On some mathematical topics in classical synchronization. A tutorial

A few mathematical problems arising in the classical synchroniza-tion theory are discussed; especially those relating to complex dynam-ics. The roots of the theory originate in the pioneering experiments by van der Pol and van der Mark, followed by the theoretical stud-ies done by Cartwright and Littlewood. Today we focus specifically on the problem on a periodically forced stable limit cycle emerging from a homoclinic loop to a saddle point. Its analysis allows us to single out the regions of simple and complex dynamics, as well as to yield a comprehensive description of bifurcational phenomena in the two-parameter case. Of a particular value among ones is the global bifurcation of a saddle-node periodic orbit. For this bifurcation, we 1 prove a number of theorems on birth and breakdown of nonsmooth invariant tori