Hyperbolic and Bi-hyperbolic solutions in the planar restricted $(N+1)$-body problem

Consider the planar restricted $(N+1)$-body problem with trajectories of the $N(\ge 2)$ primaries forming a collision-free periodic solution of the $N$-body problem, for any positive energy $h$ and directions ${\theta }_{\pm }\in [0,2\pi )$, we prove that starting from any initial position $x$ at any initial time $t_x$, there are hyperbolic solutions ${\gamma }^{\pm }{\mathrm{\mid }}_{[t_x,\pm \mathrm{\infty })}$ satisfying ${\gamma }^{\pm }(t_x)=x$ and $$\underset{t\to \pm \mathrm{\infty }}{\lim }{\gamma }^{\pm }(t)\mathrm{/}\mathrm{\mid }{\gamma }^{\pm }(t)\mathrm{\mid }=e^{i{\theta }_{\pm }(\text{mod }2\pi )},\underset{t\to \pm \mathrm{\infty }}{\lim }{\dot{\gamma }}^{\pm }(t)=\pm \sqrt{2h}{e}^{i{\theta }_{\pm }(\text{mod }2\pi )}\mathrm{.}$$ Moreover we also prove the existence of a bi-hyperbolic solution $\gamma {\mathrm{\mid }}_{\mathbb{R}}$ satisfying $$\underset{t\to \pm \mathrm{\infty }}{\lim }\gamma (t)\mathrm{/}\mathrm{\mid }\gamma (t)\mathrm{\mid }=e^{i{\theta }_{\pm }(\text{mod }2\pi )},\underset{t\to \pm \mathrm{\infty }}{\lim }\dot{\gamma }(t)=\pm \sqrt{2h}{e}^{i{\theta }_{\pm }(\text{mod }2\pi )}\mathrm{.}$$