Consider the system $$\begin{split} x'&=-x^3 \\ y'&=-y(x^2+z^2+1)\\ z'&=-\sin(z) \end{split}$$ Find all the equilibrium points and find stability at origin.
This is what I've done:
I found the equilibrium points and I compute the jacobian matrix but it turns out one eigenvalue has real part $0$, therefore the only way to know if the point is stable it's with a Lyapunov function.
Could anyone tell me what is the Lyapunov function that I should consider?
Please?
A possible Lyapunov function for the system is
$V(x,y,z) = \frac{1}{2}x^2 + \frac{1}{2}y^2 + \frac{1}{2}z^2$.
The derivative w.r.t. the trajectories of the system is
\begin{align} \dot V & = \dot x x +\dot y y + \dot z z\\ &= -x^4 - y^2(x^2+y^2+1) - z\,\sin(z) \\ &< 0\quad \text{for $(x,\,y,\,z)\ne0$ and $z \in (-\pi,\pi)$}. \end{align}
However, as also discussed in the comments, the equilibrium of each first order ODE is stable and attractive since in each equation the sign of the variable is always opposed to the sign of its corresponding derivative, except at the origin. (E.g. in $\dot x = -x^3$, any $x>0$ forces $x(t)$ to decrease due to $\dot x<0$, and $x<0$ yields $\dot x>0$.) The Lyapunov function approach says the same since $\dot x x<0$, $\dot y y<0$, $\dot z z<0$, $(x,\,y,\,z)\ne0$, $z \in (-\pi,\pi)$.