When we look at the changing world that we are living in, we can categorize the types changes into a few fundamental categories: growth and recession, stagnation, cyclic behavior and unpredictable, erratic fluctuations. All of these phenomena can be described with very well developed linear mathematical tools. Here linear means that the result of an action is always proportional to its cause: if we double our effort, the outcome will also double. However, as Stan Ulam had pointed out, most of nature is non-linear in the same sense as most of zoology is non-elephant zoology. The situation that most of traditional science is focusing on linear systems can be compared to the story of the person who looks for the lost car keys under a street lamp because it is too dark to see anything at the place where the keys were lost. Only recently do we have access to methods and compute power to make significant progress in the field of non-linear systems and understand, for example, seemingly simple things like dripping faucets. One whole class of phenomena which does not exist within the framework of linear theory has become known under the buzz-word of chaos. The modern notion of chaos describes irregular and highly complex structures in time and in space that follow deterministic laws and equations. This is in contrast to the structureless chaos of traditional equilibrium thermodynamics. The basic example system that might be helpful for visualization, is a fluid on a stove, the level of stress is given by the rate at which the fluid is heated. We can see how close to equilibrium there exists no spatial structure, the dynamics of the individual subsystem is random and without spatial or temporal coherence. Beyond a given threshold of external stress, the system starts to self-organize and form regular spatial patterns (rolls, hexagons) which create coherent behavior of the subsystems (``order parameters slave subsystems''). The order parameters themselves do not evolve in time. Under increasing stress the order parameters themselves begin to oscillate in an organized manner: we have coherent and ordered dynamics of the subsystems. Further increase of the external stress leads to bifurcations to more complicated temporal behavior, but the system as such is still acting coherently. This continues until the system shows temporal deterministic chaos. The dynamics is now predictable only for a finite time. This predictability time depends on the degree of chaos present in the system. It will decrease as the system becomes more chaotic. The spatial coherence of the system will be destroyed and independent subsystems will emerge which will interact and create temporary coherent structures.
In a fluid we have turbulent cascades where vortices are created that will decay into smaller and smaller vortices. Analog situations in societies can be currently studied in the former USSR and Eastern Europe. James Marti speculates: ``Chaos might be the new world order'' [22]. At the limit of extremely high stress we are back to an irregular Tohu-wa-Bohu-type of chaos where each of the subsystems can be described as random and incoherent components without stable, coherent structures.
It has some similarities to the anarchy with which we started close to thermal equilibrium. Thus the notion of ``Chaos'' covers the range from completely coherent, slightly unpredictable, strongly confined, small scale motion to highly unpredictable, spatially incoherent motion of individual subsystems. A schematic graphics of this organization structure is displayed in fig.1:
Fig. 1:
Organization of Chaos and Turbulence.
There is frequent confusion between chaos and randomness. There are some similarities in the nature of chaotic and random system, but there are also some fundamental differences. Some of them are listed in fig. 2:
Fig. 2:
Discrimination table between Order, Chaos, and Randomness. Planets used to be
representations of a divine order. Chaotic signals can show spectra in the full
range from pure tones to very noisy. The dimension of a dynamical system indicates
the number of independent variables. An attractor determines the geometrical structure,
towards which a system will evolve.
The game of Roulette is an interesting example that might illustrate the distinction between random and chaotic systems: If we study the statistics of the outcome of repeated games, then we can see that the sequence of numbers is completely random. That led Einstein to remark: ``The only way to win money in Roulette is to steal it from the bank.'' On the other hand we know the mechanics of the ball and the wheel very well and if we could somehow measure the initial conditions for the ball/wheel system, we might be able to make a short term predict ion of the outcome. Exactly this has been done by a group of Santa Cruz students who called themselves ``Eudemonic Enterprises''. Their story how they used chaos theory to conquer the casinos of Las Vegas and Atlantic City is described in [2].