So, basically I am given the following exercise:
Exercise. Study de following system null solution stability with the Lyapunov function method. \begin{equation*} \begin{cases} x' = x^3-y \\[5pt] y' = x+y^3 \end{cases} \end{equation*} Suggestion: Consider $V(x,y) = x^2 + y^2$.
My resolution. To check if $V$ is indeed a Lyapunov function, I do the following: \begin{equation*} V \in C^1 \quad\text{in $\mathbb{R^2}$} \\[5pt] V(0,0) = 0 \\[5pt] V(x,y) > 0, \forall (x,y) \in \mathbb{R^2}\backslash\{(0,0)\} \end{equation*} And the last condition is where the problem comes. In this case, take $f(x,y) = (x^3-y,x+y^3)$. Thus, it urges the following: \begin{equation*} f_1(x,y)\frac{\partial V}{\partial x}(x,y) + f_2(x,y)\frac{\partial V}{\partial y}(x,y) = 2(x^4+y^4) \geq 0 \end{equation*} So, $V$ doesnt' satisfy the conditions to be a Lyapunov function. (the last equation sould be $\leq 0$). I don't see where I am making a mistake here? Is the exercise wrong? (by this I mean the suggestion I am given).
Thanks for all the help in advance.
What you have shown is that any trajectory $(x(t),y(t))$ (except the constant solution $(x,y)=(0,0)$) increases its distance from the origin as $t$ increases. So the origin is an unstable equilibrium.