Chaos: Making a New Science

“As the revolution in chaos runs its course, the best physicists find themselves returning without embarrassment to phenomena on a human scale. They study not just galaxies but clouds.”

Another hallmark of chaos science, the focus on visible and tangible phenomena (rather than quarks or gluons) reveals one of the underlying objectives of this field: to understand how nature works in reality rather than in theory. As such, chaos science employs the skills of experimenters alongside theorists in order to illuminate observable phenomena.

“Implicitly, the mission of many twentieth-century scientists—biologists, neurologists, economists—has been to break their universes down into the simplest atoms that will obey scientific rules. In all these sciences, a kind of Newtonian determinism has been brought to bear.”

Chaos theory disrupted the tendency within science to study the individual components of a system rather than examining the whole. Scientists working within the chaos field are interested in determining the behavior of the whole and how systems relate to other systems. This leads to a greater understanding of the universal laws that seem to govern vastly different systems.

“The Butterfly Effect was the reason. For small pieces of weather—and to a global forecaster, small can mean thunderstorms and blizzards—any prediction deteriorates rapidly. Errors and uncertainties multiply, cascading upward through a chain of turbulent features, from dust devils and squalls to continent-sized eddies that only satellites can see.”

Perhaps the most well-known concept in chaos science, the butterfly effect explains how small events can impact larger systems. It illuminated why the goals of long-term forecasting had continually been stymied. The way a larger system functions depends on these smaller initial conditions. Even the smallest elements within a system cannot be discounted.

“Chaos has become not just theory but also method, not just a canon of beliefs but also a way of doing science.”

This refers to the field’s interdisciplinary tendencies, as well as its dependence on both physical experiments and theoretical concepts. It also refers to the field’s early use of computers to model systems and ideas. For example, the new understanding of geometry, inspired by the Mandelbrot set, relies heavily on computer modeling for its “trial-and-error geometry” (227).

“The tradition of looking at systems locally—isolating the mechanisms and then adding them together—was beginning to break down. For pendulums, for fluids, for electronic circuits, for lasers, knowledge of the fundamental equations no longer seemed to be the right kind of knowledge at all.”

The focus on the local at the expense of the global falls away in the development of chaos science. When examining the function of a system as a whole, one must also examine all of its parts, as well as how those parts interact with each other and within the larger system. Eventually, by investigating entire dynamic systems, chaos scientists observed that the laws governing these disparate systems all followed some natural preferences.

“[James] Yorke’s paper was important on its merits, but in the end its most influential feature was its mysterious and mischievous title: ‘Period Three Implies Chaos.’ His colleagues advised him to choose something more sober, but Yorke stuck with a word that came to stand for the whole growing business of deterministic disorder.”

Yorke’s paper inadvertently gave the new science its name. Still, the name provoked controversy even in the mid-1980s, when Gleick wrote his book. Some scientists objected because the name implies randomness, while they felt that their work revealed significant patterns within nature. Others argued that the name was too limited for the complexity of their work.

“Biologists had overlooked bifurcations on the way to chaos because they lacked mathematical sophistication and because they lacked the motivation to explore disorderly behavior. Mathematicians had seen bifurcations but had moved on. [Robert] May, a man with one foot in each world, understood that he was entering a domain that was astonishing and profound.”

May’s groundbreaking work within the field of ecology exemplifies the deeply interdisciplinary nature of chaos science. He was instrumental in exporting the ideas of chaos theory to other disciplines, like genetics and economics. That is, not only did his own work recognize the necessity of interdisciplinary science but he also recognized that his theories applied to other fields.

“Clouds are not spheres, Mandelbrot is fond of saying. Mountains are not cones. Lightning does not travel in a straight line. The new geometry mirrors a universe that is rough, not rounded, scabrous, not smooth. It is a geometry of the pitted, pocked, and broken up, the twisted, tangled, and intertwined.”

Euclidean geometry represents theoretical and regular geometric shapes. In contrast, Mandelbrot’s geometry engages with the roughness and irregularity of the natural world. Much of traditional science had rejected the irregular or the erratic in favor of a theoretical ideal. Chaos science was invested in the complex shapes and dynamics of the real.

“The patterns that people like Robert May and James Yorke discovered in the early 1970s, with their complex boundaries between orderly and chaotic behavior, had unsuspected regularities that could only be described in terms of the relation of large scales to small. The structures that provided the key to nonlinear dynamics proved to be fractal. And on the most immediate practical level, fractal geometry also provided a set of tools that were taken up by physicists, chemists, seismologists, metallurgists, probability theorists and physiologists. These researchers were convinced, and they tried to convince others, that Mandelbrot’s new geometry was nature’s own.”

The applications of chaos science in numerous other fields became increasingly apparent over time. By engaging with the ways that dynamic systems work as a whole, as well as how dynamic systems function in actual natural systems, chaos science—in this case, Mandelbrot geometry in particular—becomes a method rather than a discipline. Its discoveries apply across scientific disciplines.

“Theorists conduct experiments with their brains. Experimenters have to use their hands, too. Theorists are thinkers, experimenters are craftsmen.”

Despite the dichotomy that this passage conveys (something that chaos science works to unravel), the author points out that, by the 20th century, most scientists were either theorists or experimenters. Mathematicians and theoretical physicists did not engage in the same kind of science as engineers or mechanical physicists did. Chaos science needed both kinds of scientists to advance its understanding of how dynamic systems worked in reality. When Albert Libchaber’s experiments proved the theories of Mitchell Fiegenbaum, the universal understanding of how nature functions increased.

“The picture became quite dramatic: evidence of complete disorder mixed with the clear remnants of order, forming shapes that suggested ‘islands’ and ‘chains of islands’ to these astronomers.”

In investigating the irregularity of orbits, scientists discovered the strange attractor. Theoretically speaking, the paths of certain planetary orbits should result in their flying off into space rather than returning to another cycle. However, the underlying patterns within the apparent disorder reveal that something else is at work. The strange attractor organizes the behavior within the phase space.

“Part of the beauty lay in its universality. [Leo] Kadanoff’s idea gave a backbone to the most striking fact about critical phenomena, namely that these seemingly unrelated transitions—the boiling of liquids, the magnetizing of metals—all follow the same rules.”

In looking at the ways that dynamic systems interact at a holistic level, chaos science reveals universal laws that determine how systems behave. That is, phase transitions of different kinds “follow the same rules” across different scales. This indicates that nature favors particular patterns within seemingly chaotic transitions.

“[Mitchell] Fiegenbaum’s universality was not just qualitative, it was quantitative; not just structural, but metrical. It extended not just to patterns, but to precise numbers. To a physicist, that strained credulity.”

This passage further emphasizes the importance of theory coupled with the function of real numbers. That is, chaos science reveals that actual systems not only behave in universal ways but also that the ways in which these behaviors are universal are actually measurable. This is akin to measuring irregularity, or quantifying the erratic.

“The universality of shapes, the similarities across scales, the recursive power of flows within flows—all sat just beyond reach of the standard differential-calculus approach to equations of change.”

This is what Albert Libchaber’s experiment, “Helium in a Small Box,” attempted to prove. He wanted to show a link between motion—the “flows within flows”—and the idea of universality. That is, he held that motion and form are not mutually exclusive: The moving shape of a flame and the stable shape of a leaf, for example, had an intrinsic commonality.

“According to the new theory, the bifurcations should have produced a geometry with precise scaling, and that was just what Libchaber saw, the universal Feigenbaum constants turning in that instant from a mathematical ideal to a physical reality, measurable and reproducible.”

In his experiment, Libchaber witnessed what the author calls “an infinite cascade, rich with structure” (211). Like Mandelbrot’s geometry, the infinite contains the finite; order exists within disorder. The coalescing of theory and experiment reveals that irregularity and infinity are actually measurable phenomena.

“Unlike the traditional shapes of geometry, circles and ellipses and parabolas, the Mandelbrot set allows no shortcuts. The only way to see what kind of shape goes with a particular equation is by trial and error, and the trial-and-error style brought the explorers of this new terrain closer in spirit to Magellan than to Euclid.”

The role of computers became central to some of the explorations that chaos scientists undertook. The computer could model various inputs to determine the shape of the Mandelbrot set, ad infinitum. In addition, this quotation conflates chaos scientists with famous explorers: They are intrepid and hearty, ready to tackle the unknown. It aggrandizes their work and dedication.

“By using the chaos game instead, [Michael] Barnsley made pictures that began as fuzzy parodies and grew progressively sharper. No refinement process was necessary: just a single set of rules that somehow embodied the final shape.”

Barnsley’s chaos game effectively showed that fractals had natural limits. Nature prefers certain patterns over others. Fractal shapes are a result of both a dynamic, deterministic system and the natural limitations imposed on randomness. Although theoretically speaking, leaves can arrange themselves in infinite ways, a finite number of patterns appears within nature itself.

“It was as if the system had an orderly impulse and a disorderly one together, and they were decoupling. As one impulse led to random unpredictability, the other kept time like a precise clock. Both impulses could be defined and measured.”

In chaotic mixing, the University of California, Santa Cruz scientists revealed the work of strange attractors. When properties are stretched and folded within a phase space, they can lead to either instability or stability, mimicking Stephen Smale’s horseshoe. Thus, the same dynamic system can contain chaotic mixing and ordered patterns, and both are measurable phenomena.

“Strange attractors, conflating order and disorder, gave a challenging twist to the question of measuring a system’s entropy. Strange attractors served as efficient mixers. They created unpredictability. They raised entropy. And as [Robert] Shaw saw it, they created information where none existed.”

One of the central paradoxes of chaos science concerns the presence of order within disorder, or patterns within chaos. As strange attractors increase entropy—the tendency toward disorder—they simultaneously reveal measurable patterns. In Shaw’s formulation, more randomness within any given transmission paradoxically creates more information. The two impulses, order and disorder, cannot be disentangled.

“The paragon of a complex dynamical system and to many scientists, therefore, the touchstone of any approach to complexity is the human body.”

While it took some time for biology and medicine to enter the field of chaos science, many scientists recognized that the combination was fitting. Characterized by constant motion, the human body is often considered the paragon of dynamic systems. Still, the study of the human body was often aggressively divided into constituent parts, each organ justifying its own branch of biological exploration (i.e., cardiologists and pulmonologists and gastroenterologists). Chaos science looks at issues in medicine in more holistic ways, focusing on the interconnectedness of systems.

“With or without chaos, serious cognitive scientists can longer model the mind as a static structure. They recognize a hierarchy of scales, from neuron upward, providing an opportunity for the interplay of microscale and macroscale so characteristic of fluid turbulence and other complex dynamical processes.”

The study of the brain—and, by extension, the implications for artificial intelligence—has benefited from this global approach. The brain itself exhibits a fractal structure, even though science in general has not yet fully explored the connection between the brain (a physiological entity) and the mind (a theoretical space), or how tissue engenders thought. This dynamic system appears infinitely generative.

“There were no tools for analyzing irregularity as a building block of life. Now those tools exist.”

Chaos brings to science—and to any study of complex phenomena—ways to examine and understand how the irregular and erratic are as much a part of generating life as are the ordered and patterned. Instability contributes to dynamic systems as much as stability does.

“More and more of them realized that chaos offered a fresh way to proceed with old data, forgotten in desk drawers because they had proved too erratic. More and more felt the compartmentalization of science as an impediment to their work. More and more felt the futility of studying parts in isolation from the whole. For them, chaos was the end of the reductionist program in science.”

This passage sums up the revolutionary development of chaos science. Not only did it transform the ways that many scientists from numerous fields understood their work and the world, but it also transformed the discipline of science itself. In addition, many of the basic ideas of chaos science have inspired exploration in other fields far from the realm of science, such as literature and history.