Period doubling cascade in mercury, a quantitative measurement

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 of two-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 of experiments 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 of the 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 of plexiglass. In the centre of the 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-213 PERIOD 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 of convection 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 of each 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 mode f has 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 of the cascade of pitchfork 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.