There is a hamiltonian $$H\left(z,\dot{z}\right)=\frac{\alpha\left(z\right)\dot{z}^{2}}{2}+\beta\left(z\right)$$ defined on a closed interval $\left[z_{1},z_{2}\right]\subset\mathbb{R}$ s.t. $\alpha\left(z\in\left[z_{1},z_{2}\right]\right)>0$ . The function $\beta$ has exactly one local minimum: $\exists!z_{0}\in\left(z_{1},z_{2}\right):\,\beta'\left(z_{0}\right)=0$, furthermore $\beta'\left(z\in\left[z_{1},z_{0}\right)\right)<0$, and $\beta'\left(z\in\left(z_{0},z_{1}\right]\right)>0$.
It's easy to show (using Poincare-Bendixon theorem) that the initial value problem of the corresponding hamilton's equation with initial conditions $$z\left(0\right)=z_{s}$$ $$\dot{z}\left(0\right)=0$$ has periodic solution $\forall z_s\in\left[z_{1},z_{2}\right]\backslash z_{0}$.
I looked through several books on dynamical systems, but I couldn't find the corresponding theorem. Many authors don't prove the proposition, and show this like an example.
Question: Is there a theorem like this?