# Chaos, making a new science

## Contributors

Eric Klein Goldewijk, Ferdi, Jochem Beintema, Koen, Marcus Veen, Mark Uijen de Kleijn

**Authors:** J. Gleick

**Publication Year:** 1987

**Source:** http://google.com

**Title:** Chaos, making a new science

**Publisher:** -

**Edition:** -

**ISBN:** -

**Categories:** Organizational Change, Adaptive Cycle

## Abstract

James Gleick has written the book “Chaos” from the perspective of the traditional scientific disciplines and great thinkers of the past. Gleick is tying together unrelated kinds of wilderness and irregularity from the turbulence of weather to the complicated rhythms of the human heart, form the design of snowflakes to the whorls of windswept deserts sands explaining how these scientists have investigated the chaos of these natural phenomena. They all come to the same conclusion which is that most natural behaviour can not be predicted or calculated using traditional models resulting chaos.

## Critical Reflection

Introduction

Like all complex adaptive systems, corporations must be ready for a sudden confrontation with the inexorable hazards of natural selection. Just as the body’s molecules replenish themselves by moving in and out of the system, so must an organization revitalize itself by utilizing new members and diverse ideas. The butterfly effect

Edward Lorenz is best known as the inventor of the famous “Butterfly effect”, an example of chaos theory where a butterfly’s wings creates tiny changes in the atmosphere that ultimately might cause a tornado or a hurricane to appear on the other site of the world. Lorenz was a meteorologist and mathematician who tried to develop a model that could forecast long term weather. A computer simulation that Lorenz developed in 1961 was based on a simple equation that represented turbulence. This equation, like every chaos equation includes a feedback loop (e.g. x = x² + k) that produced constantly changing results, like the weather. Then coincidently Lorenz tried to redo an earlier simulation starting with the parameter he noted. At first the model produced the same results; however results started to deviate soon after he started and results then did not look anything like the previous simulation. Lorenz wondered how the equation could produce totally different outcomes with the same initial parameters. It turned out that he notes the initial parameters with one digit less precisions. A small difference in initial conditions yields a widely diverging outcome rendering long-term prediction impossible, just like the butterfly effect. The graphical representation of the mathematical model that Lorenz developed, a butterfly or eights shape, named the Lorenz attractor (figure 1) is one of the best known images of Chaos science. Lorentz.jpg Revolution

Individual scientists in various fields like biology, electric engineering, mathematics, and physics were all making “chaos” discoveries like Lorenz’s in the 1960s. One of these scientists was Stephen Smale, a brilliant mathematician, who, to amazement of his colleagues, was working on oscillators, a problem usually studied by beginning physicists. His colleagues later realized that he was actually working on nonlinear oscillators, and seeing things physicists had learned not to see. Smale was trying to prove that dynamical systems always tend to settle to an equilibrium eventually, he was wrong, but it did put him on track of a new way to understand the full complexity of dynamical systems. His topological experiments generated the same kind of unpredictable outcomes as Lorenz’s. He visualized his ideas in the horseshoe (figure 2), that provides a topological understanding of the chaotic properties of a dynamic system. A space is stretched in one direction, squeezed in another and then folded. This process is repeated until a structure exists that looks like many layered pastry dough. Points that were initially far away can now be very close together. Smale.jpg Life’s ups and downs

One science that played a special role in the development of chaos theory is biology and especially ecology, a field of biology that stripped away the noise and color of real life and treated populations as dynamical systems. Biologist used to believe that population growth was a strictly linear process that would come to an equilibrium once it’s parameters reached certain values. It was the work on population dynamics by Robert May, initially a theoretical physicist, that launched the development of theoretical ecology and the study of biodiversity. May found that a simple equation that modeled a wildlife population would settle on a steady state when he kept the nonlinear parameter low. However, the outcome of his equation would become chaotic and never reach an equilibrium once he raised the parameter beyond a critical point. May did not understand what was happening and decided to plot the outcomes of his experiment on a bifurcation diagram, in order to make sense of it. What the diagram showed was that even with the simplest equation the region of chaos in the diagram has an orderly structure. First the bifurcations produce periods of 2,4,8,16 and then chaos begins. Windows start appearing that each turn out to represent the whole diagram (figure 3). A Geometry of Nature

Benoit Mandelbrot was a mathematician who first fled from Poland to France and later to the United States to escape from the Nazi’s. He started working for IBM in the 1960s where he had access to the most powerful computers that existed at that time. Mandelbrot had been studying non-linear equations and noticed a diagram on the blackboard that he believed was his during a guest lecture for Houthakker, a Harvard economics professor. This diagram showed cotton prices over a long period. The problem was that Houthakker and his students did not manage in any way to display the prices through time as a bell shaped diagram in one way or the other, as statistical laws would suggest. Mandelbrot went to the ministry of agriculture to get cotton price data from 1900 and stored all the data in computer punch cards. The results were astonishing. The price changes on a single scale were random, however frequencies on different scales produced symmetry, monthly prices and daily prices showed exactly the same variation. Mandelbrot visualized his finding in a fractal called the Mandelbrot set, an image where self-similarity infinitely shows the same figure when zoomed in. Mandelbrot.jpg Strange Attractors

Heisenberg declared that he had two questions for god on his deathbed, why relatively? and why turbulence? He added that he was pretty sure that god would have an answer on the first question. Turbulence is one of the most difficult disciplines in physics that only few scientists dared to take on. Harry Swinney, an American physicist, took on the challenge in 1973 while working for the city college of New York. Swinney build an apparatus that existed of an outer glass cylinder with an inner cylinder of steel. Laser lights were set up to measure the water deflection, all this to combat the messines of moving fluids, hoping to find the onset of turbulence. The results seemed to form a pattern at the beginning, but collapsed a certain point, when chaos appeared. Now the Belgian mathematical physicist David Ruelle took over to come up with some explanations. Ruelle argued that the traditional view on turbulence, where a pilling up of frequencies leads to an infitude of independent overlapping motions was wrong, and that there are actually only three motions that produce the full complexity of turbulence. The math proposed by Ruelle was actually partially wrong, but he did come up with a notion that had a lasting effect on chaos research: strange attractors. An attractor is a set towards which a variable, in a dynamical system, evolves to over time. An attractor becomes strange when it has a fractal structure like Lorenz’s attractor, May’s window and the Mandelbrot set appears. Universality

Mitchell Feigenbaum was an American physicist who got involved in the science of chaos when hearing about quadratic difference equation during a gathering with Stepen Smale in 1975. Smale used his calculator while combining analytic algebra and numerical exploration to piece together the quadratic map, concentrating on the boundary between order and chaos. It was the slowness of the calculator that allowed him to see a pattern that he otherwise would not have discovered, the numbers were converging geometrically, like telephone poles disappearing in the distance. The found that ratio of this convergence was 4.669. He then started to calculate the same convergence from research results from other scientists and found the same number. The constant that he found provided the mathematical proof for a wide class of mathematical functions, prior to the onset of chaos. His findings finally enables mathematicians to start unraveling the apparently random behavior of chaotic systems. The experimenter

Albert J. Libchaber was a genuine experimenter who started his career in mathematics at the University of Paris in 1956 and the Ecole Nationale Supérieure des Telecommunications in 1958. Albert Libchaber made major contributions in experimental condensed matter physics. He carried out the first experimental observation of the bifurcation cascade that leads to chaos and turbulence in liquid Helium. He chose Helium because it has exceedingly low viscosity so it will at the lightest push. Using a so called “bolometer” for ideal thermal conductivity he was able to observe temperature fluctuations of the helium in a rectangular cell without influence of the external the environment. Resulting he observed the bifurcations that lead to chaos: period doubling, possibly accompanied by locking of several incommensurate frequencies. The theoretical predictions of Mitchell Feigenbaum were thus entirely confirmed who stated that involving a single linear parameter exhibits the apparently random behavior known as chaos when the parameter lies within certain ranges. The experiment was so perfect that it could measure quantitatively the Feigenbaum critical exponents that characterize the cascade to chaos. For this achievement, he was awarded the Wolf Prize in Physics in 1986. Images of chaos

Michael Barnsley is a British mathematician, researcher and an entrepreneur who has worked on fractal compression which made him known. He was interested in visualizing complex behavior especially boundaries of infinite complexity. He developed simple formulas to calculate these based on Madelbrot discovery of so called Julia sets. These images are a collection of points every single one in the complex plane, in or out the complex plane using simple inerratic arithmetic’s. As a result figures of sea-horse tails or brambles bushes appear. Barsnley related this to natural images particularly of living organisms modeling natural shapes. He called this technique “the global construction of fractals by means of iterated function systems” talking about this as the “chaos game”. For these images you need a computer with a random number generator and graphic screen. You choose a starting point on the screen and invent two rules telling how to take one point to another. In essence his central insight was that Mandelbrot Julia sets and other fractal shapes, though properly viewed as the outcome of a deterministic process had a second equally valid existence as the limit of a random process. By analogy he suggested one could imagine the map of the UK drawn in chalk on the floor of a room. A surveyor would find it complicated to measure the area of these awkward shapes with fractal coastlines. But suppose you throw grains of rice in the air one by one allowing them to fall randomly on the floor counting the grains that land inside the map. As time goes on the shapes form new areas as the limit of a random process. In dynamical terms these shapes proved to be attractors. The Dynamical Systems Collective

Robert Stetson Shaw is an American physicist who was part of Eudaemonic Enterprises in Santa Cruz in the late 1970s and early 1980s. In 1988 he was awarded a MacArthur Fellowship for his work in Chaos theory. He was one of the pioneers of chaos theory and his work at University of California, Santa Cruz. His main contribution was on the first research into the relationship between predictable motion and chaos. This work was best known for its ability to probing chaotic systems for signs of order. Shaw based his work on the Lyapunov exponent number to measure in a dynamical system the quantity that characterizes the rate of separation of infinitesimally close trajectories. Quantitatively, two trajectories in phase space with initial separation diverge (provided that the divergence can be treated within the linear approximation) at a specific rate. The rate of separation can be different for different orientations of initial separation vector which refers to the largest one as the Maximal Lyapunov exponent (MLE), because it determines a notion of predictability for a dynamical system. A positive MLE is usually taken as an indication that the system is chaotic (provided some other conditions are met, e.g., phase space compactness). Note that an arbitrary initial separation vector will typically contain some component in the direction associated with the MLE, and because of the exponential growth rate, the effect of the other exponents will be obliterated over time. Based on this theory Shaw was able to explain the behavior of dynamic system collective, and predict or explain the effect of chaos on it. Inner Rhythms

Bernardo Hubberman is a theoretical biologist and physicist who has studied chaos from this professional perspective. Huberman originally worked in condensed matter physics, ranging from supersonic conductors to two-dimensional super fluids, and made contributions to the theory of critical phenomena in low dimensional systems. He was one of the discoverers of chaos in a number of physical systems, and also established a number of universal properties in nonlinear dynamical systems. Hubberman studied for his chaos research initially the eye movements of schizophrenic patients in the area of physiology. This was the prelude to more research regarding the human heart and the unpredictable failures and rhythms disturbance called arrhythmias. During this research it was increasingly recognized that the human body is a place of motion and oscillation acting as a complex system. Resulting new methods of listening to its variegated drumbeat were developed so precisely making the difference between life and death. They found invisible inner rhythms of the human body which could be correlated to chaos theory. Now some physiologists speak of dynamical diseases: disorders of systems, breakdowns in coordination or control. “Systems that normally oscillate, stop oscillating, or begin to oscillate in a new and unexpected fashion, and systems that normally do not oscillate, begin oscillating,” was one formulation. Another good summary is following “If I ask you whether your brain is an equilibrium system, all I have to do is ask you not to think of elephants for a few minutes, and you know it isn’t an equilibrium system.” Chaos and beyond

In the 70’s and 80’s physicists, mathematicians, biologists, and astronomers have created an alternative set of ideas considering chaos. Simple systems give rise to complex behaviour. Complex systems give rise to simple behaviour. And most important, the laws of complexity hold universally, caring not at all for the details of a system’s constituent atoms. Now students realize that ecology based on a sense of equilibrium seems doomed to fail. The traditional models are betrayed by their linear bias. Nature is more complicated. Instead we see chaos, “both exhilarating and a bit threatening.” From the 80’s onward William Schaffer a Professor in Ecology and Evolutionary Biology took the view on chaos a step further. His principal interests involve the application of nonlinear dynamics to biology at a variety of levels. This work has resulted in the production of the first detailed model which is able to reproduce complicated bifurcation sequences observed in careful laboratory experiments. At the physiological level, he is interested in dynamical scenarios which might account for the natural history of diseases such as epilepsy in which the pathology is intermittent, but this work is still preliminary. At the ecological level, he is concerned with complex population dynamics and with the ups and downs of human epidemics. His work on chaos in childhood diseases, in particular, sparked considerable interest in this topic.

## Contributors

Eric Klein Goldewijk, Ferdi, Jochem Beintema, Koen, Marcus Veen, Mark Uijen de Kleijn