We consider a smooth autonomous system in general form that admits a nondegenerate
periodic solution. A global family (with respect to the parameter h) of nondegenerate
periodic solutions is constructed, the law of monotonic variation of the period on the
family is derived, and the existence of a reduced second-order system is proved. For it, the
problem of stabilizing the oscillation of the controlled system, distinguished by the value of
the parameter h, is solved. A smooth autonomous control is found, and an attracting cycle is
constructed.