Consider the prey-predator system: $$\left\{\begin{array}{ll}x'=(a-by)x\\y'=(-c+dx)y\end{array}\right. $$ where $a,b,c,d>0$. Im trying to find a Liapunov function for that system. The singularitys points are $(0,0)$ and $\left(\frac{c}{d},\frac{a}{b}\right)$. There is a theorem saying that if there is some Liapunov Function for the system, the singularity is a stable point. $(0,0)$ is not a stable point, so there is no Liapunov Function in this case. But $\left(\frac{c}{d},\frac{a}{b}\right)$ is a stable point (by Poincaré-Bendixon theorem). Im trying to find a Liapunov function for the linear system: $$\left\{\begin{array}{ll}x'=-\frac{b\cdot c}{d}y\\y'=\frac{a\cdot d}{b}x\end{array}\right. $$
A Liapunov Function $V:\Omega\subset \Re^{n}\rightarrow\Re$ for a system $x'=f(x)$, $x\in\Re^{n}$ is a $C^{1}$ class function defined in $\Omega$ that contains a closed ball centered at a singularity of the system. V satisfies $V(0)=0$, $V(x)>0$ if $x\neq0$, and $\dot{V}\leq0\,\forall x$, where $\dot{V}:\Omega\rightarrow\Re$ is defined by $\dot{V}=<($grad$ V(x), f(x)>$.
At my question, $f(x,y)=\left(-\frac{bc}{d}y, \frac{ab}{b}x \right)$. My problem is trying to find a Liapunov Function for that system.
You get a first integral by separating the variables in $$ \frac{dy}{dx}=\frac{(−c+dx)y}{(a−by)x}\iff0=(d-\frac cx)dx+(b-\frac ay)dy $$ which integrates to $$ V=d\,x-c\ln x+b\,y-a\ln y. $$ So you know that solution curves are restricted to level sets of this function, which can also serve as Lyapunov function.
By the way, the linearization at the stationary point $(x_0,y_0)$ for $x=x_0+u$, $y=y_0+v$ looks like \begin{align} u'&=au-bx_0v\\ v'&=-cv+dy_0u \end{align} which is slightly different from what you established.