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Crutchfield, James P., J. Doyne Farmer, Norman H. Packard, and Robert S. Shaw. “Chaos.”

This previously published paper by James P. Crutchfield, J. Doyne Farmer, Norman H. Packard, and Robert S. Shaw is reprinted here to give a broad introduction and background to the science of chaos and complexity.

Until recently scientists assumed that natural phenomena such as the weather or the roll of the dice could in principle be predictable given sufficient information about them. Now we know that this is impossible. “Simple deterministic systems with only a few elements can generate random behavior.” Though the future is determined by the past, small uncertainties are amplified so radically by chaotic systems that, in practice, their behavior rapidly becomes unpredictable. Still there is “order in chaos,” since elegant geometrical forms underlie and generate chaotic behavior. The result is “a new paradigm in scientific modeling” which both limits predictability in a fundamental way and yet extends the domain in which nature can be at least partially predictable.

The article acknowledges the challenge to Laplacian determinism posed by quantum mechanics for subatomic phenomena like radioactive decay, but stresses that large-scale chaotic behavior, which focuses instead on macroscopic phenomena like the trajectory of a baseball or the flow of water, “has nothing to do with quantum mechanics.” In fact many chaotic systems display both predictable and unpredictable behavior, like fluid motion which can be laminar or turbulent, even though they are governed by the same equations of motion. As early as 1903, Henri Poincaré suggested that the explanation lay in the exponential amplification of small perturbations.

Chaos is an example of a broad class of phenomena called dynamical systems. Such systems can be described in terms of their state, including all relevant information about them at a particular time, and an equation, or dynamic, that governs the evolution of the state in time. The motion of the state, in turn, can be represented by a moving point following an orbit in what is called state space. The orbits of non-chaotic systems are simple curves in state space. For example, the orbit of a simple pendulum in state space is a spiral ending at a point when the pendulum comes to rest. A pendulum clock describes a cyclic, or periodic, orbit, as does the human heart. Other systems move on the surface of a torus in state space. Each of these structures characterizing the long-term behavior of the system in state space - the point, the cycle, the torus - is called an attractor since the system, if nudged, will tend to return to this structure as it continues to move in time. Such systems are said to be predictable.

In 1963, Edward Lorenz of MIT discovered a chaotic system in meteorology which showed exponential spreading of its previously nearby orbits in state space. The spreading effect is due to the fact that the surface on which its orbits lie is folded in state space. Such a surface is called a strange attractor, and it has proven, in fact, to be a fractal. The shape of a strange attractor resembles dough as it is mixed, stretched, and folded by a baker. With this discovery we see that “random behavior comes from more than just the amplification of errors and the loss of the ability to predict; it is due to the complex orbits generated by stretching and folding.”

The essay closes with some profound questions about scientific method. If predictability is limited in chaotic systems, how can the theory describing them be verified? Clearly this will involve “relying on statistical and geometric properties rather than on detailed prediction.” What about the assumption of reductionism in simple physical systems? Chaotic systems display a level of behavioral complexity which frequently cannot be deduced from a knowledge of the behavior of their parts. Finally, the amplification of small fluctuations may be one way in which nature gains “access to novelty” and may be related to our experience of consciousness and free will.

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