Superlinear indefinite equations on the real line and chaotic dynamics (Q1614731)

The existence and multiplicity of solutions to the second-order differential equation of the form \[ \ddot x + c \dot x +q(t)g(x) = 0, \quad t \in (a, b) , \eqno (1) \] where \(- \infty \leq a < b \leq + \infty, c\) is a real constant, \(q: (a, b) \to \mathbb{R}\), and \( g:\mathbb{R} \to \mathbb{R}\) are continuous functions, are considered. It is supposed that the function \(q(t)\) has infinitely many zeros in \((a, b)\) and the function \(g(x)\) is superlinear. The multiplicity is investigated by searching solutions with suitable prescribed nodal properties. The existence of solutions is studied in the intervals of negativity and positivity of the function \(q(t)\). In a particular case, when \(c=0\) and \(g(x)\) is periodic, it is shown that equation (1) exibits chaotic-like dynamics.