"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)
"Cellular automata are perfect feedback machines. More precisely, they are mathematical finite state machines which change the state of their cells step by step." (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)
"In a qualitative way of thinking, universality can be seen
to be not so surprising. There are two arguments to support this. The first
part has simply to do with nonlinearity. Just as a linear object has a constant
coefficient of proportionality between, for example, its tension and its
expansion, a similar, but nonlinear version, has an effective coefficient
dependent upon its extension. So, consider two completely different nonlinear
systems. By adjusting things correctly it is not inconceivable that the
effective coefficients of each part of each of the two systems could be set the
same so that then their behaviors could, at least initially, be identical. That
is, by setting some numerical constants (properties, so to speak, that specify
the environment, mathematically called ‘parameters’) and the actual behaviors
of these two systems, it is possible that they can do the identical thing. For
a linear problem this is ostensibly true: For systems with the same number of
parts and mutual connections, a freedom to adjust all the parameters allows one
to be adjusted to be identical (truly) to the other. But, for many pieces, this
is many adjustments. For a nonlinear system, adjusting a small number of
parameters can be compensated, in this quest for identical behavior, by an
adjustment of the momentary positions of its pieces. But then it must be that
not all motions can be so duplicated between systems."
"In its core, the deterministic credo means that the universe
is comparable to the ordered running of a tremendously precise clock, in which
the present state of things is, on the one hand, simply the consequence of its
prior state, and, on the other hand, the cause of its future state. Present,
past and future are bound together by causal relationships; and according to
the views of the determinists, the problem of an exact prognosis is only a
matter of the difficulty of recording all the relevant data."
"Linearity means that the rule that determines what a piece of a system is going to do next is not influenced by what it is doing now. More precisely, this is intended in a differential or incremental sense: For a linear spring, the increase of its tension is proportional to the increment whereby it is stretched, with the ratio of these increments exactly independent of how much it has already been stretched. Such a spring can be stretched arbitrarily far, and in particular will never snap or break. Accordingly, no real spring is linear."
"Mathematics is sometimes described as the science which generates eternal notions and concepts for the scientific method: derivatives‚ continuity‚ powers‚ logarithms are examples. The notions of chaos‚ fractals and strange attractors are not yet mathematical notions in that sense‚ because their final definitions are not yet agreed upon." (Heinz-Otto Peitgen et al, "Chaos and Fractals: New Frontiers of Science", 2004)
"Much of chaos as a science is connected with the notion of ‘sensitive dependence on initial conditions.’ Technically, scientists term as ‘chaotic’ those nonrandom complicated motions that exhibit a very rapid growth of errors that, despite perfect determinism, inhibits any pragmatic ability to render accurate long-term prediction. […] The most important fact is that there is a discernibly precise ‘moment’, with a corresponding behavior, which is neither chaotic nor nonchaotic, at which this transition occurs. Yes, errors do grow, but only in a marginally predictable, rather than in an unpredictable, fashion. In this state of marginal predictability inheres embryonically all the seeds of the chaotic behavior to come. That is, this transitional point, the legitimate child of universality, without full-fledged sensitive dependence upon initial conditions, knows fully how to dictate to its progeny in turn how this latter phenomenon must unfold. For a certain range of possible behaviors of strongly nonlinear systems - specifically, this range surrounding the transition to chaos - the information obtained just at the transition point fully organizes the spectrum of behaviors that these chaotic systems can exhibit."(Ray Kurzweil, "The Singularity is Near", 2005)
"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)
"The incommensurability of the diagonal of a square was initially a problem of measuring length but soon moved to the very theoretical level of introducing irrational numbers. Attempts to compute the length of the circumference of the circle led to the discovery of the mysterious number. Measuring the area enclosed between curves has, to a great extent, inspired the development of calculus." (Heinz-Otto Peitgen et al, "Chaos and Fractals: New Frontiers of Science" 2nd Ed., 2004)
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