Existence of uncountable many homoclinic solutions to periodic orbits in a center manifolds

Consider a Lagrangian of the form \[ L(x, \dot{x}, q, \dot{q})=\frac{1}{2}(\dot{x}^2-x^2)+ \frac{1}{2}\dot{q]^2+(1+ \delta (x))V(q), \quad x, q \in \mathbb{R}. \] Assuming $\delta$ bounded, $V$ periodic in $q$ with a strict global minimum at $q=0$, it is shown for each $\varphi \in(0, 2 \pi)$ the existence of a homoclinic solution satisfying \begin{align*} & \lim_{t \to - \infty} q(t)=0 \\ & \lim_{t \to + \infty} q(t)=2 \pi \\ & \lim_{t \to - \infty} |x(t)- R \cos t|=0 \\ & \lim_{t \to + \infty} |x(t)- R \cos (t+ \varphi)|=0. \end{align*}