International Encyclopedia of Systems and Cybernetics

"The irregular, unpredictable behavior of deterministic nonlinear systems (R. JENSEN – 1987, p.168).

"The operation mode of persistently unstable systems" (after B. JONES, 1994).

This meaning of the term "chaos" was introduced by J. YORKE, of Maryland University (1975).

It is not possible to develop, even sketchily, the concept of chaos in this dictionary, basically because:

1) it is highly complex with broad mathematical development

2) it is still in the making

According to A.B. CAMBEL, chaos theory is "a heterogeneous amalgam of different techniques of mathematics and science. Systems that upon analysis are found to be nonlinear, nonequilibrium, deterministic, dynamic and that incorporate randomness so that they are sensitive to initial conditions, and have strange attractors are said to be chaotic" (1993).

J. GLEICK introduces chaos in the following terms: "To some physicists chaos is a science of process rather than state, of becoming rather than being…

"Chaos breaks across the lines that separate scientific disciplines. Because it is a science of the global nature of systems, it has brought together thinkers from fields that had been widely separated…

"Chaos poses problems that defy accepted ways of working in science. It makes strong claims about the universal behavior of complexity (1987, p.5).

As to its real importance, GLEICK comments on R. MAY opinion: "Chaos should be taught, he argued. It was time to recognize that the standard education of a scientist gave the wrong impression. No matter how elaborate linear mathematics could get, with its Fourier transforms, its orthogonal functions, its regression techniques, MAY argued that it inevitably misled scientists about their overwhelmingly nonlinear world: "The mathematical intuition so developed ill equips the student to confront the bizarre behavior exhibited by the simplest of discrete nonlinear systems" he wrote.

"Not only in research, but also in the everyday world of politics and economics, we would all be better off if more people realized that simple nonlinear systems do not necessarily possess simple dynamical properties" (p.80).

The chaos concept is an important part of a new set of mathematical theories at present in process of build-up. It is more or less closely related to catastrophe theory, stability theory, fractals, a renewal of the concept of cycle, general topology, thermodynamics of irreversible systems far away from equilibrium and criticality and percolation theory.

It has led to a deep questioning of the traditional – and absolute – notions of determinism and randomness. Formerly, in the words of GLEICK: "Either deterministic mathematics produced steady behavior, or random external noise produced random behavior. That was the choice.

"In the context of this debate, chaos brought an astonishing message: simple deterministic models could produce what looked like random behavior. The behavior actually had an exquisite fine structure, yet any piece of it seemed indistinguishable from noise" (p.78-9).

Also J. CASTI emphasizes: "Scientifically speaking, chaos is only the appearance of randomness, not the real thing" (1994, p.88).

I. EKELAND, in turn, cites MAXWELL: "It is a metaphysical doctrine that the same antecedents always produce the same consequents. Nobody could deny it. But it is of reduced usefulness in a world like this one, where the same antecedents can never be found and where nothing recurs twice… The physical axiom which looks alike is that similar antecedents produce similar consequents. But here we changed from exactness to similarity, from absolute precision to a more or less crude approximation" (1984, p.85).

Chaos definitely puts an end to LAPLACE's Clockwork Universe and, altogether, to the age old dream of absolute predicitability.

In any complex system, and at any moment, there is a new set of initial conditions. This leads to the practical impossibility of definite evolutive forecasts, as possibilities relentlessly open up at any future time and the internal correlations of the system become reduced. The only invariance is a global enveloppe embedding many possibilities as no future transformations can escape the historical development of archetypes.

From another viewpoint, as stated by Chr. VIDAL: "… the strange attractor concept, while undoubtly broadening the recognized domain of determinism, simultaneously establishes on quite more strong bases the need to resort to statistical methods… now imposed by the very logic of mathematics" (P. BERGÉ et al., 1984, p.288).

It should at last be recognized that "pure" determinism, as well as "pure" randomness are merely ideal limit cases of all possible behaviors of systems, corresponding, the first to strictly lineal systems and the second to completely degraded order.

As observed by E. LASZLO: "Chaos is not the opposite of order but its refinement: it is a subtle, complex, and ultrasensitive form or order" (1992, p.243).

This is probably the reason why science started to discover it just one century ago (POINCARÉ's three bodies problem) and to explain it no more than 30 years ago (E. LORENZ).