Nonlinear Maps And Chaos

Type of physical science: Mathematical methods

Field of study: Algebra

Chaos is the science of dynamical systems, processes in nature and within living organisms that may be orderly but that may undergo predictable and sometimes unpredictable change. Orderly systems sometimes become disordered (chaotic) and disorderly systems sometimes become ordered, as a result of minute disturbances.

Overview

Chaos is the relatively young science of nonlinear dynamics, the description of physical, chemical, and biological processes that normally follow orderly cyclical patterns but that occasionally become disordered, or chaotic. Most observed processes in the universe appear to be ordered, although some disorder appears now and then. This occasional disorder is to be expected, according to the second law of thermodynamics, which maintains that the disorder (entropy) of the universe is steadily increasing. In other words, all systems in the universe are decaying gradually. Another view of the second law of thermodynamics is the impossibility of constructing a machine that is 100-percent efficient. Waste is an inevitable by-product of all systems.

Within the universe, numerous systems exist that exhibit orderly behavior, including the orbiting clusters of stars that form spiral galaxies, the periodicity of sunspots over time, the orbiting patterns of planets around the sun, the yearly shifting of seasons on several planets (such as Earth and Mars), the motions of ocean and air currents on Earth, the circadian (twenty-four-hour) biological cycles of animals, the regular beating of a mammal's four-chambered heart, and many other examples. Many of these same processes, however, can exhibit disorderly, or chaotic, behavior. Some galaxies explode or collide with other galaxies, thereby ripping their order into disorder. Stars explode, becoming supernovas. The asteroid belt of the solar system is a very disorderly array of fragments that represent the failure of a planet to form gravitationally and/or stay together. Storm systems on Earth form, move, and dissipate unpredictably. Living organisms age. Old animals die because their hearts suddenly and mysteriously stop beating.

In 1961, Edward Lorenz, a meteorologist at the Massachusetts Institute of Technology, performed computer simulations of weather patterns in an attempt to improve weather forecasting. Because the universe operates upon the simple principle of cause and effect, specified conditions of temperature, air pressure, and humidity should make weather prediction feasible over the course of many days. Lorenz provided input data for his weather-forecasting program. The computer printer output was a series of curves on an abscissa/ordinate (X/Y) graph. When Lorenz repeated the computer simulation with the same input data, initial results showed the same pattern of curves. After a short period of time on these duplicate computer runs, however, the graphs began to show increasingly deviant behavior. The supposedly predictable computer-generated weather had descended into disorder.

Lorenz soon discovered why the experimental curves had not remained identical. On his initial experimental run, he had punched in numbers to, for example, the millionths decimal place. During the repeat runs, however, he had rounded the same numbers to, for example, the thousandths decimal place. This rounding of numbers produced a disturbance in the system that caused the weather curves to deviate, to become chaotic.

Lorenz had demonstrated that small effects add up to produce big effects. The smallest disturbance to a system, even a number in a computer program that is rounded to the thousandths decimal place, can convert an orderly process into a disorderly one. This phenomenon, called the butterfly effect, according to a weather scenario, maintains that a butterfly beating its wings in Japan can produce enough eddy currents to affect local air currents contributing to weather patterns halfway around the world. In other words, the smallest disturbance to a system can have major consequences.

Because of this important discovery, it is obvious that a process can remain predictable only if it is completely isolated from outside influences. This is impossible for any system, including weather. Therefore, precise weather prediction over long periods is impossible.

The butterfly effect is related to a phenomenon in quantum mechanics called the Heisenberg uncertainty principle, which maintains that the exact location of a given particle (an electron, for example) at a given time in a system of many particles cannot be precisely determined. The butterfly effect is also a direct result of the cause-and-effect interactions of particles within the universe.

In the universe, orderly systems become disordered and disorderly systems become ordered in a never-ending cycle. Mathematical modeling of various ordered/disordered processes generates a pattern for each process that centers on a single point called a strange attractor. Such a Lorenz attractor was constructed from Lorenz's computer simulations. For a strange attractor, the orderly and disorderly behaviors of a given process can be graphed. In most cases, the orderly/disorderly behavior loops about within defined boundaries, with no single cycle of the process repeating any other cycle exactly. The orderly/disorderly cycle flows about a single point in space as though the entire process were attracted to this particular area and could not escape.

Strange attractors and chaotic behavior are very characteristic of many dynamical, or constantly changing, conditions. Thermodynamics, hydrodynamics, and aerodynamics deal with the flow of energy, water, and air, respectively. These three types of processes are constantly changing. In aerodynamics, ideal jet flight involves streamlined airflow over the jet's wings. The slightest upward or downward tilt of the wings, however, can produce disruptive eddy currents, or turbulence.

In atmospheric weather phenomena, hurricanes, tornadoes, and the jet stream are good examples of orderly/disorderly processes under study. The Great Red Spot of Jupiter and the Great Dark Spot of Neptune represent enormous hurricanes larger than the earth, which have been circling the atmospheres of these two planets for hundreds of years. These two hurricanes must be very ordered to have endured so long. Close examinations of video transmissions from the Voyager 2 spacecraft, however, show that both of these hurricanes feed off of smaller disruptive storms in their wake. Therefore, these two massive hurricanes represent order emerging from surrounding disorder. Furthermore, each of these storms is a strange attractor.

The solar system's asteroid belt represents an example of chaotic, nonlinear dynamics.

This belt consists of perhaps a million rock fragments of various shapes and sizes, many as small as pebbles, others as big as small moons. Most asteroids orbit the sun between the orbits of Mars and Jupiter. Not all asteroids, however, follow a predictable orbit within this belt. In fact, most do not. The gravitational attractions of Mars and Jupiter upon these asteroids are constantly altering the solar orbits of all asteroids, not to mention the gravitational attraction of asteroids for one another. The pull of Mars and Jupiter occasionally flings asteroids into the outer or inner solar system, where their chaotic orbits are affected further by the gravitational pull of Earth and other planets. Eventually, some asteroids will collide with planets.

On Earth, chaos is evident not only in weather processes but also within life itself. Life has evolved over the last 3.8 billion years, often in long, uneventful periods when one major group of species dominated the planet. Nevertheless, these long, uneventful periods were interrupted by major catastrophes during which many life-forms became extinct, only to be replaced by an explosion of new life-forms. There is no question that certain disturbances led to mass extinctions in the earth's geological past. One such extinction, the Cretaceous extinction, occurred approximately 65 million years ago and ended the dominance of reptiles on Earth. One hypothesis for the Cretaceous extinction is that an asteroid or comet struck the earth.

The biological environment is very sensitive to sudden, drastic changes. Pollution and the destruction of wilderness areas by humans have resulted in the disruption of the orderly lives of thousands of species, some of which have become extinct, many others of which have become endangered. Very complex food webs and species interactions exist in the environment, with hundreds of different species critically dependent upon one another, human beings included.

In medicine, various diseases and disorders have been diagnosed as being of chaotic influence, in which the human body's normal rhythms are somehow disrupted, often because of stress, improper diet, infections, and the like. Even the chaotic malfunctioning of a heart during a heart attack is under intensive study, especially in terms of the heart's neuroelectrical activity.

Every aspect of daily life is the result of cause-and-effect orderly/disorderly behavior.

The most minor events trigger a succession of other events that collectively lead to major consequences, both good and bad. Cause-and-effect events are integral parts of dynamic, nonlinear processes.

The orderly/disorderly behavior of these chaotic, nonlinear processes tends to produce specific patterns that are repeated over and over again within these systems. Such repeatable, or reiterated, patterns are termed fractals. Fractals serve as geometrical models for describing self-similar processes and systems in nature, phenomena that repeat specific geometrical forms.

Fractals are mathematical models that are very similar to cellular automatons, self-replicating models of systems that mirror living processes. Fractals and cellular automata, like chaos, are found throughout nature.

One common fractal pattern that emerges throughout nature is the spiral, a structure that appears in such diverse chaotic systems as spiral galaxies, ocean currents, hurricanes, seashells, and proteins. Other fractal patterns that are reiterated throughout nature include triangles, ellipses, parabolas, hyperbolas, stars, and sine and cosine waves. The Lorenz attractor resembles a pair of eyes or butterfly wings (hence, the butterfly effect).

The fact that fractal patterns and nonlinear processes repeat throughout nature is a concept known as universality. Certain patterns reemerge in seemingly different processes. The goal of fractal and cellular automata mathematical models is to reveal specific patterns that are prevalent in various chaotic, nonlinear processes. A geometry of nature is envisioned that could greatly enhance the knowledge of all of science.

Applications

Understanding what makes a predictable, orderly process suddenly become chaotic is a subject of major concern to physicists, chemists, and biologists. It is important to understand how chaos occurs in various systems so that it can be controlled whenever it arises. It is extremely unlikely that a chaotic condition in any system could ever be prevented. Such a scenario would contradict the second law of thermodynamics. Natural processes that have an impact on everyday life are subject to occasional disorderly behavior.

The mathematical mapping of chaotic, nonlinear processes is critical to understanding their properties. The fractal is the principal mathematical tool. An illustration of how fractal geometry can be applied to an orderly/disorderly system is the common example of coastline length. How long is the coastline of, for example, Florida? With an accurate map and ruler, one can measure an approximate length in miles or kilometers. If one wanted an exact length, however, one would have to walk or sail around the Florida coastline with a measuring tape.

Furthermore, one would have to measure around every island, inlet, and river delta to get a more exact measurement. To be even more precise, one would have to measure every consecutive atom of the entire coastline with an incredibly small microruler. As one measured the Florida coastline with smaller and smaller scaled rulers, the length of the Florida coastline would become longer and longer. The smallest differences add up to big effects, or accurate measurements, which is the essence of chaos.

With such an experiment, it is impossible to measure at a sufficiently small scale. If one were to keep trying smaller and smaller scales, the coastline of Florida would become infinitely long. Furthermore, if one consults a map of Florida's coastline, it follows an irregular pattern, as do all coastlines. If one magnifies any portion of the coastline, the same approximate irregular pattern emerges. No matter what scale is observed, the same approximate coastline patterns are reiterated. Therefore, the chaotic coastline of Florida, or anywhere else, is a fractal.

Little effects add up to produce big effects. This is seen in Lorenz's weather simulations. It is also seen in the orbits of planets, comets, and asteroids, in the evolution of life on Earth, in the normal functioning of the human body, and in disease.

In humans and in other mammals, the electrical activity of the heart, measured by an electrocardiogram, follows a rhythmic pattern that repeats itself approximately once every second for an entire lifetime. When a person suffers a heart attack, this rhythmic pattern is grossly distorted. The cardiac team at a hospital emergency room must administer the proper medications and use the proper techniques (CPR, defibrillation) to restore seminormal heart activity in the heart attack patient. What are the normal electrical nerve activities of the heart that are needed to maintain a normal heartbeat? What chemicals or hormones may be involved? What slight disturbances can drastically affect the proper functioning of the heart? These are questions that cardiac physiologists ask when they study the orderly or disorderly behavior of the heart.

In other areas of physiology, periodic cycles that can become disordered include the endocrine system hormones, nerve-muscle interactions, sleep/wake cycles, and blood-cell production rates, among others. A major aspect of animal physiology is homeostasis, the maintenance of constant conditions within the body's internal environment. If constant conditions are not maintained for a given bodily process, severe illness or death can result. Homeostasis is maintained by positive and negative feedback mechanisms.

Certain diseases follow chaotic patterns. For example, the common cold virus often affects individuals following chaotic changes in environmental temperatures during the fall and spring months, when successive days can have drastic extremes of high and low temperatures.

Furthermore, many bodily illnesses are linked to stress, which disrupts the normal orderly cycles of the body. Stress-related illnesses include allergies, susceptibility to colds and flu, fatigue, and even cancer.

In ecology, the disruption of environmental ecosystems wrecks the balance of species.

Some organisms may benefit from the disruption, but others will suffer and become extinct. The environmental management of endangered species and wilderness areas will involve the careful analysis of the way in which human activity affects wildlife. The slightest changes in the environment can have devastating effects.

Many physicists are actively working to construct mathematical models of chaotic behavior in living systems. Among these researchers are Benoit B. Mandelbrot of International Business Machines (IBM) and Yale University, Michael Barnsley of the Georgia Institute of Technology, and Michael Mackey of McGill University. Their elaborate fractal diagrams of chaotic behavior in various systems, both living and nonliving, include bifurcation (period-doubling) diagrams in which a process doubles at specified time intervals, Mandelbrot sets, Cantor sets, Sierpinski carpets, Julia sets, and so forth.

Context

The science of chaos has opened up an entirely new approach to viewing and describing structures, systems, and processes in the universe, both living and nonliving. Events in nature that appear to be dissimilar are shown to contain many of the same patterns fractally reiterated. Chaos also demonstrates that the precise predictability of any event is impossible.

Nothing within the observable universe is isolated; all matter and energy interact. Therefore, any orderly process eventually will be disturbed to such an extent that it will become disordered.

Likewise, any disorderly process can be influenced to become orderly. The science of chaos attempts to explain how these deviations from steady-state behavior occur.

The eminent American-British quantum physicist David Bohm has described the physical state of the universe as an explicate order, an order that has unfolded from the hidden principles upon which the universe is based, the implicate order. The fractal analysis of orderly and disorderly processes reveals common patterns between the two processes that explain their interconvertibility. Fractal analysis may reveal an implicate order of nature in certain reiterated patterns.

Order becomes disorder, and disorder becomes order. Processes encompassing every aspect of life are subject to chaotic behavior, from biological rhythms to digestion to disease to weather to earthquakes to computer and machine functioning or malfunctioning. A better understanding of the triggers that disrupt these events will help to control their effects. This knowledge will assist in the mathematical modeling of biological, chemical, and physical processes, all for the benefit of medicine, agriculture, science, and industry.

In many ways, chaos represents Murphy's law: "Whatever can go wrong will go wrong." Little effects contribute to big effects in either direction, right or wrong. The entropic decay of the universe makes chaos more and more likely to occur.

Principal terms

BUTTERFLY EFFECT: a phenomenon in chaos theory in which the slightest disturbance to a system changes that system's behavior in an unpredictable fashion

CELLULAR AUTOMATON: a self-replicating system or process that mimics living cells and that is used to model physical and biological processes; the basis of intelligent machines

ENTROPY: disorder; the tendency for any system to degenerate over time in accordance with the second law of thermodynamics

FRACTAL: a pattern or geometric form that is reiterated, or repeated, over and over again within the structure or periodic behavior of any object or process

NONLINEARITY: a reference to the behavior of processes that do not follow predictable, periodic patterns; processes that are chaotic

STRANGE ATTRACTOR: a point about which a system or process orients itself in an orderly or disorderly pattern

TURBULENCE: a thermodynamic term used to describe chaotic deviations from the normal, smooth periodic behavior of a process

UNIVERSALITY: the tendency for various seemingly unrelated processes in the universe to follow similar patterns from microscale to macroscale, and vice versa

Bibliography

Devaney, Robert L., and Linda Keen, eds. CHAOS AND FRACTALS: THE MATHEMATICS BEHIND THE COMPUTER GRAPHICS. Providence, R.I.: American Mathematical Society, 1989. This short book is a collection of papers presented by various chaos researchers at this American Mathematical Society short course devoted to fractal geometry. The papers are somewhat detailed, with mathematical equations, but they provide beautiful illustrations and the basics of fractal computer graphics.

Feder, Jens. FRACTALS. New York: Plenum Press, 1988. This excellent book, written by a Norwegian physicist, is an outstanding discussion of fractals and their uses. Numerous examples are presented along with simple mathematical explanations. Chapter 2, "The Fractal Dimension," discusses the problem "How long is the coast of Norway?"

Glass, Leon, and Michael C. Mackey. FROM CLOCKS TO CHAOS: THE RHYTHMS OF LIFE. Princeton, N.J.: Princeton University Press, 1988. This comprehensive book, written by two leading chaos researchers, discusses the principles behind orderly and disorderly systems. Numerous examples are presented, especially from the biological sciences (for example, the chaotic behavior of the heart, nervous system impulses, and cancer). The book has numerous illustrations and is extensively referenced.

Gleick, James. CHAOS: MAKING A NEW SCIENCE. New York: Viking Press, 1987. Gleick's popular book is one of the most thorough, exciting surveys of cellular automata, fractal geometry, and chaos that is available for the general public. He describes the development of research in these fields and cites numerous important experiments. Chapter 1, "The Butterfly Effect," is an excellent discussion of Lorenz's chaotic computer weather experiments.

Hartley, Karen. "Solar System Chaos." ASTRONOMY 18 (May, 1990): 34-39. This review article is a very clear discussion of chaotic events within the solar system, particularly with reference to the gravitational effects of planets upon one another and upon comets and asteroids. Data from computer simulations of the solar system are described.

Hiley, B. J., and F. David Peat, eds. QUANTUM IMPLICATIONS: ESSAYS IN HONOUR OF DAVID BOHM. New York: Routledge & Kegan Paul, 1987. This unique collection of philosophical and scientific essays by a diverse spectrum of thinkers is dedicated to quantum physicist David Bohm. Many essays deal with chaotic and fractal topics ranging from physics to biology. Essay 2, "Hidden Variables and the Implicate Order," is Bohm's own contribution to the collection, a summary of major problems in quantum mechanics and the chaos of nature.

Pickover, Clifford A. COMPUTERS, PATTERN, CHAOS, AND BEAUTY: GRAPHICS FROM AN UNSEEN WORLD. New York: St. Martin's Press, 1990. This excellent book is a step-by-step introduction to fractal geometry and computer graphics. It is clearly written, beautifully illustrated, and somewhat philosophical in scope. Numerous fractal-generating algorithms are presented.

Ruelle, David. CHAOTIC EVOLUTION AND STRANGE ATTRACTORS. New York: Cambridge University Press, 1989. This very short book is a deep mathematical treatment of chaos theory for the serious chaos researcher. Strange attractors, entropy, and methods for identifying chaos in physical systems are discussed. Numerous examples, diagrams, and references are provided.

Stevens, Roger T. FRACTAL PROGRAMMING IN C. Redwood City, Calif.: M&T Books, 1989. This outstanding "how-to" book describes fractal structures and tells the reader how to write fractal programs. A knowledge of C-language programming is assumed, along with operational knowledge of appropriate computer graphics equipment.

The Physics of Weather