"[…] the equation of small oscillations of a pendulum also holds for other vibrational phenomena. In investigating swinging pendulums we were, albeit unwittingly, also investigating vibrating tuning forks." (George Pólya, "Mathematical Methods in Science", 1977)
"Where chaos begins, classical science stops. For as long as the world has had physicists inquiring into the laws of nature, it has suffered a special ignorance about disorder in the atmosphere, in the fluctuations of the wildlife populations, in the oscillations of the heart and the brain. The irregular side of nature, the discontinuous and erratic side these have been puzzles to science, or worse, monstrosities." (James Gleick, "Chaos", 1987)
"When loops are present, the network is no longer singly connected and local propagation schemes will invariably run into trouble. [...] If we ignore the existence of loops and permit the nodes to continue communicating with each other as if the network were singly connected, messages may circulate indefinitely around the loops and process may not converges to a stable equilibrium. […] Such oscillations do not normally occur in probabilistic networks […] which tend to bring all messages to some stable equilibrium as time goes on. However, this asymptotic equilibrium is not coherent, in the sense that it does not represent the posterior probabilities of all nodes of the network." (Judea Pearl, "Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference", 1988)
"Now, the main problem with a quasiperiodic theory of turbulence (putting several oscillators together) is the following: when there is a nonlinear coupling between the oscillators, it very often happens that the time evolution does not remain quasiperiodic. As a matter of fact, in this latter situation, one can observe the appearance of a feature which makes the motion completely different from a quasiperiodic one. This feature is called sensitive dependence on initial conditions and turns out to be the conceptual key to reformulating the problem of turbulence." (David Ruelle, "Chaotic Evolution and Strange Attractors: The statistical analysis of time series for deterministic nonlinear systems", 1989)
"All physical objects that are 'self-similar' have limited self-similarity - just as there are no perfectly periodic functions, in the mathematical sense, in the real world: most oscillations have a beginning and an end (with the possible exception of our universe, if it is closed and begins a new life cycle after every 'big crunch' […]. Nevertheless, self-similarity is a useful abstraction, just as periodicity is one of the most useful concepts in the sciences, any finite extent notwithstanding." (Manfred Schroeder, "Fractals, Chaos, Power Laws Minutes from an Infinite Paradise", 1990)
"Nature abounds with periodic phenomena: from the motion of a swing to the oscillations of atoms, from the chirping of a grasshopper to the orbits of the heavenly bodies. […] Of course, nothing in nature is exactly periodic. All motion has a beginning and an end, so that, in the mathematical sense, strict periodicity does not exist in the real world. Nevertheless, periodicity has proved to be a supremely useful concept in elucidating underlying laws and mechanisms in many fields." (Manfred Schroeder, "Fractals, Chaos, Power Laws Minutes from an Infinite Paradise", 1990)
"If we have several modes, oscillating independently, the motion is, as we saw, not chaotic. Suppose now that we put a coupling, or interaction, between the different modes. This means that the evolution of each mode, or oscillator, at a certain moment is determined not just by the state of this oscillator at that moment, but by the states of the other oscillators as well. When do we have chaos then? Well, for sensitive dependence on initial condition to occur, at least three oscillators are necessary. In addition, the more oscillators there are, and the more coupling there is between them, the more likely you are to see chaos." (David Ruelle, "Chance and Chaos", 1991)
"What we now call chaos is a time evolution with sensitive dependence on initial condition. The motion on a strange attractor is thus chaotic. One also speaks of deterministic noise when the irregular oscillations that are observed appear noisy, but the mechanism that produces them is deterministic." (David Ruelle, "Chance and Chaos", 1991)
"One reason nature pleases us is its endless use of a few simple principles: the cube-square law; fractals; spirals; the way that waves, wheels, trig functions, and harmonic oscillators are alike; the importance of ratios between small primes; bilateral symmetry; Fibonacci series, golden sections, quantization, strange attractors, path-dependency, all the things that show up in places where you don’t expect them [...] these rules work with and against each other ceaselessly at all levels, so that out of their intrinsic simplicity comes the rich complexity of the world around us. That tension - between the simple rules that describe the world and the complex world we see - is itself both simple in execution and immensely complex in effect. Thus exactly the levels, mixtures, and relations of complexity that seem to be hardwired into the pleasure centers of the human brain - or are they, perhaps, intrinsic to intelligence and perception, pleasant to anything that can see, think, create? - are the ones found in the world around us." (John Barnes, "Mother of Storms", 1994)
"Real dynamical problems typically involve nonlinear differential equations of second order, but these often simplify greatly if we investigate small oscillations about a position of equilibrium. Coupled oscillators are particularly interesting, an early example being the double pendulum, first studied by Euler and Daniel Bernoulli in the 1730s."
"Systems which exhibit chaotic oscillations typically do so for some ranges of the relevant parameters but not for others, so one matter of obvious interest is how the chaos appears (or disappears) as one of the parameters is gradually varied."
No comments:
Post a Comment