Positive solutions with a complex behavior for superlinear indefinite ODEs on the real line

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.