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.

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    $\begingroup$ The Poincaré-Bendixson theorem gives sufficient conditions for the $\omega$-limit set of a trajectory to be a limit cycle, and one cannot conclude from it anything about the stability of the equilibrium $\left(\frac{c}{d},\frac{a}{b}\right)$. Further, if you expect your questions to be answered you should formulate them in an orderly way. For example, are you interested in the Liapunov function for the nonlinear system, or for the linear system? What is the relation of the linear system to the nonlinear system? It looks like the linearization at $\left(\frac{c}{d},\frac{a}{b}\right)$. $\endgroup$
    – user539887
    May 10 '18 at 20:41
  • $\begingroup$ However, the existence of a Liapunov function for the linearization does not necessarily imply the existence of a Liapunov function for the original system. $\endgroup$
    – user539887
    May 10 '18 at 20:46
  • $\begingroup$ Regarding the original system, this is probably a duplicate of Lyapunov function for predator-prey system. $\endgroup$
    – user539887
    May 10 '18 at 20:52

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.


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