I once read a paper, in which:

  • a fluid in a container was heated from below,
  • after reaching temperature $T_1$, a circular motion (convection) was clearly distinguishable, in form of cylinder,
  • after reaching temperature $T_2$, the circular motion splitted into two circular convections, side by side two cylinders,
  • after reaching temperatures $T_3, T_4, \ldots$, etc. waves appeared on cyliders' flat-sides (where the height is measured),
  • the frequency of the waves doubled at temperatures $T_4, T_5, \ldots$.

Ratio of every temperature pair $T_1/T_2, T_2/T_3, \ldots$ was the same as in logistic map ratios of parameter $r$, and in every other map - the Feigenbaum number, the period-doubling route to chaos.

I cannot find the paper again.. Does anyone remember such paper? Or maybe other?


This is a famous period doubling experiment which is reprinted in Cvitanovic's "Universality in Chaos", along with other foundational papers on the period doubling route to chaos.

The paper you read is probably due to Libchaber and Maurer "A Rayleigh Bernard Experiment: Helium in a small box" Proceedings of the NATO Advanced Studies Institute on Nonlinear Phenomena at Phase Transitions p. 259 (1982).

or else Giglio, Musazzi, Perini "Transition to Chaotic Behavior via a Reproducible Sequence of Period Doubling Bifurcations" PRL 47, 243 (1981)

or else Libchaber, Laroche, Fauve "Period Doubling Cascade in Mercury, a Quantitative Measurement" J. Phys. Lett 43 L211 (1982)

I just copied these from Cvitanovic's table of contents, I read this paper you are talking about and it is one of these.

I should point out that it isn't surprising theoretically that the period doubling cascade is the same as any other one-dimensional map, because this is the most likely way for a system to make a period doubling cascade--- to have two directions unstable at once is measure zero, so you have a one-dimensional instability from a fixed point, and this is modelled by Feigenbaum, and it is universal, so universally applicable.


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