Subordination to periodic processes and synchronization

We study the subordination to a process that is periodic in the natural time scale, and equivalent to a clock with N states. The rationale for this investigation is given by a set of many interacting clocks with N states. The natural time scale representation corresponds to the dynamics of an individual clock with no interaction with the other clocks of this set. We argue that the cooperation among the clocks of this set has the effect of generating a global clock, whose times of sojourn in each of its N states are described by a distribution density with an inverse power law form and power index μ < 2. This is equivalent to extending the widely used subordination method from fluctuation-dissipation processes to periodic processes, thereby raising the question of whether special conditions exist of perfect synchronization, signaled by regular oscillations, and especially by oscillations with no damping. We study first the case of a Poisson subordination function. We show that in spite of the random nature of the subordination method the procedure has the effect of creating damped oscillations, whose damping vanishes in the limiting case of N ≫ 1, thereby suggesting a condition of perfect synchronization in this limit. The Bateman's mathematical arguments [H. Bateman, Higher Transcendental Functions, vol. III, Robert K Krieger, Publishing Company, Inc. Krim.Fr. Drive Malabar, FL;