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


By Steven H. Strogatz

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At the heart of the universe is a steady, insistent beat, the sound of cycles in sync. Along the tidal rivers of Malaysia, thousands of fireflies congregate and flash in unison; the moon spins in perfect resonance with its orbit around the earth; our hearts depend on the synchronous firing of ten thousand pacemaker cells. While the forces that synchronize the flashing of fireflies may seem to have nothing to do with our heart cells, there is in fact a deep connection. Synchrony is a science in its infancy, and Strogatz is a pioneer in this new frontier in which mathematicians and physicists attempt to pinpoint just how spontaneous order emerges from chaos. From underground caves in Texas where a French scientist spent six months alone tracking his sleep-wake cycle, to the home of a Dutch physicist who in 1665 discovered two of his pendulum clocks swinging in perfect time, this fascinating book spans disciplines, continents, and centuries. Engagingly written for readers of books such as Chaos and The Elegant Universe, Sync is a tour-de-force of nonfiction writing.



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

Steven Strogatz


To Art Winfree

Mentor, inspiration, friend


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.

For decades, no one could come up with a plausible theory. Even as late as 1961, Joy Adamson, in her sequel to Born Free, marveled at an African version of the same phenomenon, the first ever described on that continent:

a great belt of light, some ten feet wide, formed by thousands upon thousands of fireflies whose green phosphorescence bridges the shoulder-high grass . . . The fluorescent band composed of these tiny organisms lights up and goes out with a precision that is perfectly synchronized, and one is left wondering what means of communication they possess which enables them to coordinate their shining as though controlled by a mechanical device.

By the late 1960s, the pieces of the puzzle began to fall into place. One clue was so obvious that nearly everyone missed it. Synchronous fireflies not only flash in unison—they flash in rhythm, at a constant tempo. Even when isolated from one another, they still keep to a steady beat. That implies that each insect must have its own means of keeping time, some sort of internal clock. This hypothetical oscillator is still unidentified anatomically but is presumed to be a cluster of neurons somewhere in the firefly’s tiny brain. Much like the natural pacemaker in our hearts, the oscillator fires repetitively, generating an electrical rhythm that travels downstream to the firefly’s lantern and ultimately triggers its periodic flash.

The second clue came from the work of the biologist John Buck, who did more than anyone else to make the study of synchronous fireflies scientifically respectable. In the mid-1960s, he and his wife, Elisabeth, traveled to Thailand for the first time, in hopes of seeing the spectacular displays for themselves. In an informal but revealing experiment, they captured scores of fireflies along the tidal rivers near Bangkok and released them in their darkened hotel room. The insects flitted about nervously, then gradually settled down all over the walls and ceiling, always spacing themselves at least 10 centimeters apart. At first they twinkled incoherently. As the Bucks watched in silent wonderment, pairs and then trios began to pulse in unison. Pockets of synchrony continued to emerge and grow, until as many as a dozen fireflies were blinking on and off in perfect concert.

These observations suggested that the fireflies must somehow be adjusting their rhythms in response to the flashes of others. To test that hypothesis directly, Buck and his colleagues later conducted laboratory studies where they flashed an artificial light at a firefly (to mimic the flash of another) and measured its response. They found that an individual firefly will shift the timing of its subsequent flashes in a consistent, predictable manner, and that the size and direction of the shift depend on when in the cycle the stimulus was received. For some species, the stimulus always advanced the firefly’s rhythm, as if setting its clock ahead; for other species, the clock could be either delayed or advanced, depending on whether the firefly was just about to flash, whether it was halfway between flashes, and so on.

Taken together, the two clues suggested that the flash rhythm was regulated by an internal, resettable oscillator. And that immediately suggested a possible synchronization mechanism: In a congregation of flashing fireflies, every one is continually sending and receiving signals, shifting the rhythms of others and being shifted by them in turn. Out of the hubbub, sync somehow emerges spontaneously.

Thus we are led to entertain an explanation that seemed unthinkable just a few decades ago—the fireflies organize themselves. No maestro is required, and it doesn’t matter what the weather is like. Sync occurs through mutual cuing, in the same way that an orchestra can keep perfect time without a conductor. What’s counterintuitive here is that the insects don’t need to be intelligent. They have all the ingredients they need: Each firefly contains an oscillator, a little metronome, whose timing adjusts automatically in response to the flashes of others. That’s it.

Except for one thing. It’s not at all obvious that the scenario can work. Can perfect synchrony emerge from a cacophony of thousands of mindless metronomes? In 1989 my colleague Rennie Mirollo and I proved that the answer is yes. Not only can it work—it will always work, under certain conditions.

For reasons we don’t yet understand, the tendency to synchronize is one of the most pervasive drives in the universe, extending from atoms to animals, from people to planets. Female friends or coworkers who spend a great deal of time together often find that their menstrual periods tend to start around the same day. Sperm swimming side by side en route to the egg beat their tails in unison, in a primordial display of synchronized swimming. Sometimes sync can be pernicious: Epilepsy is caused by millions of brain cells discharging in pathological lockstep, causing the rhythmic convulsions associated with seizures. Even lifeless things can synchronize. The astounding coherence of a laser beam comes from trillions of atoms pulsing in concert, all emitting photons of the same phase and frequency. Over the course of millennia, the incessant effects of the tides have locked the moon’s spin to its orbit. It now turns on its axis at precisely the same rate as it circles the earth, which is why we always see the man in the moon and never its dark side.

On the surface, these phenomena might seem unrelated. After all, the forces that synchronize brain cells have nothing to do with those in a laser. But at a deeper level, there is a connection, one that transcends the details of any particular mechanism. That connection is mathematics. All the examples are variations on the same mathematical theme: self-organization, the spontaneous emergence of order out of chaos. By studying simple models of fireflies and other self-organizing systems, scientists are beginning to unlock the secrets of this dazzling kind of order in the universe.

The question about self-organization that Rennie and I explored was originally posed by Charlie Peskin, an applied mathematician at New York University’s Courant Institute. A soft-spoken man with a neatly trimmed beard and an easy smile, Peskin is one of the world’s most creative mathematical biologists. He loves to use math and computers to plumb the mysteries of physiology: how the molecules and tissues and organs of the body perform their exquisite functions. Whether he’s trying to work out how the retina can detect the dimmest light imaginable, or how molecular motors generate the forces in muscles, his trademark is his versatility. He seems willing to try anything, whatever is required to gain insight. If the math he needs does not exist, he’ll invent it. If the problem requires a supercomputer, he’ll program it. If existing procedures are too slow, he’ll devise faster ones.


On Sale
Feb 14, 2012
Page Count
352 pages
Hachette Books

Steven H. Strogatz

About the Author

Steven Strogatz is the Schurman Professor of Applied Mathematics at Cornell University. His honors include MIT's highest teaching prize, a lifetime achievement award for the communication of mathematics to the general public, and membership in the American Academy of Arts and Sciences. His research on a wide variety of nonlinear systems—from synchronized fireflies to small-world networks—has been featured in the pages of Scientific American, Nature, Discover, Business Week, and The New York Times.

Mitchal Dichter is an instructor of math at Campus Learning Assistance Services at the University of California, Santa Barbara.

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