I have to find Lyapunov functions to stablish the stability of equilibrium points for the equation $\begin{cases}x' = -xy^4 \\ y' = yx^4\end{cases}$ which coincide with the axis of the plane $\{(x,y) \in \mathbb{R}^2. xy = 0\}$. Note that this points can't be assymptotically stable and that the version of Chetaev theorem I have is not sufficient to stablish instability of any of them (since $\dot V = 0$).
Using a traditional strategy for $p = (0,0)$ I searched for a Lyapunov function of the form $V(x,y) = V_1(x) + V_2(y)$ and I found that $V(x,y) = x^4+y^4$ is a proper function.
What about the rest of the points how can I show their stability?
Best Answer
You established that any solution will be contained in the level surfaces of $V=x^4+y^4$. Next look at the movement along the curve, how it moves along the tangent direction $(-y^3,x^3)$ (oriented counterclockwise) giving $$ -y^3\dot x+x^3\dot y=xy(x^6+y^6). $$ By the sign of this scalar product, this gives a movement toward the $y$ axis inside all of the quadrants, making the stationary points on the $y$-axis stable (but not asymptotically stable) and on the $x$-axis outside the origin unstable.