Synchronization: A Universal Concept in Nonlinear Sciences , Arkady Pikovsky , Michael Rosenblum , and Jürgen Kurths Cambridge U. Press, New York, 2001. $100.00 (411 pp.). ISBN 0-521-59285-2
As budding physicists, we were all fed a steady diet of oscillators. Starting with the simple harmonic motion of a mass on a spring, we soon swallowed a large helping of electrical circuits and resonance, and washed it down with normal modes and beat phenomena. The same food group seemed to be offered at every meal for the rest of our education. Quantum mechanics dished out the harmonic oscillator again, though in barely recognizable form. Electromagnetic theory served up cavity modes; solid-state physics had phonons.
Fair enough. Harmonic oscillators are fundamental, and every physicist needs to master them. But relentless ingestion of purely conservative, purely linear oscillators is now recognized to be unhealthy, because such oscillators are so unrepresentative of many found in nature and technology. To understand the dynamics of lasers or Joseph-son junctions, or heart cells or neurons, one has to come to grips with oscillators that are both nonlinear and nonconservative. In particular, many real oscillators are self-sustained, meaning that they oscillate with a distinctive waveform at a preferred amplitude that reflects a balance between energy inflow and dissipation. (In contrast, harmonic oscillators can cycle at any amplitude, a decidedly nongeneric feature.)
When self-sustained oscillators are coupled, they behave very differently from harmonic oscillators. Instead of beating with an admixture of all the mode frequencies, they often lock onto a single frequency. In other words, they synchronize. For example, our hearts beat on command of the sinoatrial node, a natural pacemaker built out of about 10 000 cells, each an autonomous electrical oscillator. By exchanging ionic currents, all the cells manage to fire in sync and thereby trigger the rest of the heart to contract and pump blood. Inanimate oscillators can also synchronize spontaneously. As Christiaan Huygens discovered in 1665 (while lying ill in his bedroom), two pendulum clocks hanging from the same board will swing in perfect antiphase, thanks to the imperceptible vibrations they induce in their common support. The injection locking of lasers and the synchronous rotation of thousands of generators in the nationwide power grid are modern instances of the same principle.
For many years, the analysis of self-synchronizing systems was handled on a case-by-case basis, with little communication between the disparate fields in which those phenomena arise. Synchronization is the first book to treat this subject in a unified fashion. In an unconventional but excellent pedagogical choice, Arkady Pikovsky, Michael Rosenblum, and Jürgen Kurths begin in part 1 by discussing synchronization without resorting to equations. Using pictures, graphs, history, and conceptual explanations, they clarify its scientific importance through carefully chosen experimental examples from a dazzling array of disciplines. This part of the book is intended to be accessible to scientists in all fields. It succeeds brilliantly.
In part 2, the authors rework the same material from a more mathematical perspective. Again the treatment is clear and logical, organized in ascending order of structural complexity, from a single self-sustained oscillator driven by an external force, to two mutually coupled oscillators, and then on to the effects of noise, the dynamics of oscillatory media, and discrete populations of globally coupled oscillators, in which mean-field methods from statistical mechanics come into play. Much of this part reads like an extended review article, and a very good one at that.
Part 3 is about synchronization in chaotic systems, a hot research topic since 1990, when Lou Pecora and Tom Carroll of the US Naval Research Laboratory showed how to synchronize a pair of identical chaotic systems by driving one copy of the system with a small subset of signals from the other. Pecora and Carroll suggested that one could use such a pair in communication by adopting one system in the pair as a transmitter and the other as a receiver. As an additional benefit, the unique features of chaos (its random appearance and broad spectrum) could be used to enhance the privacy of an encoded message. Subsequent researchers developed that intriguing idea into a working technique in electronic circuits and optical communications, while others found ways to defeat the encryption. Very little of that story is discussed in part 3. I found the omission a bit surprising, given that the authors are experts in the area. Instead, they focus on the mathematics and downplay the applications of synchronized chaos. Fortunately, their treatment of the fundamentals is very well done and more readable than much of the original literature on this still-fashionable subject.
Steven Strogatz, at Cornell University in Ithaca, New York, works on coupled oscillators and complex networks in physics and biology. He is the author of the forthcoming trade book Sync: The Emerging Science of Spontaneous Order (Theia/Hyperion, 2003).
