A Family of Oscillations That Connects Equilibria

We study a mechanical system subject to the action of positional forces. It is assumed that, on a fixed set of the system, the acting force is nonzero everywhere except for the equilibrium points. We study symmetric periodic motions (SPMs). The general theorem on the global bilateral extension of the SPM to the boundary of the region of existence of the SPMs is proved. The global continuation of the Lyapunov family with the inheritance of a monotonic change in the period is given. It is shown that when the period decreases, the family goes to infinity, accompanied by a period tending to zero. An increase in the period on the family occurs unlimitedly. In this case, the family either goes to infinity, or adjoins a saddle-type equilibrium. In this way the center and the saddle are connected by a family of symmetric oscillations. Poincare law on the change in the nature of equilibria extends to a system with n > 1 degrees of freedom. All families of DNA base pair oscillations that bind equilibria are found.