  We show the existence of infinitely many positive solutions, defined on the
real line, for the nonlinear scalar ODE \[ \ddot u + (a^+(t) - \mu a^-(t)) u^3
= 0, \] where $a$ is a periodic, sign-changing function, and the parameter
$\mu>0$ is large. Such solutions are characterized by the fact of being either
small or large in each interval of positivity of $a$. In this way, we find
periodic solutions, having minimal period arbitrarily large, and bounded
non-periodic solutions, exhibiting a complex behavior. The proof is
variational, exploiting suitable natural constraints of Nehari type.
