Encyclopedia of Nonlinear Science
At first glance, the words chaos and order seem to be contradictory. Indeed they are contradictory in terms of common definitions of the words, but not in terms of their mathematical usages. The definition of chaos in the Oxford American Dictionary (1986) is “great disorder or confusion.” In contrast, the mathematical notion of chaos explains that data that seems to be random may have been generated by a deterministic and ordered process. In this sense, the field might have profitably been called “order theory” instead of “chaos theory.” There are two common usages of the phrase “order from chaos.” The first refers to seeking deterministic “chaotic” models to explain complex phenomenon, thus “order in chaos.” The second refers to the fact that some chaotic dynamical systems can also exhibit regular islands of simplicity within their phase space, or “order from chaos.” We will discuss each in turn.
Order in Chaos
The fact that seemingly simple and deterministic evolution rules can give rise to extremely complicated motion dates to Henri Poincaré (1892), in the setting of celestial mechanics. Perhaps the most famous example of unpredictable behavior is due to Edward Lorenz (1963), who discovered that even the most simplified models of the weather could produce data for which it is impossible to make long-term forecasts. Lorenz identified the “butterfly effect,” or “sensitive dependence to initial conditions,” whereby even small measurement errors quickly grow to swamp the signal. In 1975, Tien Yien Li and James A.Yorke published a paper entitled, “Period Three Implies Chaos,” (Li & Yorke, 1975). The most famous impact of this paper was that it coined the word chaos, the field of study describing the nonlinear effect in which even the simple systems of Poincaré, Lorenz, and others can display sensitive dependence in a bounded domain. The chaos theory has attracted a great deal of popular attention, partly, in fact, due to its sexy title. Part of the appeal is also due to the scientific tradition since the time of Newton of searching for mechanistic explanations of reality.
The concept of phase space is useful for mapping the behavior of a dynamical system. Phase space could be described as the set of all possible relevant variables, which creates a closed description of the time evolution of the system (See Phase space). For example, for a simple pendulum, we need to specify all possible angular position and angular momentum states to uniquely define all solutions; the phase space for the periodic oscillation is a closed curve, a circle. For an electronic tank oscillator (LRC) circuit, we need to specify capacitor charge and current through the inductor. The dimension of the system is the dimension of the phase space. A billiard table’s worth of balls, for example, requires many variables for a complete description—two variables for position and two for momentum for each of the balls.
As a highly interdisciplinary field, chaos theory has been successful in finding deterministic chaos, or order, in what was once thought to be mere noise, or at
least an extremely high-dimensional effect. There are numerous explicit examples of chaos in many areas, including biology, electrical engineering, chemistry, celestial mechanics, and brain and heart physiology. Thus, the old idea that extremely complicated data must always be due to extremely complicated effects is false (see Figure 1). One popular misuse of these ideas is a belief that all complicated data must have underlying order in chaos. This can be phrased as “Is what seems complicated always really simple?” The answer, of course, is no; it remains true that noise and other high-dimensional effects can also be responsible for complexity. What is true is that what seems complicated may sometimes be simple, in the sense of having a low-dimensional chaotic model. To cite one popular question, the stock market prices unarguably constitute extremely complicated data, but is there underlying order in chaos here, or is the explanation due to intractably high-dimensionality?
Order from Chaos
Some chaotic systems have an occasional tendency to exhibit simple behavior. This is often described as regular islands in an otherwise chaotic phase space. The classic mathematical example of regular islands arises in Hamiltonian systems, which can exhibit a phase space of solutions in which chaotic solutions are both intermingled and bounded by nested islands around islands around islands, and so on, of KAM-like tori (circle-like integrable solutions) (Arrowsmith & Place, 1990; Meiss, 1992).
Put differently, “order from chaos” and “regular islands” are terms commonly used to refer to the presence of an “attractor” (an imaginary point in the phase space about which the trajectories appear to orbit) in a system that one thinks should behave in a complicated manner. For example, consider a thought experiment of a game of pool in which we assume no friction so that the balls never stop moving (a two-dimensional Lorentz gas). Following the path of one specific ball, say the seven-ball, is an extremely complicated problem displaying sensitive dependence, since a small change in the velocity or position of the ball affects the next collision with the wall or with the next ball, an effect that multiplies upon subsequent collisions. However, there are several attractors near which the ball’s motion becomes quite simple—the pocket holes. Once within the rim of a pocket (its basin of attraction), falling in becomes inevitable, as it would require a relatively large energy perturbation to prevent it from falling in. In this sense, we can say that the regions of phase space, corresponding to being stuck in the billiards pocket, are regular islands in a chaotic sea.
As analogies, such descriptions of islands in chaotic seas have been extended by some to explain the emergence of a coherent phenomenon from complicated processes. There is perhaps no more intriguing question than the origins of life. Proponents of emergence theories suggest that life in the original Hadean seas (the “primordial” seas on early Earth) gave rise to life through a capture process such as the billiards game (Waldrop, 1992; Kauffman, 1995). While randomly chosen initial conditions in the billiards game may each individually be unlikely to lead to capturing a ball in a pocket (and certainly any analogous attractors in chemical processes must be exceedingly unlikely), the way to win an unlikely bet is to play very quickly, over and over. This may have been the process that led to complex organic molecules, according to proponents of the emergence theory.
Emergence and order from chaos have also been used to describe processes whose attractors are surprisingly complicated themselves, as sets, but arise from surprisingly simple rules. Michael Barnsley has called the following “The Chaos Game” (Barnsley, 1993). First label the vertices of a triangle, 1, 2, 3. Using a random number generator (such as a six-sided die), assign probabilities to each triangle vertex, say die sides 1–2 to vertex 1, die sides 3–4 to vertex 2, and die sides 5–6 to vertex 3. Roll the die to randomly select one of the vertices, and record its planar coordinates (x, y). Then roll the die again to randomly select another vertex, and record the point halfway between the resulting new vertex and the current (x, y) as the new (x, y). Repeat indefinitely. For each newly defined (x, y), we record a dot for a pictorial record. Most people guess that the result will uniformly fill the triangle with a smattering of dots, but the mathematical fact is surprising. The result is an extremely intricate structure, a fractal called the Sierpinski gasket (see Figure 2). Because this simple rule gives rise to such a complicated structure, the argument goes, perhaps many of the other intricacies we see around us might have emerged from other simple rules.
Finally, we mention the meaning of “order from chaos,” as developed from theories of the Nobel prize-winning physicist Ilya Prigogine, whose work on dissipative structures of systems held from thermal
equilibrium has been used to study self-organizing systems (Prigogine, 1984). Prigogine has defined complexity as the ability to switch between different modes of behavior as the environmental conditions vary. Thus, he has described a phenomenon in which far-from-equilibrium systems transition “from being to becoming,” which some describe as order from chaos.
ERIK M.BOLLT
See also Chaotic dynamics; Emergence; Kalmogorov-Arnol’d-Moser theorem; Phase space
Further Reading
Arrowsmith, D.K. & Place, C.M. 1990. An Introduction to Dynamical Systems, Cambridge and New York: Cambridge University Press
Barnsley, M. 1993. Fractals Everywhere, 2nd edition, San Francisco: Morgan Kaufmann
Kauffman, S. 1995. At Home in the Universe, the Search for the Laws of Self-Organization and Complexity, Oxford and New York: Oxford University Press
Lorenz, E.N. 1963. Deterministic nonperiodic flow. Journal of Atmospheric Science, 20:130–141
Li, T.Y. & Yorke, J.A. 1975. Period three implies chaos. American Mathematics Monthly, 82:985–992
Meiss, J.D. 1992. Symplectic maps, variational principles, and transport. Reviews of Modern Physics, 64:795
Poincaré, H. 1892–99. Les méthodes nouvelles de la mécanique céleste, Paris: Gauthier-Villars, 3 vols; as New Methods of Celestial Mechanics, New York: American Institute of Physics, 1993
Prigogine, S. 1984. Order out of Chaos: Man’s New Dialogue with Nature, New York: Bantam Books
Waldrop, M. 1992. Complexity: The Emerging Science at the Edge of Chaos and Order, New York: Touchstone Press
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