Consider the $2$-nd order dynamic system defined as:
$\dot{x}_1=x_2, \qquad\dot{x}_2=-g(x_1)$
where $g$ is a continuously differentiable function and $\xi\cdot g(\xi)>0, \forall \xi \in (-a,a),$ with $a$ being a known positive scalar. Defining the following energy function:
$V(x)=\frac{1}{2}x_2^2 + \int_{0}^{x_1}g(\xi) d\xi$
Is it true that, for sufficiently small $||x(0)||$, every solution is periodic?
(We can easily show that $\dot{V}(x)=0$ over any phase trajectory, namely $V(x)=constant$. However, before analysing whether a solution is periodic, I do not know how to show that the curve $V(x)=constant$ is closed).
Thanks in advance!