From: dtd@world.std.com (Don Davis) Subject: Re: History of chaos theory Date: Mon, 07 Aug 2000 08:23:08 -0500 Newsgroups: sci.math To: dtd@world.std.com, Dave Rusin Summary: [missing] (A copy of this message has also been posted to the following newsgroups: sci.math) In article <965624724.11624.0.nnrp-10.9e989aee@news.demon.co.uk>, "Dan Goodman" wrote: >> what's new is the math & experimental physics that >> proves the idea to be real, inevitable, & widespread >> in nature (cf. the ruelle-takens-newhouse theorem). Dan Goodman replied: > Interesting, what is the ruelle-takens-newhouse theorem? > Also, in what sense is the idea inevitable? also, david rusin replied: > Can you summarize what the RTN theorem says? > (I was sort of under the impression that Chaos theory was > limited to statements of the form, "sometimes there is a great > dependence on initial conditions", if you'll excuse the hyperbole.) "Let v be a constant vector field on the torus T^n = R^n/Z^n. if n >= 3, every C^2 neighborhood of v contains a vector field v' with a strange Axiom A attractor. If n >= 4, we may take C^oo instead of C^2." - S. Newhouse, D. Ruelle, and F. Takens, "Occurence of strange Axiom A attractors near quasi-periodic flows of T^m m>= 3," Commun. Math. Phys. 64 (1978) pp. 35-40. for n >= 3, this theorem was proven by ruelle & takens. newhouse proved the n >= 4 case. in '79, fenstermacher, gollub, and swinney showed that taylor-couette flow displays NRT behavior: a metal cylinder rotates inside a concentric glass cylinder, with water in a narrow gap between them. the rotation of the inner cylinder sets up a convective flow in the water, where the convection cells are toroidal in shape, like a stack of bicycle in- ner tubes. when the rotation is slow, these tori all have fixed minor radii, and the fluid rotates with just two incommensurate velocities: the fluid flows around the circumference of the gap between the cylinders, and the toroidal cells convect radially at the same time. this flow is steady and non-turbulent. when the inner cylinder rotates faster, the toroidal convection gains a third non-harmonic motion, because each cell takes on a new shape. instead of resembling a simple torus, each cell becomes swollen in some places and narrow in others, so that the boundary between two cells looks sinusoidal. note that the theorem's torus T^n is a phase space, and doesn't represent the convection cells' shape. as the rotational speed is increased more, after this third flow velocity shows up in the flow's power spectrum, some turbulence per se appears in the flow, and at high rota- tional speeds, the turbulence destroys the cells' struc- ture. if the experimental setup is kept vibration-free, then a fourth non-harmonic flow velocity can be seen, just before the turbulence sets in with increasing cylinder speed. this experiment was one of two experimental confirmations that chaotic transition to turbulence could be modelled mathematically. the other experiment, which was done at about the same time by experimenters in europe, used a tiny convection cell containing liquid helium, and the convection was driven by a minute amount of heat applied electrically to the bottom of the cell. in that case, as more heat was applied, the flow velocity changed from a constant value to a periodic value, then the period doubled steadily, until turbulence eventually set in. period-doubling is a different mathematical formalism from NRT, and has been more widely popularized, probably because period-doubling can be described without talking about phase space and incommensurate frequencies in power spectra. B^) that the NRT theorem should apply to the taylor-couette experiment is supported by recent work (early '90's) on the navier-stokes equation. roger temam told me that it has been shown that N-S solutions can show Axiom A behavior, if one assumes that flow velocities must be bounded. boundedness is an issue, because 1/r vortices abound in fluid flow. a good survey of early source papers in nonlinear dynamics, with math and physics reprints, can be found in a book by cvitanovic: "universality in chaos" -- http://bookmark.iop.org/bookpge.htm?book=294p http://snowmass.phys.nwu.edu/~predrag/papers/preprints.html#books_chaos - don davis, boston -