UK Nonlinear News Review

Synchronization: From Simple to Complex
A. Balanov, N. Janson, D. Postnov, O. Sosnovtseva

Series: Springer Series in Synergetics, 2009
ISBN 978-3-540-72127-7
Hardcover, 426 pages

Reviewed by M.V. Ivanchenko

When a specialist hears the word 'synchronization' it is typically the images of Huygens's pendulums or flashing fireflies that spring to mind. Ironically, few would think of less exotic things like broadcast, mobile phones, or satellite navigation, so ingrained in our daily life are they now. The era of classical theory of synchronization, however, dates back to the time when the mechanics of clocks had long been established but nobody yet realized it had anything to do with the animate nature. Much forgotten now, it was driven by the demand of electronic engineering to design and control stable frequency generators in 20-40s of XX. Frequency and phase locking phenomena were laid in the functional basis of analogue receivers and transmitters, and their digital successors stay indispensable today. Noise robustness and signal processing challenges led to complex filters and the development of a profound theory to describe their dynamics. One cannot but marvel at the masterpieces of mind which the works of Andronov, van der Pol, Vitt, Cartwright, Stratonovich, Tikhonov (to name a few) are.

Historical twists in the studies of synchronization that followed cross-fertilizing insights into synchronization in biology (Winfree, Kuramoto) and synchronization of chaotic oscillations (Fujisaka, Yamada, Pikovsky, Rabinovich, Pecora, Carrol, Kurths) have started an ongoing story. A concurrent rise in computational resources, while having allowed for an unprecedented progress, has also tempted many to forget or ignore the methods and results for synchronization of periodic signals. Fortunately, the capacity of extensive machine-led research and rediscovery seems to come to exhaustion. There is an increasing understanding that little further progress can be achieved without the deep knowledge of the fundamentals, and a striving need for the literature emerges.

This book responds to the demand. The authors, a cohort from the renowned Saratov scientific school in Nonlinear Dynamics, guide a reader through precise classical synchronization analysis to its modern applications making a persuasive case for the classical principles underlying complex collective dynamics. The book is divided in two parts. The first one, 'General mechanisms of synchronization' presents a scrupulous study of the basic mechanisms of synchronization (forced, mutual, homoclinic, on rational frequencies, noise, and chaotic). The authors pledge to avoid 'can be shown/easy to show/can be found in' style and make the methods and proofs delightfully complete and accessible. A special value is added by the topics well-known in Russian Nonlinear Dynamics community, but published in English for the first time. Experiment-minded will enjoy oscilloscope pictures giving a second breath before plunging back into formulae. The second part, 'Case studies in synchronization', employs the theory to describe particular classes of systems and couplings. It demonstrates how principles of synchronization can be implemented in a full and rigorous analysis of such complex systems as neural ensembles, microbiological oscillators, kidney autoregulatory circuit, where the numerical approach is often assumed to be the only way around. Imaginative funny illustrations throughout the book create a unique atmosphere.

The book takes a virtually empty niche of advanced reference and textbooks in the theory of synchronization, whereas many cover an introductory and intermediate level. In time, it can well take a place among the renowned classics in the field like Synchronization: A Universal Concept in Nonlinear Sciences by Pikovsky, Rosenblum and Kurths and SYNC: The Emerging Science of Spontaneous Order by Strogatz. It can be strongly recommended for postgraduate students or advanced undergrads for personal reading or serve a basic textbook for an elective, the prerequisite being nonlinear dynamics. It will definitely be a desk book for a specialist who will find it to be a comprehensive directory and a serendipitous source of surprisingly many things to learn. I have already.