On Some Mathematical Topics in Classical Synchronization.: A Tutorial

Abstract

A few mathematical problems arising in the classical synchronization theory are discussed; especially those relating to complex dynamics. The roots of the theory originate in the pioneering experiments by van der Pol and van der Mark, followed by the theoretical studies 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 is the global bifurcation of a saddle-node periodic orbit. For this bifurcation, we prove a number of theorems on birth and breakdown of nonsmooth invariant tori.

Evaluating the Spectrum of Unlocked Injection Frequency Dividers in Pulling Mode

“…This does not exhaust the possible bifurcations. From the theory of dynamical systems we know that systems like the one in Figure 1b exhibit a multitude of dynamical regimes that occur in different parameter regions and, consequently, different bifurcations may occur [13][14][15]. In particular, period-doubling bifurcations, which generally happen in Arnold's tongues for strong amplitude of the driving signal, are noteworthy because in the limit of their sequence a chaotic behavior occurs.…”

“…In particular, period-doubling bifurcations, which generally happen in Arnold's tongues for strong amplitude of the driving signal, are noteworthy because in the limit of their sequence a chaotic behavior occurs. The structure of the bifurcation diagrams, the possible synchronization regimes, and the connection between desynchronization and chaos are reported elsewhere, together with the study of chaos in terms of the topological entropy [13][14][15][16][17].…”

“…However, to find the spectrum components of ( ) V t under the approximation 0 v   , we need to find the spectrum of exp[ ( )] i t  , according to Equation (9). To this ends, it is convenient to use the relationship (15) …”

“…This result [9] has been reported in literature to explain experimental observations of pulling in microwave solid-state oscillators [10], in a unijunction transistor based oscillator [11], and in many papers dealing with the study of plasma instabilities and with periodically driven oscillating plasma systems (see [11,12], and references therein). These systems are well modeled by the van der Pol equation and exhibit a variety of dynamical phenomena observed in forced oscillators of van der Pol type [13][14][15]. In particular, mode locking and periodic pulling, bifurcations between quasi-periodic and frequency entrained states have been observed, as well as perioddoubling bifurcations as a route to deterministic chaos [12], for which the study of chaotic dynamics and the derivation of lower bounds on their topological entropy is yet an attractive problem [13][14][15][16][17].…”

“…These systems are well modeled by the van der Pol equation and exhibit a variety of dynamical phenomena observed in forced oscillators of van der Pol type [13][14][15]. In particular, mode locking and periodic pulling, bifurcations between quasi-periodic and frequency entrained states have been observed, as well as perioddoubling bifurcations as a route to deterministic chaos [12], for which the study of chaotic dynamics and the derivation of lower bounds on their topological entropy is yet an attractive problem [13][14][15][16][17].…”