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  SYNC

  How Order Emerges from Chaos in the Universe, Nature, and Daily Life

  Steven Strogatz

  Dedication

  To Art Winfree

  Mentor, inspiration, friend

  Contents

  Preface

  Part I: Living Sync

  Chapter One: Fireflies and the Inevitability of Sync

  Chapter Two: Brain Waves and the Conditions for Sync

  Chapter Three: Sleep and the Daily Struggle for Sync

  Part II: Discovering Sync

  Chapter Four: The Sympathetic Universe

  Chapter Five: Quantum Choruses

  Chapter Six: Bridges

  Part III: Exploring Sync

  Chapter Seven: Synchronized Chaos

  Chapter Eight: Sync in Three Dimensions

  Chapter Nine: Small-World Networks

  Chapter Ten: The Human Side of Sync

  Epilogue

  Notes

  Index

  About the Author

  Praise for Sync

  Copyright

  Preface

  AT THE HEART OF THE UNIVERSE IS a steady, insistent beat: the sound of cycles in sync. It pervades nature at every scale from the nucleus to the cosmos. Every night along the tidal rivers of Malaysia, thousands of fireflies congregate in the mangroves and flash in unison, without any leader or cue from the environment. Trillions of electrons march in lockstep in a superconductor, enabling electricity to flow through it with zero resistance. In the solar system, gravitational synchrony can eject huge boulders out of the asteroid belt and toward Earth; the cataclysmic impact of one such meteor is thought to have killed the dinosaurs. Even our bodies are symphonies of rhythm, kept alive by the relentless, coordinated firing of thousands of pacemaker cells in our hearts. In every case, these feats of synchrony occur spontaneously, almost as if nature has an eerie yearning for order.

  And that raises a profound mystery: Scientists have long been baffled by the existence of spontaneous order in the universe. The laws of thermodynamics seem to dictate the opposite, that nature should inexorably degenerate toward a state of greater disorder, greater entropy. Yet all around us we see magnificent structures—galaxies, cells, ecosystems, human beings—that have somehow managed to assemble themselves. This enigma bedevils all of science today. Only in a few situations do we have a clear understanding of how order arises on its own. The first case to yield was a particular kind of order in physical space involving perfectly repetitive architectures. It’s the kind of order that occurs whenever the temperature drops below the freezing point and trillions of water molecules spontaneously lock themselves into a rigid, symmetrical crystal of ice. Explaining order in time, however, has proved to be more problematic. Even the simplest possibility, where the same things happen at the same times, has turned out to be remarkably subtle. This is the order we call synchrony.

  It may seem at first that there’s little to explain. You can agree to meet a friend at a restaurant, and if both of you are punctual, your arrivals will be synchronized. An equally mundane kind of synchrony is triggered by a reaction to a common stimulus. Pigeons startled by a car backfiring will all take off at the same time, and their wings may even flap in sync for a while, but only because they reacted the same way to the same noise. They’re not actually communicating about their flapping rhythm and don’t maintain their synchrony after the first few seconds. Other kinds of transient sync can arise by chance. On a Sunday morning, the bells of two different churches may happen to ring at the same time for a while, and then drift apart. Or while sitting in your car, waiting to turn at a red light, you might notice that your blinker is flashing in perfect time with that of the car ahead of you, at least for a few beats. Such sync is pure coincidence, and hardly worth noting.

  The impressive kind of sync is persistent. When two things keep happening simultaneously for an extended period of time, the synchrony is probably not an accident. Such persistent sync comes easily to us human beings, and, for some reason, it often gives us pleasure. We like to dance together, sing in a choir, play in a band. In its most refined form, persistent sync can be spectacular, as in the kickline of the Rockettes or the matched movements of synchronized swimmers. The feeling of artistry is heightened when the audience has no idea where the music is going next, or what the next dance move will be. We interpret persistent sync as a sign of intelligence, planning, and choreography.

  So when sync occurs among unconscious entities like electrons or cells, it seems almost miraculous. It’s surprising enough to see animals cooperating—thousands of crickets chirping in unison on a summer night; the graceful undulating of schools of fish—but it’s even more shocking to see mobs of mindless things falling into step by themselves. These phenomena are so incredible that some commentators have been led to deny their existence, attributing them to illusions, accidents, or perceptual errors. Other observers have soared into mysticism, attributing sync to supernatural forces in the cosmos.

  Until just a few years ago, the study of synchrony was a splintered affair, with biologists, physicists, mathematicians, astronomers, engineers, and sociologists laboring in their separate fields, pursuing seemingly independent lines of inquiry. Yet little by little, a science of sync has begun coalescing out of insights from these and other disciplines. This new science centers on the study of “coupled oscillators.” Groups of fireflies, planets, or pacemaker cells are all collections of oscillators—entities that cycle automatically, that repeat themselves over and over again at more or less regular time intervals. Fireflies flash; planets orbit; pacemaker cells fire. Two or more oscillators are said to be coupled if some physical or chemical process allows them to influence one another. Fireflies communicate with light. Planets tug on one another with gravity. Heart cells pass electrical currents back and forth. As these examples suggest, nature uses every available channel to allow its oscillators to talk to one another. And the result of those conversations is often synchrony, in which all the oscillators begin to move as one.

  Those of us working in this emerging field are asking such questions as: How exactly do coupled oscillators synchronize themselves, and under what conditions? When is sync impossible and when is it inevitable? What other modes of organization are to be expected when sync breaks down? And what are the practical implications of all that we’re trying to learn?

  I’ve been fascinated by such questions for 20 years, first as a graduate student at Harvard University and then as a professor of applied math at the Massachusetts Institute of Technology and Cornell University, where I now teach and do research on chaos and complexity theory. My interest in cycles goes back even further than that, to an epiphany I had as a freshman in high school. For one of the first experiments in Science I, Mr. diCurcio gave each of us a stopwatch and a little toy pendulum, a tricky gadget with an extensible arm that could be lengthened or shortened in discrete steps, like one of those old telescopes you see in pirate movies. Our assignment was to clock the pendulum’s period—the time it takes for one swing back and forth—and to figure out how its period depends on its length: Does a longer pendulum swing faster, slower, or stay the same? To find out, we set our pendulums to the shortest length, timed its period, and plotted the result on a piece of graph paper. Then we repeated the experiment for progressively longer pendulums, always stretching the arm one click at a time. As I drew the fourth or fifth dot on the graph paper, it suddenly dawned on me that a pattern was emerging: The dots were falling on a parabolic curve. The same parabolas that I was learning about in Algebra II were secretly governing the motions of these pendulums. An enveloping sensation of wonder and fear came over me. In that moment of revelation, I became aware of a hidden but beautiful world that can be seen only through mathematics. It was a moment
from which I have never really recovered.

  Thirty years later, I’m still captivated by the mathematics of nature, especially as manifested by things that move in cycles, like the periodic swaying of the pendulum. But instead of a single cycle, my research has taken me to the study of many of them working together all at once—to the study of coupled oscillators. My training leads me to make simple models, to replace the bewildering complexity and richness of real fireflies or superconductors with idealized sets of equations that mimic their group behavior. I try to use calculus and computers to see how order emerges from chaos. What makes these puzzles so much fun is that they lie at the edge of known mathematics. Two coupled oscillators would be no challenge—their behavior has been understood since the early 1950s. But for questions involving hundreds or thousands of oscillators, we’re still in the dark. The nonlinear dynamics of systems with that many variables is still beyond us. Even with the help of supercomputers, the collective behavior of gigantic systems of oscillators remains a forbidding terra incognita.

  Still, over the past decade, thanks to the combined efforts of mathematicians and physicists around the world, one special case has finally been worked out, opening the door to a deeper understanding of sync. If we assume that all the oscillators in a given group are nearly identical, and that they are all coupled equally to one another, the dynamics become mathematically tractable. In Parts I and II of this book, I tell the story of how my colleagues and I solved this class of theoretical problems, and what their solutions imply for sync in the real world: in Part I for living oscillators (cells, animals, and people) and then in Part II with reference to inanimate oscillators (pendulums, planets, lasers, and electrons). Part III deals with the frontiers of sync, when we cast aside our earlier simplifying assumptions. This realm is still largely unexplored, and includes situations where the oscillators are replaced by chaotic systems, or where they are coupled in less symmetrical ways—to their neighbors in three-dimensional space, or in intricate networks that transcend geography.

  Sync is an attempt to synthesize a vast body of knowledge on this subject created by scientists working across disciplines, continents, and centuries. The science needed to understand sync draws on the work of some of the greatest minds of the twentieth century, many of whom are household names and others who should be—the physicists Albert Einstein, Richard Feynman, Brian Josephson, and Yoshiki Kuramoto; the mathematicians Norbert Wiener and Paul Erdös; the social psychologist Stanley Milgram; the chemist Boris Belousov; the chaos theorist Edward Lorenz; and the biologists Charles Czeisler and Arthur Winfree.

  My own research runs through the story, not because I have any illusions about my place in history, but because I want to give a feel for what it’s like to be working in the trenches of science—the blind alleys, the twists and turns, the exhilaration of discovery, the metamorphosis from student to colleague to mentor. To convey the vitality of mathematics to a broad spectrum of readers, I’ve avoided equations altogether, and rely instead on metaphors and images from everyday life to illustrate the key ideas.

  My hope is that you’ll come to share some of my excitement about the breathtaking diversity of synchronization in the natural world, and the power of mathematics to explain it. Sync is both strange and beautiful. It is strange because it seems to defy the laws of physics (though in fact it relies on them, often in curious ways). It is beautiful because it results in a kind of cosmic ballet that plays out on stages that range from our bodies to the universe as a whole. And it is also critically important. Our basic understanding of sync has already spawned such technological wonders as the global positioning system; the laser; and the world’s most sensitive detectors, used by doctors to pinpoint diseased tissues in the brains of epileptics without the need for surgery, by engineers to search for tiny cracks in airplane wings, and by geologists to locate oil buried deep underground. By investigating what happens when sync unravels, mathematicians are helping cardiologists track down the cause of fibrillation, a deadly arrhythmia that kills hundreds of thousands of people every year, suddenly and without warning, even those with no history of heart disease. And this is just a sample of what we are able to do today, thanks to our growing but still rudimentary knowledge of sync.

  I am deeply grateful for the opportunity to have worked with so many brilliant and creative minds throughout my career. The research described here was a joint effort with my advisers Art Winfree, Richard Kronauer, Chuck Czeisler, and Nancy Kopell; my collaborators Rennie Mirollo, Paul Matthews, Kurt Wiesenfeld, Jim Swift, Kevin Cuomo, Al Oppenheim, and Tim Forrest; and my former students Shinya Watanabe and Duncan Watts. Thanks for being such wonderful companions on our journeys into the wilds of sync.

  Other scientists helped improve the book in various ways. Jack Cowan shared his affectionate memories of Norbert Wiener at MIT in the late 1950s and enlightened me with the untold but very human story behind the double-dip spectrum. Lou Pecora provided a blow-by-blow account of how he and Tom Carroll were led to the discovery of synchronized chaos. Jim Thorp answered my questions about the power grid with his usual wisdom and good humor. Cedric Langbort kindly translated Huygens’s correspondence about the sympathy of clocks. Joe Burns, Erik Herzog, Chris Lobb, Charlie Marcus, Raj Roy, and Joe Takahashi offered insightful comments on early drafts of the manuscript. Margy Nelson prepared the illustrations with her distinctive blend of scientific judgment and artistic flair. I’m especially grateful to Art Winfree for sharing his playfulness and his mastery of sync, and, above all, for his heroic and amazingly generous effort in reading the manuscript from cover to cover, even under the most difficult circumstances.

  Thank you to Lindy Williams, Stephen Tien, Herbert Hui, Tom Gilovich, and all my other friends who so patiently endured my tribulations in the early stages; Karen Dashiff Gilovich, who helped me find my voice; and Alan Alda, a terrifically stimulating partner in brainstorming sessions, who taught me a lot about how to approach the creative process. (Though I never did manage to follow his best piece of advice, about writing the first draft in one long, happy belch. Maybe next time.)

  My colleagues at Cornell, especially Richard Rand and my department chairman, Tim Healey, have provided encouragement and support throughout the exhausting process of writing this book and have been patient with me whenever my mind seemed to be elsewhere. Thanks for being so understanding.

  My literary agents Katinka Matson and John Brockman have been enthusiastic and helpful at every turn. John suggested the main title for the book within a millisecond of hearing my description of it. Katinka gently coached me through all aspects of the book-writing process, from proposal to publication.

  A writer could not ask for a better publication team than the staff at Hyperion Books. In particular, editorial assistant Kiera Hepford was always gracious, upbeat, and efficient. Art director Phil Rose designed a cover that captures the essence of sync memorably and beautifully. And thanks especially to my editor, Will Schwalbe, whose keen eye, good taste, and sense of structure improved the book in so many ways, and whose unflagging excitement about this project spurred me on when I needed it most.

  Thanks to my family for their love and encouragement, and especially to my dad, who has—as always—been on my side, quietly cheering, smiling, urging me on. The incredible selflessness of my mother-in-law, Shirley Schiffman, made it possible for me to work for long stretches without feeling guilty about neglecting my baby girls. Thank you to my daughters: Leah, for bringing me back down to earth by being a toddler; and Joanna, for not being born too early or too late. My wife, Carole, has shown her love in countless ways—listening, reading, coaxing, forgiving, teaching me how to create, how to loosen up, how to let go. Her generosity of spirit gave me the freedom to be consumed by a sometimes needy, always present obsession.

  Finally, thank you to the citizens of the United States for your trust and farsightedness. By supporting the American research enterprise through agencies like the National Science Foundation, your taxes
give scientists the most precious gift we could hope for—the chance to follow our imaginations wherever they may lead. I hope you take as much pleasure in our discoveries as we do.

  Part I

  Living Sync

  • One •

  Fireflies and the Inevitability of Sync

  “Some twenty years ago I saw, or thought I saw, a synchronal or simultaneous flashing of fireflies. I could hardly believe my eyes, for such a thing to occur among insects is certainly contrary to all natural laws.”

  SO WROTE PHILIP LAURENT IN THE JOURNAL Science in 1917, as he joined the debate about this perplexing phenomenon. For 300 years, Western travelers to Southeast Asia had been returning with tales of enormous congregations of fireflies blinking on and off in unison, in displays that supposedly stretched for miles along the riverbanks. These anecdotal reports, often written in the romantic style favored by authors of travel books, provoked widespread disbelief. How could thousands of fireflies orchestrate their flashings so precisely and on such a vast scale? Now Laurent felt certain he had solved the enigma: “The apparent phenomenon was caused by the twitching or sudden lowering and raising of my eyelids. The insects had nothing whatsoever to do with it.”

  In the years between 1915 and 1935, Science published 20 other articles on this mysterious form of mass synchrony. Some dismissed the phenomenon as a fleeting coincidence. Others ascribed it to peculiar atmospheric conditions of exceptional humidity, calm, or darkness. A few believed there must be a maestro, a firefly that cues all the rest. As George Hudson wrote in 1918, “If it is desired to get a body of men to sing or play together in perfect rhythm they not only must have a leader but must be trained to follow such a leader. . . . Do these insects inherit a sense of rhythm more perfect than our own?” The naturalist Hugh Smith, who had lived in Thailand from 1923 to 1934 and witnessed the displays countless times, wrote in exasperation that “some of the published explanations are more remarkable than the phenomenon itself.” But he confessed that he too was unable to offer any explanation.