CHAOS

CHAOS , in the history of religions, refers primarily to the primordial condition, precosmic period, or personified being found in many oral and literary mythologies. It is commonly, although not always legitimately, taken to mean the horribly confused state, muddled matrix, vacuous condition, or monstrous creature preceding the foundation of an organized world system. By extension, the idea of chaos in myth and ritual may also apply to any anomalous condition, event, or entity outside conventionally sanctioned codes of order. The meaning and significance of chaos in world mythology has, moreover, a special thematic relationship with the idea of the beginnings of the world, or of any structured condition. The word beginning is, in fact, etymologically connected with the Old English on-ginnan and the Old Norse gina, in both of which lurks the mythological image of the cosmogonic Ginnungagap, the primordial void that spawned the giant Ymir (the Primordial Man) in ancient Norse tradition. From a cross-cultural perspective, the image of chaos is therefore especially prominent in cosmogonic and anthropogonic myths, as well as in many types of origin myths and passage rituals concerned with some transitional situation in human life or with some significantly altered state of affairs, whether for well or ill. Chaos appears, for example, within the context of the condition of death or the dream time of sleep, flood mythology, apocalyptic imagery in general, or foundational legends and rites pertaining to a new sociopolitical tradition.

The English word chaos derives directly from the Greek chaos, which in Hesiod's Theogony (c. eighth century bce) denoted a cosmogonic "yawning gap, chasm, or void," from which generated the successive worlds of the gods and mankind. Hesiod, who drew upon earlier mythological sources, rather neutrally depicted the original chaos as merely the empty, dark space that allowed for the penetrating movement of erotic desire and for the appearance of Earth (Gaia) as the secure home for all subsequent created forms and beings. But the Theogony also displays the mythological premise for a more negative evaluation of chaos, since the earliest generations of Titanic gods, most closely identified with the untrammeled passion and anarachy of the primitive chaos condition, must be violently defeated by Zeus to insure the permanence and universality of the Olympian order. The primal chaos is itself only the blind abyss necessary for the creation of the physical world, but chaos here also refers to the mythic period—and, by implication, to a kind of "chaos-order" or condition—of the pre-Olympian gods who struggle against the imposition of Zeus's all-encompassing rule.

Because of their general impact on the colorations of meaning popularly conjured up by the modern use of the term chaos, it is worth citing two other ancient Western documents. In the priestly tale of Genesis (c. fourth century bce) found in the Hebrew scriptures, chaos is reduplicatively called tohu va-vohu, a dark, watery, formless waste or "limbo-akimbo" that must be wrested into order by the willful fiat of a god completely separate from the stuff of creation. In a somewhat similiar vein, although more somberly stressing a hostile jumble of primal matter over blank vacuity, Ovid (43 bce–18 ce), in his poetic compendium of mythology known as the Metamorphosis, describes chaos as "all ruse and lumpy matter … in whose confusion discordant atoms warred." As in the biblical version, Ovid's creation requires a nameless god, or "Kindlier Nature," who brings order out of the formless chaos.

The above-mentioned accounts serve to exemplify the commonplace tendency to dichotomize the meaning of existence into the negative-positive polarities of chaos and cosmos, confusion and order, death and life, evil and good, or, more theologically, into some dualistic distinction between the absolutely sacred and creative being of a transcendent "kindlier" God, on the one hand, and the utterly profane nothingness and nonbeing of a passively neutral or actively belligerent chaos. Clearly, these distinctions have both ontological and moral implications, so that it may also be said that the polarized evaluation of the mythic chaos is the backdrop for the theological and philosophical elaborations on such problems as creatio ex nihilo and theodicy. Indeed, the overall issue here directly affects the modern academic understanding of religion, since a whole tradition of Western scholarship defines religion as the contrast between the sacred and the profane, or, to use Peter Berger's more straightforward sociological formulation, as the "establishment through human activity of … a sacred cosmos that will be capable of maintaining itself in the ever-present face of chaos" (The Sacred Canopy, New York, 1967, p. 51).

A comparative assessment of world mythology shows, however, that such pat divisions are not always warranted, so that, for instance, the apparently fundamental contrast between chaos and cosmos may reveal more of a dialectical relationship. This ambivalence is at least suggested by the observation that the Greek root of the term kosmos does not so much refer to the creation of an absolute, universal, and final world order (although with Pythagoras in the sixth century bce it will take on this sense as its basic meaning) as to the more relative and transitive idea of the "cosmetic" alteration of some more natural, plain, and primitive condition. Cosmos in this sense is the differentiated, deferential, and ornamental order; it is the painted and tattooed body of chaos—pretty and pleasing primarily to the eye of the beholder. One tradition's chaos, in other words, is another's cosmos, and vice versa. It depends on the vantage point, or, at least, on whether the original cosmogonic chaos is conceived advantageously. In the broadest sense, chaos stands for the root "otherness" and "strangeness" of existence and the ironic indeterminacy of all human constructs.

"Creation out of chaos," in like manner, may not just refer to the appearance of order and reality out of the void, but the creative possibility of many different orders and worlds. As the hidden sum of all potential kosmoi, chaos is intrinsically linked to the transformative nature of phenomenal and cultural existence. The tensed relationship of "chaos and cosmos," then, usually has sociopolitical as well as metaphysical implications, and this, it would seem, has much to do with the interpretation of chaos seen in particular myths and cultural traditions. This, of course, begs the question as to the ultimate premise of world construction, but it is exactly the fundamental existential puzzles of "something from nothing" and the interrelationship of unity and multiplicity, plenitude and limitation, that give rise to a moot diversity of possible answers and that are always addressed to some extent in chaos mythology.

Images of Chaos

Any excessive or transitional aspect of the natural world (e.g., the untamed vegetation of a jungle, the blurring of light and dark at twilight, the frenzied winds of a storm, and so on) may be taken as a cipher for the mythological chaos; but, as already suggested by the biblical allusions, the most prevalent natural metaphor for chaos is water. Given water's infinite fluidity, its protoplasmic vitality, as well as its lethal and regenerative potencies, it is hardly surprising that images of a vast ocean, a turbulent sea, or some other murky, cloudy, frothy, and misty mixture of air and water is used in many myths to depict the original broth of creation.

Common also are references to the moist darkness and foggy gloom of the precosmic condition, along with various depictions of a swirling vortex or whirlpool that links the water imagery with the more abstract ideas of the abysmal void. Other traditions speak of a primal muddle of earth and water, as in the "earth diver" myths that tell of a fragment of muddy soil (often imagined as a central hillock or mountain) rescued from the depths by some animal god or sky deity. Both in tribal cultures and in ancient civilizations there are descriptions of a watery, labyrinthine underworld of the dead, and accounts of limitless seas and rivers surrounding and penetrating the inhabited world; these imply that the dark waters of creation continue to flow around, through, and beneath the hollows of the established cosmos. Finally, there are the worldwide myths of a great flood (or of excessive heat and drought) in the distant past or apocalyptic future that are clearly charged with the cosmogonic idea of a regression to an initial state of total solvency. In these myths, the twin potency of chaos comes to the fore, since the deluge is devastating to the existing world yet simultaneously establishes the necessary precondition for a new creation.

Another important category of chaos symbolism concerns the universal imagery of an embryonic condition or womblike form. This is especially exemplified in the so-called cosmic egg myths (along with the analogous myths involving a bloody lump of flesh or a creatively fertile yet "empty" vessel, such as a pot, sack, gourd, cocoon, or drum), that stress the preexistence of some ovarian matrix within which is mysteriously harbored the structured multiplicity of all cosmic forms. While they sometimes involve an external agent of creation responsible for the production and development of the cosmic zygote, these myths often emphasize the organic conjunction of cosmogonic unity and phenomenal duality as well as the spontaneous self-sufficiency of the creation. It is in this way that the undifferentiated unity and implicate order of the cosmic egg can be said to come before any divine chicken, or, in the words of Samuel Butler, "a hen is only an egg's way of making another egg."

Theriomorphic and anthropomorphic personifications of chaos are often imagined as the gigantic and misshapen offspring of the primal waters, embryonic condition, or dark void. Recalling Hesiod's portrait, such creatures represent the individualized embodiments of chaos within the differentiated world. The actions of these creatures, moreover, show that chaos has a "history" that continuously impinges upon human history. The primary denizens of the chaos time can be categorized as: (1) dragon-serpent figures, often having composite avian-aquatic features (e.g., Vṛtra in Vedic mythology); (2) animal or hybrid man-animal trickster figures (e.g., Raven and Crow in North American Indian tradition); (3) a female demoness, a terrible mother, or chaos hag who has associations with the primal waters and "mother earth" imagery (e.g., Tiamat in Babylonian myth); (4) cosmic giant figures who, while theoretically androgynous, are often more male than female (e.g., Pangu in Chinese tradition); and (5) an incestuous brother and sister couple or a set of divine twins of ambiguous sexuality (e.g., Izanagi and Izanami in Japanese mythology). All of these chaos creatures are particularly related to cosmogonic and other origin myths, are often combined within a single mythic tradition, and are frequently portrayed in rituals concerned with significant seasonal and social transitions (e.g., worldwide celebrations of the New Year). While they may be suppressed, hidden, and transformed in various ways, all of them have popular folkloric surrogates (e.g., the revenant, demon, witch, and fool) and continue their ambivalent careers at the margins of the human world.

The underlying logic that emerges from this rapid cross-cultural survey suggests that chaos is both prior to the world as its cosmogonic source and existentially interstitial to the world as its transformative ground. Because it lies before and between any single order, or always "in relation" to any explicit world, the religious meaning of chaos remains profoundly ambiguous. By its very nature, then, chaos can be variously imagined as simply before and other than, as negatively destructive of, or as creatively challenging to, some ordered world system. Given this queasy multivalence, it is also possible to see why in cultural history any single cosmological tradition will most often seek to deny the relativity of its own vision of order by officially upholding a predominantly negative image of chaos.

Negative and Positive Evaluations

The negative evaluation of chaos commonly takes the form of a mythic and ritual scenario of combat, which was first delineated in relation to ancient Near Eastern and biblical materials. But this pattern is not restricted to the ancient Near East; it is clearly found in many diverse oral and literate traditions. While the combat pattern of myth displays many permutations depending on the particular cultural context, the basic plot is typified by the Babylonian Enuma elish (dating back, in part, to the second millennium bce), which tells of the struggle between a chaos monster associated with the primal waters (i.e., the demoness Tiamat and her forces) and a triumphant sky deity responsible for some significant cultural innovation (the warrior god Marduk, who slays Tiamat, divides her carcass to form the world, and establishes the central temple and righteous rule of the city-state of Babylon).

In the Enuma elish, Tiamat and her monstrous cohorts represent the older, otiose generation of gods that resist the noisy, rambunctious creative activity of the younger gods championed by Marduk. From this perspective—and it is an interpretive judgment found in many other versions of the combat myth—chaos and its first generation of creatures refer to the impotency of a form that, in time, is drained of its initial creative energy. Tiamat, in other words, stands for the dangerous principle of entropy, the negative, polluting force that seeks to dissolve all new life forms back into the silent slumber and amniotic inertia of death.

The law of cosmic life in this sense is the organic rule of chaotic disintegration that is necessary for new life. Such a cyclic return to chaos may be delayed, but even the younger gods, as part of a cosmos connected by origins with the principle of chaos, are still ultimately subject to the cosmic weariness and senility first displayed by Tiamat. One way to resolve this dilemma, which is seen prominently in monotheistic traditions, is to discover the reality of a dualistic separation between chaos and an absolutely transcendent, wholly spiritual or divine order impervious to the inevitable temporal change and collapse of all cosmic forms. Monistic theories asserting the fundamental unreality or illusory nature of chaos/cosmos represent another strategy.

The combat pattern is also witnessed in many tribal traditions, although the intensity of the antagonistic relationship between chaos and the human order (and consequently the dualistic translation of this as the polarity between death and life, evil and good, demon and god) is ordinarily heightened within the context of the classical or historical religions. Thus, such traditions often suggest that ritual remembrances of the mythic skirmish primarily function to celebrate the victory over chaos and the heroic finality of some authoritarian order. There is an emphasis here on the permanent suppression of chaos, or at least a denial that the primordial enemy possesses any positive attributes. The problem, as previously indicated, is that chaos is never completely overcome in ordinary cosmic life, although for some religions a postmortem heavenly existence (as well as a climactic apocalyptic purification of the cosmos) can be interpreted as a final and total victory over chaos.

While it is true that festivals of licensed folly are found in both tribal and classical traditions, the former tend to accept more readily the instrinsic value and positive ambiguity of a periodic ritual return to a chaotic or "liminal" condition. The danger perceived by such peoples is not so much chaos in the sense of the end of order and life but rather the social entropy and tension of too much deadening order. Chaos in this "primitive" sense is the pivot of cosmic and social equilibrium, and refers to the ritual reappearance of unstructured freedom and sheer potentiality. To refresh life, chaos must be disciplined and periodically embraced, not simply defeated.

The contemporary American satirist Peter De Vries has perversely suggested that if "in de beginning was de void, and de void was vit God," then it is probably the case that one "mustn't say de naughty void" (Blood of the Lamb, Boston, 1962, p. 181). In like manner, even when an implacably vile and naughty chaos is portrayed—as in some versions of the combat myth—there is often the contradictory implication that the divine champions are finally congenerous with their primordial foes. The forces of chaos and the watery void are always, it seems, the enemies of righteous order yet originally and simultaneously "vit God."

Chaos, it must be said, is both naughty and nice, or to borrow appropriately enough from Rudolf Otto's classic definition of the sacred (The Idea of the Holy, [1917] 1958), chaos is both repulsive and attractive in its awful appeal to the religious imagination. Its repugnant aspects are clearly seen in the many worldwide adumbrations of the combat scenario, but it remains to indicate the somewhat more muted allure of chaos as a positive and beneficial religious principle. Thus, there are what might be called "pro-chaos" religious traditions that in different ways espouse chaos as a goal. Of these there are, in general, three—sometimes overlapping—possibilities: (1) chaos may symbolize the final attainment of, and fusion with, some perfectly unconditioned unity and bliss totally beyond cosmic existence—a "nothingness that glistens with plentitude" (E. M. Cioran, The Temptation to Exist, Chicago, 1968, p. 155); (2) chaos may be experienced as a stage, threshold, or "dark night of the soul" at the ultimate edge of cosmic reality that leads to a distinct and higher vision of the absolutely transcendent Divine; and (3) chaos may represent the experience of a more paradoxical state, or coincidentia oppositorum, merging transcendent unity and cosmic multiplicity and functioning not as an end, but as a healing way station for a more harmonious inner and social life.

Such options obviously relate to "mystical" forms of world religions, but it should be recalled that a positive attitude toward chaos was already forecast by tribal rituals that periodically welcomed a twilight zone back into the human fold. Because of this sympathy, mystical forms of religion (along with other types of shamanistic-ecstatic, individualistic, and revolutionary religious movements) often manifest a kind of "primitive" sentiment toward chaos that contrasts and challenges the more one-sidedly negative evaluations seen in institutionalized religion.

One instance of these contrasting interpretations within a single tradition is found in India, where some radical forms of Upanisadic, Buddhist, and bhakti mysticism seek a supreme integration with the sacred "emptiness" of chaos. These forms can, in turn, be distinguished from the dharmic system of rigidly differentiated castes seen in Vedic and Puranic Hinduism. Within Western tradition—and frequently in tension with mainstream Christian, Jewish, and Islamic institutions—there are also movements that stress the mystical conjugation of the divine and chaos. Conceptions arising from such movements include, in Christianity, the Ungrund (the "unground" or abyss that gives rise to God's self-consciousness), described by Jakob Boehme (1575–1624); the qabbalistic idea of tsimtsum, the creative "gap" within God, in sixteenth-century Judaism; and the alchemical massa confusa (the prime matter, often imagined as an egg or coiled snake) in both Christian and Islamic esoteric circles after the Renaissance. All allude in some degree to the mythological chaos as a strangely positive image.

There are other cases of this pro-chaos persuasion, but one of the more striking examples is found in Chinese tradition. In ancient China during the Eastern Zhou period (c. eighth through second centuries bce) the mythological chaos was called hundun, which connoted the image of a Humpty-Dumpty-like, closed, embryonic condition or creature. Confucian thought and the ancient classics stressed the role of a succession of semidivine Sage-Kings who disciplined the chaotic forces of the natural world and carved the hierarchical order of the Middle Kingdom out of the carcass of the primitive condition of hundun. As a counterpoint to this point of view, the early Daoist texts suggest the existence of a veritable cult of chaos, since for these works the attainment of an authentically spontaneous and harmonious life required the rejection of conventional standards of propriety and recommended a return to an experience of primitive unity by means of the mystical "arts of hundun." Thus, in early Daoist texts (as distinct from the later institutionalized Daoist religion), the mythical hundun can be identified with the ultimate principle of the Dao as the rhythmic source and ground of life. Because of his periodic journeys in mind and heart back to the time of chaos, the Daoist mimics the seasonal regeneration of nature and the ritual regeneration of primitive cultural life and is able, therefore, to remain fresh and whole in the world.

Affirmations of the saving power of chaos have had a significant, although largely unorthodox, role to play in the history of religions; and, as broadly protesting all conventionalized truth, the cult and cultivation of chaos can be said to have inspired a whole spectrum of countercultural irruptions, "interstitial events," or "liminoid phenomena" throughout history. Because it rubs against the customary order of things, the religious, philosophical, artistic, and political "art of chaos" is always a risky enterprise, as indicated by the checkered careers of assorted Daoist mystics, Zen monks, holy fools, clownish alchemists, utopian Ranters, Romantic poets, Nietzschian nihilists, frenzied surrealists, neo-pagan anarchists, the Maoist "Gang of Four," and deconstructionist critics.

Conclusion

Perhaps the most responsible way to end an investigation of chaos is to refuse the temptation to parse a subject so hopelessly ironic. It is, after all, the principle of chaos that opens the abyss of indeterminacy and undecidability in all interpretive endeavors. Suffice it to say that, despite its decidedly negative public reputation, the image of chaos may be found in fact to have positive religious value. Even more important is the recognition that the idea of chaos represents one of the honored ways religions have tried to imagine the ambiguous origins and equivocal nature of existence. For this very reason the meaning of chaos in the history of religions maintains its imaginative integrity by remaining chaotic. Respecting the root topsy-turviness of chaos should not, however, prevent careful reflection on its imaginative history since, even in its most negative guise, the phantomlike shapes of chaos are directly related to the way particular religions have envisioned reality. The nature and significance of chaos, therefore, touches upon a number of issues that are central to the overall interpretive understanding and definition of religion.

See Also

Cosmogony; Dragons; Egg; Water.

Bibliography

Concerning the general theoretical background to the religious and philosophical understanding of chaos and order, see The Concept of Order, edited by Paul G. Kuntz (Seattle and London, 1968), for an eclectic selection of articles—especially pertinent are James K. Feibleman's "Disorder" (pp. 3–13) and, for the religious context, Charles Hartshorne's "Order and Chaos" (pp. 253–267) and Joseph M. Kitagawa's "Chaos, Order, and Freedom in World Religions" (pp. 268–289). See also David L. Hall's Eros and Irony (Albany, 1982), which provocatively analyzes cultural history in relation to varying conceptions of creation, chaos, and cosmology.

Barbara C. Sproul's Primal Myths: Creating the World (San Francisco, 1979) is a convenient sourcebook for the more important creation myths and includes some abbreviated, but helpful, commentary on the different religious images of chaos. More valuable for their discussions of the relation between the ideas of "creation" and "chaos" are Mircea Eliade's Myth and Reality (New York, 1963), which investigates the mythological and ritual meaning of chaos as correlated with the author's theory concerning the "prestige" of cosmogony, and Charles H. Long's Alpha: The Myths of Creation (New York, 1963), which includes some of the important mythological source materials as well as an extensive comparative examination of the structural significance of chaos imagery. For the ritual themes of "liminality" and "pollution" as suggestively analogous to the mythic idea of chaos, especially among tribal traditions, see Victor Turner's The Ritual Process (Ithaca, N.Y., 1977) and Mary Douglas's Purity and Danger: An Analysis of Concepts of Pollution and Taboo (New York, 1966).

The myth and ritual theme of combat that promotes a negative and dualistic evaluation of chaos is classically presented for ancient Near Eastern and biblical materials by Hermann Gunkel's Schöpfung und Chaos in Urzeit und Endzeit (Göttingen, 1895). But see also the more recent studies by Bernhard W. Anderson, Creation versus Chaos (New York, 1967), which emphasizes the biblical context; Mary K. Wakeman, God's Battle with the Monster (Leiden, 1973), which comparatively reexamines the ancient Near Eastern documentation; and Joseph Fontenrose, Python: A Study of Delphic Myth and Its Origins (Berkeley, 1959), which focuses on Greek tradition but draws upon a broad assortment of cross-cultural materials (i.e., Indo-European, ancient Near Eastern, East Asian, American Indian, etc.).

For studies that examine the more positive and ambiguous dimensions of chaos symbolism, along with related imagery, see, for India, Wendy Doniger O'Flaherty's Women, Androgynes, and Other Mythical Beasts (Chicago and London, 1980) and, for the ancient Chinese theme of hundun, my own Myth and Meaning in Early Taoism: The Theme of Chaos (Berkeley and London, 1983). In addition to their primary subject areas, both of these works comparatively cite a broad range of cross-cultural materials. Finally it is worth noting, among other possibilities, William Willeford's The Fool and His Scepter (London, 1969) and Mircea Eliade's Mephistopheles and the Androgyne: Studies in Religious Myth and Symbol (New York, 1965). The former is a fascinating literary study of the folkloric and popular embodiments of chaos in the Western tradition of the fool or jester, and the latter is a rich comparative study of different symbolic themes touching on the religious ideas of duality and the "coincidence of opposites."

Norman J. Girardot (1987)

Chaos

Revising the Newtonian world view

Current research

Chaos may depend on initial conditions and attractors

Resources

Chaos theory is used to model the overall behavior of complex systems. Despite its name, chaos theory is used to identify order in complex and otherwise seemingly unpredictable systems. As a mathematical concept, chaos is defined as any system that varies according to precise, pre-determined laws, even when their appearance seems random. Chaos theory, therefore, is defined as the study of complex non-linear dynamic systems, with the term complex meaning that the systems contain many variables and equations, non-linear meaning that equations are not of the first degree, and dynamic meaning that the system is never static (is always changing).

Chaos theory is used to understand explosions, complex chemical reactions (e.g., the Belousov-Zhabotinsky oscillating reaction that yields a red solution that turns blue at varying intervals of time), and many biological and biochemical systems. Chaos theory is now an important tool in the study of population trends and in helping to model the spread of disease. Epidemiologists use chaos theory to help predict the spread of epidemics.

Deterministic dynamical systems are those systems that are predictable based on accurate knowledge of the conditions of the system at any given time. When systems are, however, sensitive to their initial conditions, they eventually become unpredictable. In particular, chaos theory deals with complex nonlinear dynamic (i.e., nonconstant, nonperiodic, etc.) systems. Nonlinear systems are those described by mathematical recursion and higher algorithms. Deterministic chaos is mainly devoted to the study of systems, the behavior of which can, in principle, be calculated exactly from equations of motion.

Non-linear dynamic systems include systems of activities (weather, turbulence in fluids, the stock market) that cannot be visualized in a graph with a straight line. Although dictionaries usually define chaos as complete confusion, scientists who study chaos have discovered deep patterns that predict global stability in dynamic systems in spite of local instabilities.

Revising the Newtonian world view

English physicist and mathematician Sir Isaac Newton (1642–1727), and the physicists of the eighteenth and nineteenth centuries who built upon his work, showed that many natural phenomena could be accounted for in equations that would predict outcomes. If enough was known about the initial states of a dynamic system, then, all things being equal, the behavior of the system could be predicted with great accuracy for later periods, because small changes in initial states would result in small changes later on. For Newtonians, if a natural phenomenon seemed complex and chaotic, then it simply meant that scientists had to work harder to discover all the variables and the interconnected relationships involved in the physical behavior. Once these variables and their relationships were discovered, then the behavior of complex systems could be predicted.

But certain kinds of naturally occurring behaviors resisted the explanations of Newtonian science. The

weather is the most famous of these natural occur-rences, but there are many others. The orbit of the Moon around the Earth is somewhat irregular, as is the orbit of the dwarf planet Pluto around the Sun. Human heartbeats commonly exhibit minor irregularities, and the 24-hour human cycle of waking and sleeping is also irregular.

In 1961, American mathematician and meteorologist Edward Norton Lorenz (1917–) discovered that one of the crucial assumptions of Newtonian science is unfounded. Small changes in initial states of some systems do not result in small changes later on. The contrary is sometimes true: small initial changes can result in large, completely random changes later. Lorenz’s discovery is called the butterfly effect: named for the example of a butterfly beating its wings in China creating small turbulences that eventually affect the weather in New York City.

Lorenz, of Massachusetts Institute of Technology, made crucial discoveries in his research on the weather in the early 1960s. Lorenz had written a computer program to model the development of weather systems. He hoped to isolate variables that would allow him to forecast the weather. One day he introduced an extremely small change into the initial conditions of his weather prediction program: he changed one variable by one one-thousandth of a point. He found that his prediction program began to vary wildly in later stages for each tiny change in the initial state. This was the birth of the butterfly effect. Lorenz proved mathematically that long-term weather predictions based upon conditions at any one time would be impossible.

American mathematical physicist Mitchell Jay Feigenbaum (1944–) was one of several people who discovered order in chaos. He showed mathematically that many dynamic systems progress from order to chaos in a graduated series of steps known as scaling. In 1975, Feigenbaum discovered regularity even in orderly behavior so complex that it appeared to human senses as confused or chaotic. An example of this progression from order to chaos occurs if you drop pebbles in a calm pool of water. The first pebble that you drop makes a clear pattern of concentric circles. So do the second and third pebbles. However, if the pool is bounded, then the waves bouncing back from the edge start overlapping and interfering with the waves created by the new pebbles that you drop in. Soon the clear concentric rings of waves created by dropping the first pebbles are replaced by a confusion of overlapping waves.

American mathematician James A. Yorke (1941–) applied the term chaos to non-linear dynamic systems in the early 1970s, but before Yorke gave non-linear dynamical systems their famous name, other scientists had been describing the phenomena now associated with chaos.

Current research

Chaos theory has a variety of applications. One of the most important of these is the stock market. Some researchers believe that they have found non-linear patterns in stock indexes, unemployment patterns, industrial production, and the price changes in Treasury bills. These researchers believe that they can reduce to six or seven the number of variables that determine some stock market trends. However, the researchers concede that if there are non-linear patterns in these financial areas, then anyone acting on those patterns to profit will change the market and introduce new variables that will make the market unpredictable.

Population biology illustrates the deep structure that underlies the apparent confusion in the surface behavior of chaotic systems. Some animal populations exhibit a boom-and-bust pattern in their numbers over a period of years. In some years, there is rapid growth in a population of animals; this is followed by a bust created when the population consumes all of its food supply and most members die from starvation. Soon the few remaining animals have an abundance of food because they have no competition. Since the food resources are so abundant, the few animals multiply rapidly, and some years later, the booming population turns bust again as the food supplies are exhausted from overfeeding. This pattern, however, can only be seen if much data have been gathered over many years. Yet this boom-and-bust pattern has been seen elsewhere, including disease epidemics. Large numbers of people may come down with measles, but in falling ill, they develop antibodies that protect them from future outbreaks. Thus, after years of rising cases of measles, the cases will suddenly decline sharply because their antibodies naturally protect so many people. After a period of reduced cases of measles, the outbreaks will rise again and the cycle will start over, unless a program of inoculation is begun.

Chaos theory can also be applied to human biological rhythms. The human body is governed by the rhythmical movements of many dynamical systems: the beating heart, the regular cycle of inhaling and exhaling air that makes up breathing, the circadian rhythm of waking and sleeping, the saccadic (jumping) movements of the eye that allow humans to focus and process images in the visual field, and the regularities and irregularities in the brain waves of mentally healthy and mentally impaired people as represented on electroencephalograms. None of these dynamic systems is perfect all the time, and when a period of chaotic behavior occurs, it is not necessarily bad. Healthy hearts often exhibit brief chaotic fluctuations, and sick hearts can have regular rhythms. Applying chaos theory to these human dynamic systems provides information about how to reduce sleep disorders, heart disease, and mental disease.

Chaos may depend on initial conditions and attractors

It is now understood that chaotic behavior may be characterized by sensitive dependence on initial conditions and attractors (including, but not limited to strange attractors). A particular attractor represents the behavior of the system at any given time. The actual state of any system (i.e., measured characteristics) depends upon earlier conditions. If initial conditions are changed, even to a small degree, the actual results for the original and altered systems become different (sometimes drastically different) over time, even though the plot of the attractor for both the original and changed systems remains the same. In other words, although both systems yield different values as measured at any given time, the plots of their respective attractors (i.e., the overall behavior of the system) look the same.

See also Mathematics; Quantum mechanics; Physics.

Resources

BOOKS

Abraham, Ralph, and Toshisuke Ueda. The Chaos Avant-garde: Memories of the Early Days of Chaos Theory. Singapore and River Edge, NJ: World Scientific, 2000.

Banks, John. Chaos: A Mathematical Introduction. Cambridge, UK: Cambridge University Press, 2003.

KEY TERMS

Antibody —A molecule created by the immune system in response to the presence of an antigen (a foreign substance or particle). It marks foreign microorganisms in the body for destruction by other immune cells.

Boom-and-bust cycle —A recurring period of sharply rising activity (usually economic prosperity) that abruptly falls off.

Circadian rhythm —The rhythmical biological cycle of sleep and waking that, in humans, usually occurs every 24 hours.

Dynamics —The motion and equilibrium of systems that are influenced by forces, usually from the outside.

Electroencephalogram (EEG) —An electronic medical instrument used to measure brain activity in the form of waves printed on a sheet of paper.

Newtonian world view —The belief that actions in the physical world can be predicted (within a reasonable margin of error) according to physical laws, which only need to be discovered, combined appropriately, and applied accurately to determine what the future motions of objects will be.

Nonlinear —Something that cannot be represented by a straight line: jagged, erratic.

Population biology —The branch of biology that analyses the causes and (if necessary) solutions to fluctuations in biological populations.

Quantum —The amount of radiant energy in the different orbits of an electron around the nucleus of an atom.

Scaling —A regular series or progression of sizes, degrees, or steps.

Hilborn, Robert C. Chaos and Nonlinear Dynamics: An Introduction for Scientists and Engineers. Oxford and New York: Oxford University Press, 2000.

OTHER

Trump, Matthew A. “What is Chaos?” The University of Texas. (accessed October 4, 2006) <http://order.ph.utexas.edu/chaos/index.html>>.

Patrick Moore

Chaos

The word "chaos" is used in mathematics to mean something other than what it means in everyday English. In English, we speak of chaos as a state in which everything has gone awry, there is complete disorder, and there are no rules governing this behavior. In mathematics, chaotic systems are well defined and follow strict mathematical rules. Although chaotic systems are unpredictable, they do have certain patterns and structure, and they can be mathematically modeled, often by an equation or a system of equations.

Chaos theory is the study of systems that change over time and are inherently unpredictable. Some systems, such as Earth's orbit about the Sun, are very predictable over long periods of time. Other systems, such as the weather, are notoriously unpredictable. Chaos theory builds a mathematical framework to account for unpredictable systems, like the weather, which behave in a seemingly random fashion.

The last two decades of the twentieth century saw tremendous advances in our understanding of chaotic systems. Chaos theory has been applied* to the study of fibrillation in human hearts, the control of epileptic seizures, the analysis of how the brain processes information, the dynamics of weather, fluctuations in the stock market, the formation of galaxies, the flow of traffic during rush hour, and the stability of structures during earthquakes, to name a few. Wherever there are unstable or unpredictable systems, chaos scientists can be found attempting to reveal the underlying order in the chaos.

*In 1993, Goldstar Company created a washing machine whose pulsator motion exploited chaos theory.

Early History of Chaos Theory

Although chaos theory is very much a late-twentieth-century development, most chaos scientists consider the great nineteenth-century mathematician, Henri Poincaré, to be the true father of the discipline. The fundamental phenomenon common to all chaotic systems is sensitivity to initial conditions , and Poincaré was the first to articulate this sensitivity in his study of the so-called three-body problem.

Poincaré had described this problem in 1890 when he attempted to calculate the orbits for three interacting celestial bodies. Poincaré found that Newton's equations for two celestial objects were useless after a short time. The orbits became so tangled that he gave up hope of trying to predict where they would go. No one quite knew what to do about Poincaré's problem and so it remained essentially a mystery for the next eight decades.

Chaos in the Twentieth Century

In the 1950s meteorologists had great hopes that accurate weather prediction was about to become a reality. The advent of computers that could analyze vast amounts of data was seen as a great breakthrough in weather forecasting. Unfortunately, by the end of the decade, the optimism had begun to fade. Even with the rapid analysis of data from weather stations all over the world, forecasting did not seem to improve.

Meteorologist and mathematician Ed Lorenz of the Massachusetts Institute of Technology wondered why and began attempting to model the weather with equations programmed into his computer. The computer used the equations to simulate several months of weather, producing a graph that would rise and fall according to changes in the variables of Lorenz's weather equations.

At one point Lorenz wanted to take a closer look at a certain part of his graph, so he went back to an earlier point on the plot, inserted the values for that point from a printout from the computer, and waited to see a repetition of the graph he had seen before. At first he did, but after a few minutes he was startled to see a new plot diverging from the old plot, creating an entirely different version of the weather. After checking his computer for a malfunction and finding none, he discovered the cause of this unexpected behavior. When he had restarted the simulation, he had entered values from the printout that were rounded off to three decimal places, whereas the computer was using six place decimals for the simulation. This small variation in the initial values had produced a completely different version of the simulated weather.

By accident, Lorenz had rediscovered Poincaré's sensitivity to initial conditions. He realized that this meant that unless one has infinitely precise knowledge of all the initial conditions in a weather system, one cannot accurately predict the weather very far into the future. Lorenz called this finding "The Butterfly Effect"* because his results implied that the flapping of a butterfly's wings in Brazil could stir up weather patterns that might ultimately result in a tornado in Texas.

*The now-popular phrase "If a butterfly flaps its wings"—which many people use in connection with global ecology—actually came from the study of mathematical chaos.

Fish Population Studies. In 1970, the biologist Robert May was studying the growth of fish populations using a well-known mathematical model for populations with an upper bound to growth. This upper bound is sometimes called the "carrying capacity " of the environment. The model was the logistic difference equation P (t + 1) = rP (t )(1 − P (t )), where r is a number greater than 1 representing the growth rate, P (t ) is the population as a percentage of carrying capacity at time t, and the factor (1 - P (t )) represents the fact that as P (t ) gets closer to 1 (closer to 100 percent of the carrying capacity), the growth rate of the population slows to almost zero.

May had noticed that fish populations sometimes maintained the same population from one year to the next and sometimes would fluctuate from year to year. He decided to study what happened with the logistic equation when he varied the values of the parameter r, which essentially represent the growth rates of the population. The logistic difference equation is an iterator . This means that one inputs a certain initial population called P (0) into the equation, does the calculation, and receives as the output P (1), which is the population at the next time interval. Then this output value P (1) becomes the new input into the equation, yielding a new output P (2), and so on.

Using a simple hand calculator, May carried out hundreds of iterations (repetitions) using different values for the initial population, P (0), and different values of the growth parameter r. He discovered that when the value of r was between 1 and 3, the iteration produced a string of numbers that would ultimately settle on a single value no matter what initial value was used. This implied a stable population year after year. When r was raised slightly above 3, however, the output of the iteration cycled back and forth between two values.

This remained true for all initial values until May raised the parameter beyond 3.45 and observed that the values began oscillating among four values, again independent of the initial population. At about 3.54, the oscillation doubled again to eight distinct values, then to sixteen when r was increased passed 3.556, then to thirty-two and sixty-four, and so on. When the parameter value reached about 3.56994, this "period-doubling cascade," as May would call it, ended and the values seemed to jump about randomly.

At this point May noticed that the values produced by an iteration for a given r value were no longer independent of the initial value. For r = 3.6, for example, if the initial value were 0.1, the twentieth iteration was about0.7977, but if the initial value were 0.09, the twentieth iteration was about0.8635. The discrepancy became greater with additional iterations. Once again Poincaré's sensitivity to initial conditions had been rediscovered. In this case, May had discovered chaos in the logistic difference equation.

Feigenbaum's Constant. May's period-doubling cascade became the subject of intense study by the physicist Mitchell Feigenbaum in the mid-1970s. Whereas May had concerned himself with the qualitative aspects of the logistic system, Feigenbaum wanted to understand the quantitative basis of this period-doubling route to chaos, as it would come to be called. Starting with a handheld calculator and eventually making use of more powerful computers, Feigenbaum discovered a completely unexpected pattern associated with the parameter (r ) values at which each new period doubling occurred. He called these bifurcation values .

Taking the computations far beyond where May had taken them, Feigenbaum came up with exceptionally high accuracy for the bifurcation values. For example, the first six are b ₁ = 3, b ₂ = 3.449490…, b ₃ = 3.544090…, b ₄ = 3.556441…, b ₅ = 3.568759…, and b ₆ = 3.569692….So, for example, when the parameter is raised past b ₅ = 3.568759, the logistic equation's output changes from an oscillation among sixteen values to an oscillation among thirty-two values. Feigenbaum had the computer draw a graph showing parameter values on the horizontal axis and population values on the vertical axis. The result, known as the Feigenbaum plot, is now one of the icons of chaos theory.

The Feigenbaum plot shows how the bifurcations come increasingly closer as the parameter is increased from left to right. When the parameter value passes 3.56994…, the period doublings are no longer regular. This is the onset of chaos.

As fascinating as this was, Feigenbaum found something even more amazing in these numbers. He found that if he calculated ratios of the form (b _{k + 1} – b _k )/(b _k – b _{k – 1}) for larger and larger values of k, these ratios would approach the number 4.669201609…. Furthermore, he found that if heused other functions, such as sine or cosine , completely unrelated to the logistic equation, the result would be the same. The number 4.669201609… is now known as Feigenbaum's constant and is considered to be one of the fundamental constants of mathematics.

The Link to Fractals. The Feigenbaum plot exhibits self-similarity, meaning that any one of the branches looks like the entire plot. Figures that exhibit self-similarity are called "fractals ," a term invented by Benoit Mandelbrot in 1975. Just as Euclidean geometry is the geometry of Newtonian mechanics, fractal geometry is the geometry of chaos theory.

Geometrically, the dynamics of chaotic systems are described by figures called "attractors," of which the Feigenbaum plot is one example. Although these dynamics are unpredictable due to sensitivity to initial conditions, they are geometrically bound to a certain structure, the attractor. It is in this sense that chaos scientists are said to seek order in chaos.

see also Fractals.

Stephen Robinson

Bibliography

Devaney, Robert L. A First Course in Chaotic Dynamical Systems. New York: Addison-Wesley, 1992.

Gleick, James. Chaos: Making a New Science. New York: Penguin Books, 1987.

Gulick, Denny. Encounters with Chaos. New York: McGraw-Hill, 1992.

Kellert, Stephen. In the Wake of Chaos. Chicago: University of Chicago Press, 1993.

Peitgen, Heinz-Otto, Hartmut Jurgens, and Dietmar Saupe. Chaos and Fractals: New Frontiers of Science. New York: Springer-Verlag, 1992.

Peitgen, Heinz-Otto, and Peter Richter. The Beauty of Fractals. Heidelberg: Springer-Verlag, 1986.

Stewart, Ian. Does God Play Dice? The Mathematics of Chaos. Cambridge, MA: Blackwell Publishers, 1989.

Internet Resources

Fraser, Blair. The Nonlinear Lab. <http://www.apmaths.uwo.ca/~bfraser/nonlinearlab.html>.

The Mathematics of Chaos. Think Quest. <http://library.thinkquest.org/3120/>.

Trump, Matthew. What Is Chaos? The Ilya Prigogine Center for Studies in Statistical Mechanics and Complex Systems. <http://order.ph.utexas.edu/chaos/index.html>.

FEIGENBAUM'S UNIVERSAL CONSTANT

Although initially limited to studies of chaos theory, Feigenbaum's constant is now regarded as a universal constant of nature. Since the 1980s, scientists have found period-doubling bifurcations in experiments in hydrodynamics, electronics, laser physics, and acoustics that closely approximate this constant.

Chaos

Chaos theory is used to model the overall behavior of complex systems. Despite its name, chaos theory is used to identify order in complex and otherwise seemingly unpredictable systems.

Chaos theory is used to understand explosions, complex chemical reactions (e.g., the Belousov-Zhabotinsky oscillating reaction that yields a red solution that turns blue at varying intervals of time), and many biological and biochemical systems. Chaos theory is now an important tool in the study of population trends and in helping to model the spread of disease . Epidemiologists use chaos theory to help predict the spread of epidemics.

Deterministic dynamical systems are those systems that are predictable based on accurate knowledge of the conditions of the system at any given time. When systems are, however, sensitive to their initial conditions they eventually become unpredictable. In particular, chaos theory deals with complex nonlinear dynamic (i.e., nonconstant, nonperiodic, etc.) systems. Nonlinear systems are those described by mathematical recursion and higher algorithms. Deterministic chaos, is mainly devoted to the study of systems the behavior of which can, in principle, be calculated exactly from equations of motion .

Chaos theory is the study of non-linear dynamic systems, that is, systems of activities (weather , turbulence in fluids, the stock market) that cannot be visualized in a graph with a straight line. Although dictionaries usually define "chaos" as "complete confusion," scientists who study chaos have discovered deep patterns that predict global stability in dynamic systems in spite of local instabilities.

Revising the Newtonian world view

Isaac Newton and the physicists of the eighteenth and nineteenth centuries who built upon his work showed that many natural phenomena could be accounted for in equations that would predict outcomes. If enough was known about the initial states of a dynamic system, then, all things being equal, the behavior of the system could be predicted with great accuracy for later periods, because small changes in initial states would result in small changes later on. For Newtonians, if a natural phenomenon seemed complex and chaotic, then it simply meant that scientists had to work harder to discover all the variables and the interconnected relationships involved in the physical behavior. Once these variables and their relationships were discovered, then the behavior of complex systems could be predicted.

But certain kinds of naturally-occurring behaviors resisted the explanations of Newtonian science. The weather is the most famous of these natural occurrences, but there are many others. The orbit of the moon around Earth is somewhat irregular, as is the orbit of the planet Pluto around the sun . Human heartbeats commonly exhibit minor irregularities, and the 24-hour human cycle of waking and sleeping is also irregular.

In 1961, Edward N. Lorenz discovered that one of the crucial assumptions of Newtonian science is unfounded. Small changes in initial states of some systems do not result in small changes later on. The contrary is sometimes true: small initial changes can result in large, completely random changes later. Lorenz's discovery is called the butterfly effect: a butterfly beating its wings in China creates small turbulences that eventually affect the weather in New York.

Lorenz, of MIT, made crucial discoveries in his research on the weather in the early 1960s. Lorenz had written a computer program to model the development of weather systems. He hoped to isolate variables that would allow him to forecast the weather. One day he introduced an extremely small change into the initial conditions of his weather prediction program: he changed one variable by one one-thousandth of a point. He found that his prediction program began to vary wildly in later stages for each tiny change in the initial state. This was the birth of the butterfly effect. Lorenz proved mathematically that long-term weather predictions based upon conditions at any one time would be impossible.

Mitchell Feigenbaum was one of several people who discovered order in chaos. He showed mathematically that many dynamic systems progress from order to chaos in a graduated series of steps known as scaling. In 1975 Feigenbaum discovered regularity even in orderly behavior so complex that it appeared to human senses as confused or chaotic. An example of this progression from order to chaos occurs if you drop pebbles in a calm pool of water . The first pebble that you drop makes a clear pattern of concentric circles. So do the second and third pebbles. But if the pool is bounded, then the waves bouncing back from the edge start overlapping and interfering with the waves created by the new pebbles that you drop in. Soon the clear concentric rings of waves created by dropping the first pebbles are replaced by a confusion of overlapping waves.

Feigenbaum and others located the order in chaos: apparently chaotic activities occur around some point, called an attractor because the activities seem attracted to it. Figure 1 illustrates an attractor operating in threedimensional space . Even though none of the curving lines exactly fall one upon the other, each roughly circular set of curves to the left and right of the vertical line seems attracted to an orbit around the center of the set of circles. None of the curved lines in Figure 1 are perfectly regular, but there is a clear, visual structure to their disorder, which illustrates the structure of a simple chaotic system.

James Yorke applied the term "chaos" to non-linear dynamic systems in the early 1970s. But before Yorke gave non-linear dynamical systems their famous name, other scientists had been describing the phenomena now associated with chaos.

Current research

Chaos theory has a variety of applications. One of the most important of these is the stock market. Some researchers believe that they have found non-linear patterns in stock indexes, unemployment patterns, industrial production, and the price changes in Treasury bills. These researchers believe that they can reduce to six or seven the number of variables that determine some stock market trends. However, the researchers concede that if there are non-linear patterns in these financial areas, then anyone acting on those patterns to profit will change the market and introduce new variables which will make the market unpredictable.

Population biology illustrates the deep structure that underlies the apparent confusion in the surface behavior of chaotic systems. Some animal populations exhibit a boom-and-bust pattern in their numbers over a period of years. In some years there is rapid growth in a population of animals, followed by a bust created when the population consumes all of its food supply and most members die from starvation. Soon the few remaining animals have an abundance of food because they have no competition . Since the food resources are so abundant, the few animals multiply rapidly, and some years later, the booming population turns bust again as the food supplies are exhausted from overfeeding. This pattern, however, can only be seen if many data have been gathered over many years. Yet this boom-and-bust pattern has been seen elsewhere, including disease epidemics. Large numbers of people may come down with measles, but in falling ill, they develop antibodies that protect them from future outbreaks. Thus, after years of rising cases of measles, the cases will suddenly decline sharply because so many people are naturally protected by their antibodies. After a period of reduced cases of measles, the outbreaks will rise again and the cycle will start over, unless a program of inoculation is begun.

Chaos theory can also be applied to human biological rhythms . The human body is governed by the rhythmical movements of many dynamical systems: the beating heart , the regular cycle of inhaling and exhaling air that makes up breathing, the circadian rhythm of waking and sleeping, the saccadic (jumping) movements of the eye that allow us to focus and process images in the visual field, the regularities and irregularities in the brain waves of mentally healthy and mentally impaired people as represented on electroencephalograms. None of these dynamic systems is perfect all the time, and when a period of chaotic behavior occurs, it is not necessarily bad. Healthy hearts often exhibit brief chaotic fluctuations, and sick hearts can have regular rhythms. Applying chaos theory to these human dynamic systems provides information about how to reduce sleep disorders , heart disease, and mental disease.

Chaos may depend on initial conditions and attractors

It is now understood that chaotic behavior may be characterized by sensitive dependence on initial conditions and attractors (including, but not limited to strange attractors). A particular attractor represents the behavior of the system at any given time. The actual state of any system (i.e., measured characteristics) depends upon earlier conditions. If initial conditions are changed even to a small degree the actual results for the original and altered systems become different (sometimes drastically different) over time even though the plot of the attractor for both the original and changed systems remains the same. In other words, although both systems yield different values as measured at any given time the plots of their respective attractors (i.e., the overall behavior of the system) look the same.

See also Mathematics; Quantum mechanics; Physics.

Resources

books

Gleick, James. Chaos: Making a New Science. New York: Viking, 1987.

Prigogine, Ilya. The End of Certainty: Time, Chaos, and the New Laws of Nature New York: Free Press, 1998.

other

Trump, Matthew A. The University of Texas. "What is Chaos?" <http://order.ph.utexas.edu/chaos/index.html> (February 5, 2003).

Patrick Moore

KEY TERMS

Antibody

—A molecule created by the immune system in response to the presence of an antigen (a foreign substance or particle). It marks foreign microorganisms in the body for destruction by other immune cells.

Boom-and-bust cycle

—A recurring period of sharply rising activity (usually economic prosperity) which abruptly falls off.

Circadian rhythm

—The rhythmical biological cycle of sleep and waking which, in humans, usually occurs every 24 hours.

Dynamics

—The motion and equilibrium of systems which are influenced by forces, usually from the outside.

Electroencephalogram (EEG)

—An electronic medical instrument used to measure brain activity in the form of waves printed on a sheet of paper.

Newtonian world view

—The belief that actions in the physical world can be predicted (within a reasonable margin of error) according to physical laws, which only need to be discovered, combined appropriately, and applied accurately to determine what the future motions of objects will be.

Nonlinear

—Something that cannot be represented by a straight line: jagged, erratic.

Population biology

—The branch of biology that analyses the causes and (if necessary) solutions to fluctuations in biological populations.

Quantum

—The amount of radiant energy in the different orbits of an electron around the nucleus of an atom.

Scaling

—A regular series or progression of sizes, degrees, or steps.

cha·os / ˈkāˌäs/ • n. complete disorder and confusion: snow caused chaos in the region. ∎  Physics behavior so unpredictable as to appear random, owing to great sensitivity to small changes in conditions. ∎  the formless matter supposed to have existed before the creation of the universe. ∎  (Chaos) Greek Mythol. the first created being, from which came the primeval deities Gaia, Tartarus, Erebus, and Nyx.