As Aaron Hoffman pointed out, all trajectories lie on the level lines of $p^2+x^2+(2/3)x^3=c$.
The LHS has two critical points: the local minimum $(0,0)$ with critical value $c=0$,
and the saddle $(-1,0)$ with critical value $c=1/3$. The behavior for large $(x,p)$ is also clear.
So it is easy to sketch all these level lines, and the conclusion is that the trajectory
starting from $(x_0,0)$ is bounded if and only if $x_0^2+(2/3)x_0^3\leq 1/3$, and closed when
this inequality is strict.
For $x_0=0.5$ we obtain that the trajectory is not really closed, but tends to $(-1,0)$ as
time goes to infinity.
You cannot detect this on computer because the point $(-1,0)$ is unstable.
It takes infinite time to approach it on the trajectory, but once you miss, no matter
how little, you will be either on a closed trajectory or escape to infty.
And it will take you long time to find out, if you miss very little.
EDIT. By the way, this system is called the classical anharmonic oscillator. An explicit solution
in elliptic functions exists, but physicists prefer to consider perturbative expansions.
See, for example, L. Landau and E. Lifshitz, Mechanics (Course of Theoretical Physics, vol. I).