4 to the stable fixed point). The smaller 1 /ΔΓ , the more energy needs to be supplied through vibration to reach the plateau of the curve γ$_{n}$ ( Γ ). In Fig. 6 we plotted the width ΔΓ of the region where γ grows. We observe that ΔΓ rises with the frequency up to f = 12 . 5 Hz and then drops almost an order of magnitude. This be- havior signals a qualitative change around f = 12 . 5 Hz . for this frequency 1 /ΔΓ is much smaller than for the other ones we measured, therefore around this partic- ular frequency we need to supply more energy to the granular bed to make it go from zero growth rate to the fastest growth. To probe the nature of the transition at low Γ , we plot in Fig. 5, τ = γ$_{max}$/γ versus ξ = ( Γ − Γ$_{c}$ ) /Γ$_{c}$ . The parameter τ can be thought as a characteristic re- laxation time for the evolution from a symmetric to an asymmetric configuration. The critical value Γ$_{c}$ of the adimensional acceleration was taken as the mean be- tween the highest Γ before γ becomes non zero, and the lowest value of Γ for which γ is different from zero. At the lower amplitudes it was observed that the critical value Γ$_{c}$ for the transition from zero growth to nonzero γ , shifted to higher values of Γ as the frequency in- creased. We can see how τ diverges as Γ approaches the critical value Γ$_{c}$ . More experiments have to be done before we can determine the nature of this transition. 6 8 10 12 14 16 18 20 f (Hz) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 0 10 20 30 40 50 60 70 Fig. 6 Plot of ΔΓ versus the frequency of vibration. The param- eter ΔΓ is a measure of the increase in energy supplied to the granular bed through vibrations, to go from zero growth to the plateau of the curve γ ( Γ ) (see fits in Fig. 3) for a given frequency. A sharp increase occurs around 12 . 5 Hz. Fig. 5 The U-tube instability seems to exhibit a continuous phase transition. Here we can see how the characteristic time τ ( ξ ) diverges as Γ approaches a critical value Γ c ≈ 0. The data shown correspond to all the non zero γ measured. A physical model that would explain our data and the above relationship for the growth rate and the adi- mensional acceleration for a given frequency is beyond the scope of this paper, but we consider that the sim- plicity of the results should help to suggest the rele- vant conditions that need to be satisfied by a reasonable model for the behavior of a vertically shaken granular bed in a partially filled U-tube. Finally, we want to concentrate on the region of nonzero growth, for each curve we fixed the frequency and changed the amplitude A of vibration, assuming that γ$_{min}$ ≈ 0, we can rewrite equation (1) in the fol- lowing form: We have made an experimental characterization of the rate of growth for the height difference Δh between the free levels of the granular columns in a partially filled U-tube subject to vertical vibrations. We have observed a continuous transition from zero growth to non zero growth rate γ . The region of finite growth was well described by a differential equation analogous to the logistic equation. We also observed an abrupt This equation is analogous to the well know logis- tic equation so we can picture γ$_{n}$ = 1 as a stable fixed point for γ$_{n}$ ( Γ ). Therefore as Γ increases the growth rate approaches its maximum value (which correspond This expression gives the value of the normalized growth rate as a function of Γ . For each frequency equation (2) is a solution of the following differential equation: where γ$_{n}$ = γ/γ$_{max}$ . γ$_{n}$ ( Γ ) = 1 1 + e ( Γ i − Γ ) /ΔΓ , (2) dγ$_{n}$ dΓ = 1 ΔΓ [1 − γ$_{n}$ ] γ$_{n}$. (3) 4 Conclusions