On Chaos VI

"Despite the disorder observed in Nature, one finds enough traces of the wisdom and power of its Author that one cannot fail to recognize Him." (Pierre L Maupertuis, "Les Loix du Mouvement et du Repos, déduites d'un Principe Métaphysique", 1746)

"Never in the annals of science and engineering has there been a phenomenon so ubiquitous‚ a paradigm so universal‚ or a discipline so multidisciplinary as that of chaos. Yet chaos represents only the tip of an awesome iceberg‚ for beneath it lies a much finer structure of immense complexity‚ a geometric labyrinth of endless convolutions‚ and a surreal landscape of enchanting beauty. The bedrock which anchors these local and global bifurcation terrains is the omnipresent nonlinearity that was once wantonly linearized by the engineers and applied scientists of yore‚ thereby forfeiting their only chance to grapple with reality." (Leon O Chua, "Editorial", International Journal of Bifurcation and Chaos, Vol. l (1), 1991)

"Most of us use the word 'chaos' rather loosely to represent anything that occurs randomly, so it is natural to think that the motion described by the erratic pendulum above is completely random. Not so. The scientific definition of chaos is different from the one you may be used to in that it has an element of determinism in it. This might seem strange, as determinism and chaos are opposites of one another, but oddly enough they are also compatible." (Barry R Parker, "Chaos in the Cosmos: The stunning complexity of the universe", 1996)

"Nonlinearity is important because it can lead to chaos. That's not to say that we get chaos all the time with nonlinear equations; in practice it only occurs under certain conditions." (Barry R Parker, "Chaos in the Cosmos: The stunning complexity of the universe", 1996)

"What chaos implies is not catastrophes, but rather our inability to make long-range predictions [...]" (Barry R Parker, "Chaos in the Cosmos: The stunning complexity of the universe", 1996)

"What is chaos? Everyone has an impression of what the word means, but scientifically chaos is more than random behavior, lack of control, or complete disorder. [...] Scientifically, chaos is defined as extreme sensitivity to initial conditions. If a system is chaotic, when you change the initial state of the system by a tiny amount you change its future significantly." (Barry R Parker, "Chaos in the Cosmos: The stunning complexity of the universe", 1996)

"According to a 'sociological' view of mathematics, a system, in general, should be able to do whatever is permitted by the laws governing it: the normal state of anarchy is chaos! From this point of view, we should expect that, in the absence of conservation laws, typical motions should be dense in the space available to them; Kolomogorov’s theorem denies this, saying that when the laws are relaxed a bit, the majority of motions stay 'pretty much' where they were, as if in fear of a non-existent police force." (John H Hubbard, "The KAM Theorem", 2004)

"An apparent paradox is that chaos is deterministic, generated by fixed rules which do not themselves involve any elements of change. We even speak of deterministic chaos. In principle, the future is completely determined by the past; but in practice small uncertainties, much like minute errors of measurement which enter into calculations, are amplified, with the effect that even though the behavior is predictable in the short term, it is unpredictable over the long term." (Heinz-Otto Peitgen et al, "Chaos and Fractals: New Frontiers of Science" 2nd Ed., 2004)

"Chaos theory, too, is occasionally in danger of being overtaxed by being associated with everything that can be even superficially related to the concept of chaos. Unfortunately, a sometimes extravagant popularization through the media is also contributing to this danger; but at the same time this popularization is also an important opportunity to free areas of mathematics from their intellectual ghetto and to show that mathematics is as alive and important as ever." (Heinz-Otto Peitgen et al, "Chaos and Fractals: New Frontiers of Science" 2nd Ed., 2004)

"Natural laws, and for that matter determinism, do not exclude the possibility of chaos. In other words, determinism and predictability are not equivalent. And what is an even more surprising rinding of recent chaos theory has been the discovery that these effects are observable in many systems which are much simpler than the weather. [...] Moreover, chaos and order (i.e., the causality principle) can be observed in juxtaposition within the same system. There may be a linear progression of errors characterizing a deterministic system which is governed by the causality principle, while (in the same system) there can also be an exponential progression of errors (i.e., the butterfly effect) indicating that the causality principle breaks down." (Heinz-Otto Peitgen et al, "Chaos and Fractals: New Frontiers of Science" 2nd Ed., 2004)