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Period doubling cascade in mercury, a quantitative
measurement
A. Libchaber, C. Laroche, S. Fauve
To cite this version:
A. Libchaber, C. Laroche, S. Fauve. Period doubling cascade in mercury, a quantitative
measurement. Journal de Physique Lettres, Edp sciences, 1982, 43 (7), pp.211-216.
�10.1051/jphyslet:01982004307021100�. �jpa-00232033�
L-211
Period doubling cascade in mercury, a quantitative measurement
A. Libchaber, C. Laroche and S. Fauve
Groupe de Physique des Solides de l’Ecole Normale Supérieure,
24, rue Lhomond, 75231 Paris Cedex 05, France
(Reçu le 21 decembre 1981, accepte le 12 fevrier 1982)
Résumé. 2014 Observation de la cascade de doublement de période dans une expérience de RayleighBénard
sur le mercure. La cellule expérimentale a un rapport d’aspect 0393 = 4 et comporte quatre rouleaux
convectifs. Un champ magnétique de 270 G est appliqué le long de l’axe des rouleaux. Le
nombre de Feigenbaum mesuré est 03B4 =
4,4. Le rapport des sous harmoniques successifs est de l’ordre
de 14 dB pour les sous harmoniques les plus bas mesurés.
Abstract 2014
Observation of the period doubling cascade in a Rayleigh-Bénard experiment in mercury.
The experimental cell has an aspect ratio 0393 = 4 and contains four convective rolls. A DC
magnetic field of 270 G is applied along the convective roll axis. The measured Feigenbaum number
is 03B4 = 4.4. The ratio of the successive subharmonics is of the order of 14 dB for the lowest measured
subharmonics.
Tome43 No 7 1 er AVRIL 1982
LE JOURNAL DE PHYSIQUE -LETTRES
J. Physique LETTRES
43 (1982) L-211 - L-216 ler AVRIL 1982,
Classification
Physics Abstracts
47.20 - 47.65
The period doubling cascade as a route to chaos is now well documented theoretically [1]]
as well as experimentally [2]. We present here new results on a Rayleigh-Benard experiment
in a cell of liquid mercury with an aspect ratio larger than in previously reported works on helium
or water (four convective rolls). The very good signal to noise ratio of the experiment allows a
precise determination of the Feigenbaum number and of the power ratio of the successive subharmonics.
In this experiment, a DC magnetic field, applied along the convective roll axis, introduces
an extra damping of the oscillators which favours the period doubling cascade.
1. An experimental dynamical system with controlled dissipation. - At low Prandtl number,
the relevant instability oftwo-dimensional convective rolls, which leads to time dependent convection
and chaos, is the oscillatory instability [3]. It was shown [2] in experiments of convection in
liquid helium that for a range of Prandtl numbers and wavenumbers of the convective pattern,
one route to chaos is a period doubling cascade, which follows the Feigenbaum scenario [1]. It is
customary to compare this type ofexperiments to a one-dimensional mapping. But this mapping
relates to strongly dissipative modes. A more realistic mapping is therefore a two-dimensional
one with two parameters, a constraint and a viscosity (area contraction in phase space). The
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyslet:01982004307021100
L-212 JOURNAL DE PHYSIQUE - LETTRES
simplest quadratic one is the Henon mapping [4] which reduces for infinite contraction to a onedimensional
mapping. In a way, we want to find a physical system depending on those two parameters.
In a Rayleigh-Benard experiment, the constraint is the Rayleigh number R. At low
Prandtl number, the damping of the modes associated with the oscillatory instability depends
on the Prandtl number and the wavenumber of the convective pattern [3]. But it is not easy to
control such parameters experimentally, and furthermore, they do not affect selectively the oscillatory
instability. On the contrary, a horizontal magnetic field, parallel to the convective roll axis,
will add a quantitative damping to the oscillatory instability, as we have recently shown [5]. This
increased damping ofthe oscillators will favour the period doubling cascade as a route to chaos [6].
2. Experimental apparatus. - Our experimental system has been described in detail elsewhere
[5, 7]. We use a parallelepipedic cell of aspect ratio r = 4 (dimensions 7 x 7 x 28 mm)
with, in the convective state, four rolls parallel to the shorter side wall (see reference five for visualization).
The top and bottom boundaries consist of two thick copper plates. The lateral boundaries
are made ofplexiglass. In the centre ofthe bottom plate, a small NTC (negative temperature
coefficient) thermistor is located on a 2 mm diameter hole and is adjacent to the convective fluid.
The temperature signal, given by the bolometer, is analysed by a 5 420 A H. P. digital analyser. An
Fig. 1. Effect
of a magnetic field parallel to the roll axis. The Rayleigh number is RI Rc = 2.9 at zero field.
The two oscillators are locked (f1/12 = 2), the amplitude of oscillator 11 about 20 dB larger than the amplitude
of oscillator f2 for H = 0. As H increases, the two oscillators tend to the same amplitude and keep
their locking ratio 2.
L-213PERIOD DOUBLING CASCADE IN MERCURY
electromagnet provides a uniform horizontal magnetic field. The experimental convective cell is
placed in a vacuum chamber.
3. Effect of a DC magnetic field, parallel to the roll axis, on the oscillating state. - Defining
the Rayleigh number for the onset ofconvection as Rc, the first time dependent instability observed
is the oscillatory instability, its onset being at R ^_r 2 Rc. As previously reported [7], the frequency
of the oscillatory mode increases linearly with the Rayleigh number. It allows, by measuring the
frequency, a precise determination ofeach bifurcation point Increasing slightly the Rayleigh number,
a second frequency f2 appears in the temperature spectrum which, within a very small range
of R, locks with fi, the ratio being it = 2 f2.
We then apply a DC magnetic field parallel to the roll axis up to a field of 270 G. We present,
in figure 1, the effect of the magnetic field on the direct local temperature recording. Two distinct
regimes appear.
4
For H > 100 G, the evolution is the one described in our previous study [5]. The frequency of
the oscillatory instability increases and so does the damping of the mode. This will tend at a
higher field to a complete inhibition of the oscillatory instability.
But for H 100 G, and the effect is already noticeable around 10 G, the relative oscillating
strength of the two locked oscillators changes. This sensitivity to a very small field is surprising.
Fig. 2. Direct
time recordings of temperature for various stages of the period doubling cascade showing
the onset of fl4 (R/R~ =
3.52), f/8 (RI Rc =
3.62), fl 16 (RI Rc =
3.65).
L-214 JOURNAL DE PHYSIQUE - LETTRES
Fig. 3. The
Fourier spectrum. Arrows indicate the peak at the frequency fl.
The net result of this phenomenon is that, at 270 G, the two modes are highly damped and have
about the same oscillating strength, whereas at zero field the modefhas an amplitude about 20 dB
larger than the mode f’r2.
4. The period doubling cascade. - From now on, we keep a constant magnetic field H = 270 G.
A typical recording is shown in figure 2 for ~//?c == 3.47. Let us note that it shows a striking
similarity with a recent Cray machine simulation done by Upson et al. [8] on a Rayleigh-Bénard
experiment in a small box. The corresponding Fourier spectrum is shown in figure 3A.
As we increase the Rayleigh number, a period doubling cascade develops up to f/32, well
resolved as far as the onset values up to fl 16.
In table I, we present the various onset values ofthe cascade ofpitchfork bifurcations. In figure 2,
the temperature recordings are presented, showing the development of the subharmonics fl4,
f/8 and ~/16, for R/ Rc = 3.52, 3.62 and 3.65. Their respective Fourier spectra are presented in
figures 3B, C, D for values of ~/~c close to the preceding ones.
Table I
L-215PERIOD DOUBLING CASCADE IN MERCURY
Fig. 4. Enlargement
of the recording showing the development of the cascade.
°
If we compute the Feigenbaum number 6 for the last three bifurcations, we get
This is to be compared with the theoretical asymptotic value [1] ð =
4.669...
We can also compute the ratio of the successive subharmonics amplitude called jM.
This ratio is measured directly on the temperature recordings. We show in figure 4 an enlargement
of the temperature signal after the f/16 bifurcation and in figure 5 an enlargement of its
Fourier spectrum. The last value of Jl measured is ~ ~ 5 to be compared with theoretical values
between 4.58 and,6.5 (the first one is given by Nauenberg and Rudnick [9], the second one by
Feigenbaum [1]). This measurement of p led to some confusion in the past, which can be understood
if we look at the Fourier spectrum in figures 3D and 5. It is clear there that the odd harmonics
of f/16 have an amplitude which is modulated and depends on the order of the harmonics.
Fig. 5. Enlargement
of the Fourier spectrum corresponding to figure 4.
L-216 JOURNAL DE PHYSIQUE - LETTRES
It is thus not clear how to calculate the ratio from the Fourier spectrum. The numbers we give in
table I are derived from the direct recording.
We have also observed beyond the accumulation point the inverse cascade. Its study and the
subtle phenomena of locking windows in the chaotic state will be presented elsewhere.
Let us add some final remarks. Recent computer experiments have shown the role of the breakdown
of some spatial symmetries on the onset of chaos [10]. In this respect, the presence of a DC
magnetic field plays an important role, such as, for example, introducing an anisotropic effective
viscosity. Clearly, for other values of the field and especially at very high fields, other routes to
chaos will be found. We must also stress the fact that a very small magnetic field affects the oscillating
modes which is somewhat surprising.
References
[1] FEIGENBAUM, M. J., Phys. Lett. 75A (1979) 375.
COLLET, P. and ECKMANN, J. P., Iterated maps on the interval as dynamical systems (Birkhaüser,
Boston) 1980.
[2] MAURER, J. and LIBCHABER, A., J. Physique Lett. 40 (1979) L-419.
LIBCHABER, A. and MAURER, J., J. Physique Colloq. 41 (1980) C3-51.
GIGLIO, M., MUZZATI, S. and PERINI, U., Phys. Rev. Lett. 47 (1981) 243.
LINSAY, P. S., Phys. Rev. Lett. 47 (1981) 1349.
[3] BUSSE, F. H., Rep. Progr. Phys. 41 (1978) 1929.
[4] HENON, M., Commun. Math. Phys. 77 (1976) 50.
[5] FAUVE, S., LAROCHE, C. and LIBCHABER, A., J. Physique Lett. 42 (1981) L-455.
[6] ARNEODO, A., COULLET, P., TRESSER, C., LIBCHABER, A., MAURER, J. and D’HUMIÈRES, D., About
the observation of the uncompleted cascade, preprint.
[7] FAUVE, S. and LIBCHABER, A., Chaos and order in nature, H. Haken Ed., (Springer) 1981.
[8] UPSON, C. D., GRESHO, P. M., SANI, R. L., CHAN, S. T. and LEE, R. L., Lawrence Livermore Lab., Preprint
UCRL 85555.
[9] NAUENBERG, M. and RUDNICK, J., UCSC preprint.
[10] LIPPS, F. B., J. Fluid Mech. 75 (1976) 113.
Mc LAUGHLIN, F. B. and ORSZAG, S. A., preprint.