2. Consider the Lyapunov function $V = x^2 + y^2,$ and set $U = \{x\ge 0,y\ge 0\}.$ We can show that $\dot{V} = -2xy(xy-1)$ is positive is a neighborhood around $(0,0)$ in set $U$. Hence using Chetaev instability theorem we can argue that $(0,0)$ is an unstable equilibrium point.

My question is, does argument 1) and 2) show that $xy=1$ is asymptotically stable?

EDIT --

To prove - For almost all initializations in a bounded set $x<R,y<R, R<\infty$ the set $S = \{xy=1\}$ is asymptotically stable, i.e. the set of initializations that lead to point $\{x=y=0\}$ is of measure zero. I am hoping to show this using Lyapunov analysis, and maybe not apply Stable Manifold Theorem.

Answer 1

The proof is exactly the same as yours (the first approach), with a little analysis added.

Let $s = x y - 1$ and $s = 0$ be the equilibrium.

Taking the time derivative of $s$ yields

$$ \begin{align*} \dot{s} &= x \dot{y} + \dot{x} y \\ \dot{s} &= - x \left[\left(x y - 1\right) x\right] - \left[\left(x y - 1\right) y\right] y \\ \dot{s} &= - s x^{2} - s y^{2} \\ \dot{s} &= - s \left(x^{2} + y^{2}\right) \\ \end{align*} $$

The result shows that $s$ will converge to $0$ so long as $x^{2} + y^{2} > 0$. This also implies we cannot have $x = 0 \land y = 0$. Continue with the Lyapunov stability proof, we have the candidate function

$$ V(s) = \frac{1}{2} s^{2} .$$

Taking the time derivative of $V$ yields

$$ \begin{align*} \dot{V}(s) &= s \dot{s} \\ \dot{V}(s) &= s \left[- s \left(x^{2} + y^{2}\right)\right] \\ \dot{V}(s) &= - \left(x^{2} + y^{2}\right) s^{2} < 0 \quad \forall \; s \neq 0 \land x \neq 0 \land y \neq 0 \\ \dot{V}(s) &= - 2 \left(x^{2} + y^{2}\right) V .\\ \end{align*} $$

The stream plot looks like this: