Given the following system:
$$\frac{dx}{dt} = x\left(2-x-y\right)$$ $$\frac{dy}{dt} = x-y$$
I found fixed points $(0,0)$ and $(1,1)$. I then want to show that this function is Lyapunov for $x>0$ :
$$V\left(x,y\right)=-\log{x}+x+\frac{1}{2}\left(1-y\right)^{2}$$
One of the characteristics of a Lyapunov function (according to Strogatz) is that $V(x^*) = 0$ where $x^*$ is a fixed point of the system. Obviously $(0,0)$ isn't valid), but plugging in the F.P. $(1,1)$ I get that $V(1,1) = 1$. I'm sure I've just made a silly error but I don't know where it is. After much checking.