The behaviour of the period of symmetrical periodic motions

Symmetrical periodic motions (SPMs) of a reversible mechanical system are considered; the motions include oscillations and rotations. The initial points for the SPM form sets Λ in phase space. It was established earlier that in a family of SPMs the period depends, as a rule, on a single important parameter. It is shown that in a region, stable to parametric perturbations of the system and contained in Λ, the period is a monotonic function of a single variable, while its derivative on the boundary of the region either vanishes or does not exist (unilateral, infinite). A formulation of the relation between the period and the parameter is also given.