Continuation of periodic motions to a reversible system in structurally unstable cases. Application to the \(N\)-planet problem (Q2714637)

The author considers an autonomous or \(2\pi\)-periodic reversible system NEWLINE\[NEWLINE \begin{aligned} \dot u &= U_0(u,v,t) +\mu U_1(\mu,u,v,t),\\ \dot v &= V_0(u,v,t) +\mu V_1(\mu,u,v,t);\end{aligned} \quad u\in {\mathbb R}^l,\quad v\in {\mathbb R}^n,\quad l\geq n,\tag{1} NEWLINE\]NEWLINE with a fixed set \( M = \{u, v\: v=0\}\). The continuation of solutions with respect to the parameter of system (1) is discussed for the nonrough case. The existence of two families of space periodic orbits in \(n\)-body problem is proved.