Construct a Lyapunov Function in order to show that the system \begin{align}\frac{dx}{dt}&=x(y^2 +1) +y \\ \frac{dy}{dt} &= x^2y +x\end{align} has no closed orbits (limit cycle) and hence no periodic solutions.

I have absolutely no experience in constructing Lyapunov functions. Can anyone please help guide me with this? I know the conditions that the Lyapunov function $L(x,y)$ must satisfy, but I do not know how to "create" the function $L$.


The given dynamical system has one equilibrium point at $x = y = 0$ and calculating the Jacobian at this point we have

$$ J_{\{0,0\}} = \left( \begin{array}{cc} 1 & 1 \\ 1 & 0 \\ \end{array} \right) $$

with eigenvalues

$$ \left\{\frac{1}{2} \left(1+\sqrt{5}\right),\frac{1}{2} \left(1-\sqrt{5}\right)\right\} $$

one positive and another negative, characterizing the origin as a saddle point. So this equilibrium point is unstable. All those explanations are to justify that the quest for a Liapunov function has sense as long as we don't now for sure that the dynamical system is not unstable. If we know that the system is unstable then no Liapunov function exists for this system as is the present case.


A Liapunov function is a positive definite function $V(x(t),y(t))$ such that $\dot V = V_x(x(t),y(t)).\dot x(t)+V_y(x(t),y(t)).\dot y(t)$ is definite negative. See definition here.


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