Chapter 4
Life's Ups and Downs: Chaos in Ecology and Population Biology
Ravenous fish and tasty plankton. Rain forests dripping with nameless reptiles, birds gliding under canopies of leaves, insects buzzing like electrons in an accelerator. The world makes a messy laboratory for ecologists, a cauldron of five million interacting species. Or is it fifty million? Ecologists do not actually know.
Mathematically inclined biologists built a discipline that stripped away the noise and color of real life, treating populations as dynamical systems. They used mathematical models but always knew these were thin approximations of the seething real world-a perspective that allowed them to see importance in what mathematicians considered mere oddities.
Unlike physicists who could derive equations from first principles, biologists had to gather data and find equations that produced similar output. What happens if you put one thousand fish in a pond with limited food? What if you add fifty sharks? Population biology learned to predict how predators interact with prey, how population density affects disease spread.
Early ecologists used the logistic difference equation: xnext = rx(1-x), where r represents growth rate and (1-x) keeps growth within bounds. Australian ecologist W.E. Ricker applied it to fisheries in the 1950s. Researchers understood that different growth-rate parameters affected a population's ultimate destiny, typically reaching equilibrium after some oscillation.
But occasionally, when using higher parameters, they must have seen chaos-though they likely dismissed erratic number sequences as calculation errors. Most importantly, ecologists assumed erratic results meant their calculations were wrong. The stable solutions were the interesting ones. Order was its own reward. No one wanted to waste time on equations producing no stability, especially when these models were already vast oversimplifications of reality.
James Yorke, a mathematician with philosophical inclinations at the University of Maryland, discovered Lorenz's 1963 paper through a fluid dynamicist colleague in 1972. Yorke recognized that physicists and mathematicians spoke different languages but saw that Lorenz's paper bridged this gap with an example physicists could understand.
Yorke believed scientists had learned not to see chaos. In daily life, sensitive dependence on initial conditions is everywhere-a man leaving home thirty seconds late might miss a bus and then a train, or avoid being hit by a flowerpot only to be run over by a truck. But science taught students to focus on solvable differential equations that behaved predictably.
As Yorke explained, "If you could write down the solution to a differential equation, then necessarily it's not chaotic." Textbooks showed only the rare nonlinear systems that yielded to linear approximations. Scientists dismissed chaotic systems as aberrations, forgetting that the orderly, linear systems were actually the aberrations. As Stanislaw Ulam remarked, calling chaos study "nonlinear science" was like calling zoology "the study of non-elephant animals."
Robert May's groundbreaking work in the 1970s revealed how simple equations could produce remarkably complex behavior. His bifurcation diagram plotted parameter values horizontally and population values vertically, showing how populations first reached equilibrium, then began oscillating between two values in a two-year cycle when the parameter passed a critical point. As the parameter increased further, the points doubled again and again in a cascade-from 2 to 4 to 8 to 16 to 32-with the doublings coming faster until suddenly breaking off at a "point of accumulation" where periodicity gave way to chaos.
In his "messianic" 1976 Nature paper, May argued that chaos should be taught widely, contending that standard scientific education with its focus on linear mathematics misled scientists about our overwhelmingly nonlinear world. "Not only in research, but also in everyday politics and economics, we would all be better off if more people realized that simple nonlinear systems do not necessarily possess simple dynamical properties."
Chapter 5
A Geometry of Nature: Fractals and the Hidden Order of the World
Benoit Mandelbrot's vision of reality began taking shape in 1960 when he recognized a familiar pattern on Harvard economist Hendrik Houthakker's blackboard. What Mandelbrot thought was his own income distribution diagram turned out to be eight years of cotton prices, yet both exhibited the same distinctive pattern-one that defied the bell-shaped curves economists typically used to model variation.
While economists assumed commodity prices followed two different patterns-orderly long-term trends driven by real economic forces and random short-term fluctuations-Houthakker's cotton price data refused to fit this model. The distribution showed too many large jumps compared to small changes, creating a "long tail" that couldn't be explained by standard Gaussian distributions.
When Mandelbrot analyzed decades of cotton prices at IBM's research center, he discovered something remarkable: the sequence of price changes was independent of scale. Daily and monthly price changes, when properly analyzed, produced identical curves. Despite sixty years of tumultuous history including two World Wars and the Depression, the degree of variation remained constant.
This unexpected order within disorder suggested a fundamental pattern that transcended specific fields. The same scaling laws that governed personal income distribution somehow applied to cotton prices. Though Mandelbrot's economic background was minimal, he recognized this scaling phenomenon as a signature worth pursuing across disciplines.
Mandelbrot's intellectual journey was shaped by his life as a refugee. Born to a Lithuanian Jewish family in Warsaw in 1924, he fled to Paris in 1936, then escaped the Nazis again during the war, surviving as an apprentice toolmaker despite his conspicuous height and education. Despite irregular schooling, he passed the rigorous entrance exams for France's elite schools through his remarkable geometric intuition-visualizing mathematical problems as shapes he could transform.
At IBM, Mandelbrot applied his unconventional thinking to practical problems, beginning with telephone line noise. Engineers were puzzled by transmission errors that clustered unpredictably. Mandelbrot discovered these errors followed a pattern resembling the Cantor set-a mathematical construction where removing the middle third of a line segment and repeating this process infinitely creates a "dust" of points that are infinitely many yet infinitely sparse.
Turning to coastlines, Mandelbrot posed his famous question: "How Long Is the Coast of Britain?" The answer depended on measurement scale-using smaller measuring tools revealed more detail, making the measured length increase without limit. This led Mandelbrot to explore fractional dimensions as a way to characterize irregular shapes that Euclidean geometry couldn't adequately describe.
Mandelbrot revolutionized our understanding of dimension by showing it depends on perspective and scale. His breakthrough was introducing fractional dimensions-a mathematical tool measuring qualities like roughness, brokenness, and irregularity. These dimensions quantify how efficiently objects fill space: a Koch curve with infinite length in finite area has dimension 1.2618-more than a line but less than a plane.
The key insight was self-similarity-symmetry across scales with pattern inside pattern. Using computers to visualize these previously "monstrous" shapes, Mandelbrot created intuition for forms that hand-drawing mathematicians couldn't fully explore. He named these new forms: ropes, sheets, sponges, foams, curds and gaskets-shapes that revealed "regular irregularity" across different scales.
While mathematicians initially dismissed these as pathological curiosities unlike anything in nature, Mandelbrot saw connections to real-world patterns. The Eiffel Tower, with its branching network of increasingly finer beams, demonstrates the engineering principle behind the Sierpinski gasket-removing weight without sacrificing structural strength. These shapes, once considered mathematical aberrations, proved to be sophisticated models of natural forms.
Chapter 6
Strange Attractors: The Patterns Within Turbulence
Turbulence represented one of physics' most profound and intractable problems. The greatest minds had contemplated it-how smooth flow breaks into whorls and eddies, how energy transfers from large-scale motions to small. A story circulated about Werner Heisenberg declaring on his deathbed that he would have two questions for God: why relativity and why turbulence-adding that God might have an answer only for the first.
Fluid dynamics had fallen from the front lines of physics to mere engineering. The practical interest in turbulence was usually one-sided: make it go away. While sometimes desirable (inside jet engines for efficient burning), turbulence typically meant disaster-destroying lift on aircraft wings and creating drag in pipes. Despite vast resources devoted to designing aircraft, engines, and submarines, the fundamental question remained: how does flow change from smooth to turbulent?
Theorists and experimenters maintained an uneasy relationship. Theorists worked in pristine conditions using only their brains; experimenters were craftsmen who needed assistants, machinists, and equipment. Though they needed each other, prestige accumulated on the theorist's side. Harry Swinney exemplified the experimental approach. As a graduate student at Johns Hopkins, he chose experimenting with phase transitions over particle physics after seeing that graduate students in the latter field merely wrote computer programs or soldered equipment.
Swinney and Gollub's desktop apparatus used a glass cylinder the size of a skinny tennis ball can with an inner steel cylinder, leaving just one-eighth inch between for water. They used laser light to measure the fluid's behavior-a technique called laser doppler interferometry-with data processed by computer, rare for a tabletop experiment in 1975.
Their goal was to confirm Landau's theory about turbulence onset. Initially, they found a well-defined transition as predicted, but at the next transition, the expected Landau sequence broke down. Instead of adding new frequencies gradually, the flow jumped directly to a chaotic state with no distinguishable cycles.
David Ruelle, a Belgian physicist at the Institut des Hautes Etudes Scientifiques outside Paris, had developed an alternative to Landau's theory of turbulence. In 1971, Ruelle and Dutch mathematician Floris Takens published "On the Nature of Turbulence," challenging Landau's view. Instead of turbulence requiring infinite overlapping motions, they proposed just three independent motions could produce full turbulent complexity. Their paper introduced the seductive concept of a "strange attractor"-a term whose origin neither author could precisely remember claiming.
Phase space became one of modern science's most powerful inventions-a way to turn numbers into pictures by abstracting essential information from any moving system. In phase space, a system's complete state at any instant collapses to a single point, with the system's evolution traced by that point's movement over time.
For a simple pendulum, phase space needs just two dimensions-position and velocity. A frictionless pendulum traces a repeating loop in phase space, while one with friction spirals inward toward a fixed-point attractor (position 0, velocity 0). These simple attractors-fixed points and limit cycles-represented steady states or repeating behaviors.
Traditional physics recognized only fixed points and limit cycles as attractors. But turbulence displayed every possible rhythm simultaneously, like white noise. Could deterministic equations produce such behavior? Ruelle and Takens proposed a new kind of attractor: stable, low-dimensional, yet never repeating itself. Geometrically, this required an infinitely long line confined to finite space-essentially, a fractal.
By the mid-1970s, strange attractors remained theoretical constructs-mathematically fascinating but experimentally elusive. When David Ruelle visited Gollub and Swinney's City College laboratory in 1974, they had only a tenuous connection between theory and experiment: mathematical ideas that were philosophically bold but technically uncertain, and cylinder experiments showing turbulent fluid behavior contradicting Landau's established theory.
Strange attractors promised to reveal fundamental properties of chaos-sensitive dependence on initial conditions, mixing behavior relevant to engineering applications, and fractal dimensions. But scientists still didn't know how to measure these properties or apply them to practical problems.
Chapter 7
Universality: The Surprising Order in Disorder
At a rushing stream near a waterfall, Mitchell Feigenbaum demonstrates an unusual technique for perceiving turbulent water patterns-rapidly turning his head from side to side to discern the structure of the churning surface. For someone with mathematical training, this chaotic natural phenomenon, like clouds with their "puffs on top of puffs," reveals patterns that resonate viscerally.
When Feigenbaum arrived at Los Alamos in 1974, he sought to transform the cliched concept of "order in chaos" into practical scientific framework. Finding his physics education useless for nonlinear problems, Feigenbaum turned to the simple quadratic equation used by Robert May in population biology. This equation-essentially a parabola-became a feedback loop where output values were fed back as inputs, creating a system that could stabilize, oscillate between values, or behave chaotically depending on a parameter value.
In summer 1975, Feigenbaum heard mathematician Steve Smale discuss the quadratic equation, particularly the boundary between periodic and chaotic behavior. Using his HP-65 calculator, Feigenbaum began exploring this transition region-the same boundary between smooth flow and turbulence in fluids. He focused on the cascade of period-doublings where cycles split into longer cycles before reaching chaos.
The calculator's slowness proved fortunate-while manually recording numbers and waiting for calculations, Feigenbaum discovered an unexpected pattern: the period-doublings were converging geometrically at a constant rate of 4.669, suggesting some hidden scaling pattern in the equation. This ratio appeared in completely different equations, revealing a universal constant that transcended specific mathematical forms.
When Feigenbaum applied his approach to a trigonometric function (x(t+1) = r sin x(t)), he discovered the identical convergence rate: 4.669. This coincidence seemed impossible-two completely different equations producing exactly the same numerical constant. Testing other functions that underwent bifurcations on the path to disorder, every one produced the same number.
The whole tradition of physics was unraveling. "You isolate the mechanisms and then all the rest flows," Feigenbaum said. "That's completely falling apart. Here you know the right equations but they're just not helpful." The microscopic pieces couldn't be extended to predict long-term behavior. This discovery meant that the specific details of functions-whether sine functions or parabolas-were irrelevant. Nature had briefly revealed unexpected order, but the question remained: why?
Suddenly you could see where the different frequencies in turbulence came from. Universality made Feigenbaum's theory both beautiful and useful-by solving simple problems, physicists could solve much harder ones with the same answers. Yet this universality that made the theory valuable also made it difficult for physicists to accept, as it claimed different systems would behave identically.
Despite later recognition through prizes and awards, Feigenbaum kept his rejection letters in a desk drawer. For two years, top academic journals deemed his breakthrough work unfit for publication. Though science isn't supposed to be subjective, one editor later admitted rejecting what became a turning point paper. Meanwhile, Feigenbaum's ideas spread through lectures and preprints, with hundreds requesting photocopies of his unpublished papers.
In summer 1977, physicists Joseph Ford and Giulio Casati organized the first conference on chaos in Como, Italy. About a hundred scientists attended, mostly physicists but also curious researchers from other fields. Feigenbaum's universality model provided the first clear, intuitive approach to chaos that everyone could understand. Scientists from disciplines ranging from astronomy to zoology discovered they weren't alone in their "eccentric" research interests. Many were "weepingly grateful" to find others working on similar complex phenomena.
Chapter 8
The Experimenter: Testing Chaos in the Laboratory
The experimenter's greatest joy comes when something conceived in the mind corresponds exactly to nature-"a great shock, and a great, great joy," as Leo Kadanoff put it.
At Ecole Normale Superieure, colleagues worried that distinguished low-temperature physicist Albert Libchaber was wasting resources on a seemingly trivial experiment. Born to Polish Jews in pre-war Paris, Libchaber survived the Holocaust by hiding in the countryside. He rose through French academia with unquestioned brilliance, though colleagues sometimes found him eccentric-a Jewish mystic among rationalists, a Gaullist among Communists, obsessed with old scientific texts. In 1977, he and engineer Jean Maurer began building an experiment to reveal the onset of turbulence-a tiny "Helium in a Small Box" setup small enough to carry in a matchbox.
Libchaber's laboratory near Pasteur's old workspace was a typical physics mess, but amid the chaos stood his precisely engineered apparatus. The experiment featured a minuscule fluid cell of stainless steel with copper bottom plate and sapphire crystal top, all housed in vacuum containers and liquid nitrogen baths. His plan was to create Rayleigh-Benard convection in liquid helium by heating the bottom plate-the classic system Lorenz had modeled, though Libchaber wasn't yet aware of Lorenz or Feigenbaum's work.
The scientific community's skepticism serves a purpose-"Science was constructed against a lot of nonsense," as Libchaber acknowledged. When colleagues called him a mystic, it wasn't always complimentary. Yet despite his experimental discipline, Libchaber had an intuitive feeling for the abstract concept of flow-shape plus change, motion plus form. He embraced Platonic ideas about hidden universal forms, noting how natural shapes like leaves or flames follow limited patterns regardless of their specific material manifestations.
Libchaber's experiment was a masterpiece of miniature precision. Using liquid helium for its exceedingly low viscosity, he created a millimeter-wide cell where convection could be triggered with just a thousandth of a degree temperature difference. The tiny size was crucial-in a larger box, even smaller temperature variations would trigger motion, making control impossible. He embedded microscopic temperature probes in the sapphire upper surface, recording output continuously.
As Libchaber increased heating in his experiment, the system revealed increasingly complex behavior. After the first bifurcation created a steady two-second wobble, the next bifurcation caused temperature to split into two different maximums and minimums-a "metawobble." On spectrum diagrams, this appeared as a new frequency at exactly half the original frequency, repeating every four seconds. With further bifurcations, new frequencies appeared at double the previous periods, creating a pattern resembling "a picket fence with alternating short and tall pickets."
Physicists initially resisted connecting simple mathematical maps to complex fluid systems. As Pierre Hohenberg explained, most physicists viewed Feigenbaum's work on maps as merely a game, too remote from physical systems with potentially infinite dimensions. The breakthrough came when experimental results matched theoretical predictions-the "miracle" that complex systems could be understood through models with few degrees of freedom.
When Feigenbaum visited Libchaber in Paris, they examined the experimental apparatus together before walking through Paris streets searching for the perfect cup of coffee. Libchaber was struck by how young and lively the theorist was-a surprising contrast to his expectations.
The connection between simple maps and complex fluid flow seemed miraculous to scientists like Jerry Gollub. Within years, this "miracle" was replicated across numerous laboratory systems-fluid cells with water and mercury, electronic oscillators, lasers, and chemical reactions. Theorists expanded Feigenbaum's techniques to discover other mathematical routes to chaos like intermittency and quasiperiodicity, which also proved universal in both theory and experiment.
Chapter 9
Inner Rhythms: Chaos in Biology and Medicine
The chaos revolution expanded into biology and medicine, challenging traditional scientific models. As John von Neumann noted, sciences mainly create mathematical models that, with verbal interpretations, describe observed phenomena-their justification being simply that they work.
At the first major conference on chaos in biology and medicine in 1986, Bernardo Huberman presented his work on schizophrenic eye movements. Since collaborating with the Santa Cruz group, Huberman had continued chaos research at Xerox's Palo Alto Research Center. He explained how schizophrenics' eyes jump disruptively, creating constant extraneous movements. Rather than assuming random brain disturbances caused this behavior, Huberman created a simple nonlinear model with terms for amplitude, frequency, inertia, damping, and error correction. His equation produced both orderly and chaotic behavior depending on parameter values, suggesting the erratic eye movements might stem from excessive nonlinearity in an otherwise normal system rather than external noise.
In the 1980s, chaos theory birthed a new kind of physiology built on mathematical tools that could help understand complex systems independent of local detail. Scientists began viewing the body as a place of motion and oscillation, developing methods to listen to its variegated drumbeat. They found rhythms invisible in static samples, studying chaos in respiratory disorders, feedback mechanisms in blood cell control, periodicity in cell growth, and multidimensional approaches to drug prescriptions. But the heart's animated rhythms, stable or unstable, dominated this new physiology, precisely measuring the difference between life and death.
Even David Ruelle, typically focused on mathematical formalism, speculated about chaos in the heart-"a dynamical system of vital interest to every one of us." He suggested that "great medical benefit might be derived from computer studies of a realistic mathematical model which would reproduce the various cardiac dynamical regimes." This challenge was taken up by research teams across North America, who began examining the rich variety of heart arrhythmias through the lens of chaos theory.
Leon Glass of McGill University, trained in physics and chemistry before turning to cardiac arrhythmias, observed that physicians typically diagnose irregular heartbeats by pattern recognition, matching electrocardiogram strips to textbook examples. "They really don't analyze in detail the dynamics of these rhythms," he noted. "The dynamics are much richer than anybody would guess from reading the textbooks." This pattern-matching approach obscured deeper dynamical causes that chaos theory could potentially reveal.
Ventricular fibrillation causes hundreds of thousands of sudden deaths yearly in the United States alone. While many cases have identifiable triggers-arterial blockage, cocaine use, stress-in many others the onset remains mysterious. Paradoxically, patients with seemingly healthy hearts are actually more likely to suffer new attacks than those with visible damage. In fibrillation, instead of contracting and relaxing rhythmically, the heart muscle writhes uncoordinated like "a bag of worms," unable to pump blood effectively.
Arthur Winfree brought a rare geometric approach to biological problems. Beginning with circadian rhythms in the early seventies, he applied nonlinear dynamics to biological clocks rather than waiting for biochemists to discover the underlying mechanisms. His mathematical approach examined the qualitative shape of data rather than quantitative details, allowing him to identify critical patterns in biological systems.
Winfree approached biological clocks through complex systems theory rather than biochemistry. He studied mosquitoes, whose natural cycle runs about twenty-three hours when isolated from environmental cues. By analyzing their responses to light stimuli topologically, he discovered a singularity in the geometry-a special point where a precisely timed burst of light could completely break down a mosquito's biological clock, leaving it in perpetual disorientation.
Physiologists began speaking of "dynamical diseases"-disorders where systems that normally oscillate stop or begin oscillating unexpectedly, or non-oscillating systems suddenly start oscillating. These include breathing disorders like panting and infant apnea, blood disorders including forms of leukemia, and possibly even schizophrenia and depression.
Harvard's Ary Goldberger proposed that healthy dynamics were characterized by fractal physical structures, like branching networks in lungs and heart, allowing a wide range of rhythms. These "information-rich" fractal processes with broadband spectra contrasted with the "monotonous, repetitive sequences" of periodic states. Treatment might depend on broadening a system's "spectral reserve"-its ability to range across many frequencies without falling into locked periodicity.
Chapter 10
Chaos and Beyond: A New Vision of Science
Two decades before, Edward Lorenz was studying the atmosphere, Michel Henon the stars, Robert May the balance of nature. Benoit Mandelbrot was an unknown IBM mathematician, Mitchell Feigenbaum an undergraduate, and Doyne Farmer just a boy in New Mexico. Scientists then shared unspoken beliefs: simple systems behave simply; complex behavior requires complex causes; and different systems behave differently. But in twenty years, everything changed. Scientists discovered that simple systems produce complex behavior, complex systems produce simple behavior, and most importantly, the laws of complexity hold universally across disciplines. For many scientists, chaos represented the end of reductionism-the futility of studying parts in isolation from the whole.
Joseph Ford of Georgia Tech remembered lecturing about chaos in the Duffing equation to a thermodynamics group in the 1970s, only to face hostile resistance. The audience was outraged, claiming generations had studied this equation without seeing such behavior. Scientists resisted the notion that nature could be so complicated, clinging to familiar linear approaches and avoiding nonlinear problems.
Scientists couldn't agree on the definition of chaos. Philip Holmes called it "complicated, aperiodic, attracting orbits." Hao Bai-Lin: "order without periodicity." Bruce Stewart: "apparently random behavior in simple deterministic systems." Roderick Jensen: "irregular, unpredictable behavior of deterministic, nonlinear dynamical systems." James Crutchfield: "dynamics with positive, but finite, metric entropy." And Joseph Ford, self-proclaimed evangelist: "dynamics freed from the shackles of order and predictability."
The Second Law of Thermodynamics states that entropy must always increase, a rule seemingly without appeal. While this principle has been metaphorically extended to explain social decay and cultural decline, such applications now seem misguided. Despite the universe's movement toward maximum entropy, complexity flourishes. The important creative laws lie beyond thermodynamics, in the realm of chaos theory, which helps explain how purposeless energy flow can produce life and consciousness.
Snowflake formation exemplifies chaos principles. As crystals grow outward from a seed, any boundary portion that gets ahead gains advantage in capturing water molecules-the "lightning-rod effect." The process balances instability (heat diffusion) with stability (surface tension). Though surface tension effects are tiny, they prove crucial at microscopic scales where they amplify ice's natural six-fold molecular symmetry. Each snowflake records its unique atmospheric journey, making combinations virtually infinite-sensitive dependence creating rather than destroying.
"Evolution is chaos with feedback," Joseph Ford declared. The universe contains randomness and dissipation, but randomness with direction produces surprising complexity. And as Lorenz discovered, dissipation actually functions as an agent of order.
"God plays dice with the universe," Ford answered Einstein. "But they're loaded dice. And the main objective of physics now is to find out by what rules were they loaded and how can we use them for our own ends."
William Schaffer, the last student of ecology pioneer Robert MacArthur, found traditional equilibrium-based ecology failing. MacArthur's models assumed natural balance and efficient resource use, but Schaffer discovered nature was more complicated. Chaos seemed "both exhilarating and a bit threatening" as it undermined ecology's most enduring assumptions.
Schaffer applied strange attractors to childhood disease epidemiology, collecting data from New York, Baltimore, Aberdeen, and across England and Wales. His dynamical model resembled a damped, driven pendulum-driven by school-return infections and damped by natural resistance. The model predicted chicken pox would vary periodically while measles would vary chaotically-exactly matching the data. Where traditional epidemiologists saw random noise in measles variations, Schaffer found a strange attractor with fractal dimension of about 2.5.
Even in the 1980s, "chaos theory" sounded oxymoronic-the words didn't seem to belong together. What was strange and alien then has now been internalized by both mainstream science and popular culture. The Butterfly Effect became a cultural cliche, appearing in films like Jurassic Park, entering Bartlett's Quotations, and spawning countless references. Chaos concepts have been adopted by management theorists and postmodern literary critics alike, while fractal geometry has inspired artists and sculptors. Today's scientists know complex systems can behave unpredictably yet remain measurable, and the pioneers of chaos have joined the scientific establishment, collecting honors and recognition for their revolutionary work.
