  A polynomial of degree $\ge 2$ with coefficients in the ring of $p$-adic
numbers $\mathbb{Z}_p$ is studied as a dynamical system on $\mathbb{Z}_p$. It
is proved that the dynamical behavior of such a system is totally described by
its minimal subsystems. For an arbitrary quadratic polynomial on
$\mathbb{Z}_2$, we exhibit all its minimal subsystems.
