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A scale law in a dripping faucet
24 February 1997
PHYSICS LETTERS A
Physics Letters A 226 ( 1997) 269-274
ELSEWIER
A scale law in a dripping faucet J.G. Marques da Silva, J.C. Sartorelli I, W.M. Gonplves, Institute de Fhica, Universidade de S&o Pa&o, C.P. 66318, 053/5-970
R.D. Pinto
SEo Pa&o. Brazil
Received 9 September 1996; revised m~usc~pt received 3 December 1996; accepted for publi~tion 3 December 1996 Communicated by C.R. Doering
Abstract The evolution to a period-l motion after an inverse secondary Hopf bifurcation, at fo = 39.976 drops/s, was characterized by the autocorrelation function. The amplitude of the autocorrelation function for f > fo decays exponentially; its characteristic correlation drop scales with /(f - fc)/fciy, where fC = 39.897 drops/s and y = -2.28 f 0.03. PACS: 47.52.fj; 05.45.+b Keywords: Leaky faucet; Drop
Recently an inverse secondary Hopf (or Neimark) bifurcation [ 1] in a leaky faucet experiment was reported. A T2 torus evolves from a period-5 frequency locking to a smaller quasiperiodic torus by increasing the water flux. Also a deviation of some experimental points from one scale law that describes the Hopf bifurcation was observed. In order to clarify these points we redid the measurements changing the nipple/detection system geometry a little. The details of the experimental apparatus can be found in Ref. [ 21. The interdrop time intervals T, are obtained by detecting the passage of the drops through a laser beam focused in a photodiode. The induced pulses in the photodiode are collected by a microcomputer parallel port and software in the C language calculates the times between successive pulses as well as their widths. Now we have put the laser beam closer to the nipple faucet (19 mm) and we took 24 longer time series with 16 384 drops each. Keeping the water reservoir level constant, the water flux was increased ’ E-mail: sa~o~lli~if.usp.br. 0375-9601/97/$17.00
by opening the faucet, from a drop rate of f M 39.27 drops/s up to f M 40.14 drops/s, a little bit before the water flux becomes continuous at the laser level. The quasiperiodic motion region evolves from a period-5 motion to a quasiperi~ic motion, as shown in Fig. 1, where 16 first return maps T,+l versus T, among the 24 are displayed. From the first up to the fifth series we have the periodic motion and the quasiperiodic behavior starts close to f = 39.69 drops/s in the seventh series giving rise to a limit circle with decreasing size with the drop rate increasing. At f = 39.93 drops/s ( 19th series) this limit circle closes and despite the decreasing size/noise ratio the profile of the observed spots shows traces of this quasiperiodic motion. The respective power spectra of T,* = T, - (T), are shown in Fig. 2. The power spectra of the first six series present a behavior similar to the one shown by the fourth series, one fundamental frequency FO and its first harmonic F2 = 2Fn. At f = 39.69 drops/s (seventh series) a new pair of frequencies Fi and F3 = FO+ Fl starts to appear, that persists until the 18th series at f = 39.90
Copyright @ 1997 Elsevier Science B.V. All rights reserved. PII SO37S-9601(96)00941-3
241 24
25
26 T,
Fig.
@=I
I. First return maps r,+l versus 7;1of f6 time (right top comer) are indicated on each frame.
series
among a sequence of
drops/s. The last six plots correspond to the closed limit circle, shown in Fig. 1, and the presence of an Fa frequency of low intensity confirms that we should have an embedded quasiperiodic regime in an apparent period-l motion characterized by a quasipe~odic ripple over a period- 1 off-set. In Fig. 3 the power spectra peak intensities are shown as a function of the drop rate showing the three regions of different regimes. A Hopf bifurcation is described [ 3,1] by twodImensiona maps (r, 8) -+ (r’, t9’>,where the radius
24.
The drop rate as well as the respective series number
of the limit circle is given by d r; = -p
fo>*
(1)
where d/a is negative since the bifurcation occurs for f < fo, and f is the drop rate and fa is a critical drop rate. ro was chosen as the size of the attractor 2ro = Tmax- Tti,,, where T,, (‘T&i,) is the mean value of T calculated in the last (first) bin of a ten-bin histogram. The rotation number, here identified as Fo, is related to the control parameter through
271
Fig. 2, Power spectra of the data shown in Fig. I. The drop rate as we11as the respective series number are indicated on each frame. Each time series was divided in four successive subseries of 4096 drops, each plot is the mean of the four power spcctm.
272
J.C. Marques
du Silvu et &/Physics
Letters A 226 (1997)
269-274
0.18
Fig. 3. The power spectra peak intensities of the 24 time series as a function of the drop rate. The continuous lines are guides for the eyes. 0.3
I
I
0.2
0.1
C 3
ro2( ms*) Fig. 5. Fn versus ri. The continuous line is the curve fit of Eq. (3) to the experimental data in the quasiperiodic region. The misaligned points are the ones in the apparent period-l region.
drop rate of fa = 39.976 drops/s was obtained. As above for fo, in the apparent period-l region, there is no circle limit, therefore the histogram technique nor the second moment is suitable to characterize rg. In Fig. 5 we have the rotation number FO as a function of r& The continuous line is the curve fitting of Eq. (3) to the experimental data for t-i 2 0.041 ms2. The misaligned points, below ri = 0.05 ms2, belong to the apparent period-l region. In order to characterize the passage from the inverse Hopf bifurcation to the apparent period-l we also calculated the normalized aut~o~eiation function [4] of the 24 experimental time series
f (drops/s) Fig. 4. I$ versus f. The continuous line is the curve fit of Eq. ( I ) to the experimental data in the quasiperiodic region. The critical drop ram is fn = 39.976 drops/s. (*) Data with lower size/noise ratio. (2) 2rrFo = c + br$
(3)
In Fig. 4 the plot of rg versus f is shown, the continuous line is the curve fitting of Eq. (1) to the experimental data in the quasiperiodic region, and a critical
that showed the respective characteristic behavior, periodic or quasiperiodic pattern, until the 18th time series. In Fig. 6 the plots of the six autocorrelation functions of the time series in the apparent period-l region (19th-24th series) are shown. The amplitudes of the autocorrelation functions decay exponentially, therefore we can define a correlation drop r by
J.G. Marques da Silva et al./Physics Latters A 226 (1997) 269-274
273
1.
0.
CJ O.
-0. -1. 20
40
60
60
100
20
40
60
60
1
k(drops) Fig. 6. The autocorrelation functions in the apparent period-l region. The dashed lines are the fit of the exponential decay, Eq. (5). to the maximum amplitude data. The correspondent correlation drop (T) as well as the drop rate and the series number are shown in each frame.
IG( k)
Imax= GOe-k/T.
(5)
By plotting T versus I( f - fc) /fcl on a log-log scale, see Fig. 7, where fc is a critical drop rate, whose value was chosen as the one that gives the minimum fitting error of Eq. (6) to the experimental data on a log-log scale (see inset of Fig. 7), fc = 39.897 drops/s and y = -2.28 f 0.03 were obtained,
It was shown that after an inverse Hopf bifurcation the evolution of the dynamical system to an apparent period-l motion is better described by the autocorrelation function. The amplitude of the autocorrelation function decays exponentially, its characteristic correlation drop scales with the drop rate.
Fig. 7. T versus I( f - fc) /fcI.The continuous line is the curve fit of Eq. (6) to the data. The inset shows the fitting error on the choice of the fc value. It should be noted that fc= 39.897 drops/s is very close to the last quasiperiodic drop rate ( 18th series).
This work was partially financed agencies Finep, CNPq and Fapesp.
by the Brazilian
274
J.G. Marques da Silva et al./Physics Letters A 2.26 (1997) 269-274
References [l] R.D. Pinto, W.M. GonGalves, J.C. Sartorelli and M.J. de Oliveira, Phys. Rev. E 52 (1995) 6896. [2] J.C. Sartorelli, W.M. Gonglves and R. Pinto, Phys. Rev. E 49 ( 1994) 3963.
131 J. Guckenhei~r
and P. Holmes, Nonlinear oscitlations, dynamical systems and bifurcations of vector fieids (Springer, Berlin, 1986). 141 H.D.I. Abarbanel, R. Brown, J.J. Sidorowich and L.S. Tsimring, Rev. Mod. Phys. 65 ( 1993) 133 I.
PHYSICS LETTERS A
Physics Letters A 226 ( 1997) 269-274
ELSEWIER
A scale law in a dripping faucet J.G. Marques da Silva, J.C. Sartorelli I, W.M. Gonplves, Institute de Fhica, Universidade de S&o Pa&o, C.P. 66318, 053/5-970
R.D. Pinto
SEo Pa&o. Brazil
Received 9 September 1996; revised m~usc~pt received 3 December 1996; accepted for publi~tion 3 December 1996 Communicated by C.R. Doering
Abstract The evolution to a period-l motion after an inverse secondary Hopf bifurcation, at fo = 39.976 drops/s, was characterized by the autocorrelation function. The amplitude of the autocorrelation function for f > fo decays exponentially; its characteristic correlation drop scales with /(f - fc)/fciy, where fC = 39.897 drops/s and y = -2.28 f 0.03. PACS: 47.52.fj; 05.45.+b Keywords: Leaky faucet; Drop
Recently an inverse secondary Hopf (or Neimark) bifurcation [ 1] in a leaky faucet experiment was reported. A T2 torus evolves from a period-5 frequency locking to a smaller quasiperiodic torus by increasing the water flux. Also a deviation of some experimental points from one scale law that describes the Hopf bifurcation was observed. In order to clarify these points we redid the measurements changing the nipple/detection system geometry a little. The details of the experimental apparatus can be found in Ref. [ 21. The interdrop time intervals T, are obtained by detecting the passage of the drops through a laser beam focused in a photodiode. The induced pulses in the photodiode are collected by a microcomputer parallel port and software in the C language calculates the times between successive pulses as well as their widths. Now we have put the laser beam closer to the nipple faucet (19 mm) and we took 24 longer time series with 16 384 drops each. Keeping the water reservoir level constant, the water flux was increased ’ E-mail: sa~o~lli~if.usp.br. 0375-9601/97/$17.00
by opening the faucet, from a drop rate of f M 39.27 drops/s up to f M 40.14 drops/s, a little bit before the water flux becomes continuous at the laser level. The quasiperiodic motion region evolves from a period-5 motion to a quasiperi~ic motion, as shown in Fig. 1, where 16 first return maps T,+l versus T, among the 24 are displayed. From the first up to the fifth series we have the periodic motion and the quasiperiodic behavior starts close to f = 39.69 drops/s in the seventh series giving rise to a limit circle with decreasing size with the drop rate increasing. At f = 39.93 drops/s ( 19th series) this limit circle closes and despite the decreasing size/noise ratio the profile of the observed spots shows traces of this quasiperiodic motion. The respective power spectra of T,* = T, - (T), are shown in Fig. 2. The power spectra of the first six series present a behavior similar to the one shown by the fourth series, one fundamental frequency FO and its first harmonic F2 = 2Fn. At f = 39.69 drops/s (seventh series) a new pair of frequencies Fi and F3 = FO+ Fl starts to appear, that persists until the 18th series at f = 39.90
Copyright @ 1997 Elsevier Science B.V. All rights reserved. PII SO37S-9601(96)00941-3
241 24
25
26 T,
Fig.
@=I
I. First return maps r,+l versus 7;1of f6 time (right top comer) are indicated on each frame.
series
among a sequence of
drops/s. The last six plots correspond to the closed limit circle, shown in Fig. 1, and the presence of an Fa frequency of low intensity confirms that we should have an embedded quasiperiodic regime in an apparent period-l motion characterized by a quasipe~odic ripple over a period- 1 off-set. In Fig. 3 the power spectra peak intensities are shown as a function of the drop rate showing the three regions of different regimes. A Hopf bifurcation is described [ 3,1] by twodImensiona maps (r, 8) -+ (r’, t9’>,where the radius
24.
The drop rate as well as the respective series number
of the limit circle is given by d r; = -p
fo>*
(1)
where d/a is negative since the bifurcation occurs for f < fo, and f is the drop rate and fa is a critical drop rate. ro was chosen as the size of the attractor 2ro = Tmax- Tti,,, where T,, (‘T&i,) is the mean value of T calculated in the last (first) bin of a ten-bin histogram. The rotation number, here identified as Fo, is related to the control parameter through
271
Fig. 2, Power spectra of the data shown in Fig. I. The drop rate as we11as the respective series number are indicated on each frame. Each time series was divided in four successive subseries of 4096 drops, each plot is the mean of the four power spcctm.
272
J.C. Marques
du Silvu et &/Physics
Letters A 226 (1997)
269-274
0.18
Fig. 3. The power spectra peak intensities of the 24 time series as a function of the drop rate. The continuous lines are guides for the eyes. 0.3
I
I
0.2
0.1
C 3
ro2( ms*) Fig. 5. Fn versus ri. The continuous line is the curve fit of Eq. (3) to the experimental data in the quasiperiodic region. The misaligned points are the ones in the apparent period-l region.
drop rate of fa = 39.976 drops/s was obtained. As above for fo, in the apparent period-l region, there is no circle limit, therefore the histogram technique nor the second moment is suitable to characterize rg. In Fig. 5 we have the rotation number FO as a function of r& The continuous line is the curve fitting of Eq. (3) to the experimental data for t-i 2 0.041 ms2. The misaligned points, below ri = 0.05 ms2, belong to the apparent period-l region. In order to characterize the passage from the inverse Hopf bifurcation to the apparent period-l we also calculated the normalized aut~o~eiation function [4] of the 24 experimental time series
f (drops/s) Fig. 4. I$ versus f. The continuous line is the curve fit of Eq. ( I ) to the experimental data in the quasiperiodic region. The critical drop ram is fn = 39.976 drops/s. (*) Data with lower size/noise ratio. (2) 2rrFo = c + br$
(3)
In Fig. 4 the plot of rg versus f is shown, the continuous line is the curve fitting of Eq. (1) to the experimental data in the quasiperiodic region, and a critical
that showed the respective characteristic behavior, periodic or quasiperiodic pattern, until the 18th time series. In Fig. 6 the plots of the six autocorrelation functions of the time series in the apparent period-l region (19th-24th series) are shown. The amplitudes of the autocorrelation functions decay exponentially, therefore we can define a correlation drop r by
J.G. Marques da Silva et al./Physics Latters A 226 (1997) 269-274
273
1.
0.
CJ O.
-0. -1. 20
40
60
60
100
20
40
60
60
1
k(drops) Fig. 6. The autocorrelation functions in the apparent period-l region. The dashed lines are the fit of the exponential decay, Eq. (5). to the maximum amplitude data. The correspondent correlation drop (T) as well as the drop rate and the series number are shown in each frame.
IG( k)
Imax= GOe-k/T.
(5)
By plotting T versus I( f - fc) /fcl on a log-log scale, see Fig. 7, where fc is a critical drop rate, whose value was chosen as the one that gives the minimum fitting error of Eq. (6) to the experimental data on a log-log scale (see inset of Fig. 7), fc = 39.897 drops/s and y = -2.28 f 0.03 were obtained,
It was shown that after an inverse Hopf bifurcation the evolution of the dynamical system to an apparent period-l motion is better described by the autocorrelation function. The amplitude of the autocorrelation function decays exponentially, its characteristic correlation drop scales with the drop rate.
Fig. 7. T versus I( f - fc) /fcI.The continuous line is the curve fit of Eq. (6) to the data. The inset shows the fitting error on the choice of the fc value. It should be noted that fc= 39.897 drops/s is very close to the last quasiperiodic drop rate ( 18th series).
This work was partially financed agencies Finep, CNPq and Fapesp.
by the Brazilian
274
J.G. Marques da Silva et al./Physics Letters A 2.26 (1997) 269-274
References [l] R.D. Pinto, W.M. GonGalves, J.C. Sartorelli and M.J. de Oliveira, Phys. Rev. E 52 (1995) 6896. [2] J.C. Sartorelli, W.M. Gonglves and R. Pinto, Phys. Rev. E 49 ( 1994) 3963.
131 J. Guckenhei~r
and P. Holmes, Nonlinear oscitlations, dynamical systems and bifurcations of vector fieids (Springer, Berlin, 1986). 141 H.D.I. Abarbanel, R. Brown, J.J. Sidorowich and L.S. Tsimring, Rev. Mod. Phys. 65 ( 1993) 133 I.