Chaos Theory

Various Types of Attractors

A system, according to chaos theory, can exhibit either some pattern of stability or order according to some range of values that its variables can take, or be driven into chaotic behavior based on another set of values of the variables. That the chaotic behavior arises as a function of the defining equations and the values of its variables illustrates the idea of deterministic chaos. The set of values for the equation that produce stable results are termed attractors. A perfectly stable system that never varies from its steady state would have its equation graphed out such that it would be a point (the trajectory of values never deviates the initial values); such a set of values is termed a point attractor. A homeostatic system, one that deviates from its original state but that is brought back to its original state, has its trajectory of values following a toroidal trajectory, forming what is known as a toroidal or doughnut attractor. In illustrating the differences in explanatory power that chaos theory offers in explaining systems behavior, what is usually depicted is the butterfly or Lorenz attractor. With this attractor, a system will follow a toroidal path for a given set of values, but small variations can cause the system to shift into an alternative toroidal trajectory. That is, the equation describes a system with two homeostatic pathways possible, with small changes at a critical moment causing them to switch between stable trajectories. The resulting graphed trajectories appear to be two adjacent and linked toroidal attractors, vaguely resembling the wings of a butterfly, hence the name butterfly attractor. Higher order attractors, with three or more stable states, are possible as well. The set of values that drive the system into chaotic behavior are termed strange attractors. The trajectories followed by the equation under the conditions of strange attractors often exhibit recursive self-similarity (fractal structure).

The Butterfly Effect

Both attractors and strange attractors illustrate another important concept in chaos theory, this being sensitivity to initial conditions. Very minute changes in value for the variables could cause an orderly system to switch between stable states, as in a Lorenz attractor. Alternatively, these minute changes in values could drive the system into chaotic behavior, as in strange attractors. This idea of sensitivity to initial conditions has been popularized as the butterfly effect, which basically states that the minute turbulence produced by a butterfly flapping its wings in one location could make the difference between pleasant weather and violent storms elsewhere on the planet. Sensitivity to initial conditions is a concept that chaos theory shares with complexity theory; although the two differ, in that chaos theory views this sensitivity to be endogenous to the system, whereas complexity theory tends to stress the sensitivity of its component variables to exogenous noise. Chaos theory is broadly concerned with describing closed systems, whereas complexity theory tends to view systems more openly.

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