Experimental Study of the Triplet Synchronization of Coupled Nonidentical Mechanical Metronomes

Triplet synchrony is an interesting state when the phases and the frequencies of three coupled oscillators fulfill the conditions of a triplet locking, whereas every pair of systems remains asynchronous. Experimental observation of triplet synchrony is firstly realized in three coupled nonidentical mechanical metronomes. A more direct method based on the phase diagram is proposed to observe and determine triplet synchronization. Our results show that the stable triplet synchrony is observed in several intervals of the parameter space. Moreover, the experimental results are verified according to the theoretical model of the coupled metronomes. The outcomes are useful to understand the inner regimes of collective dynamics in coupled oscillators.

where n and m are some integers and C is a rather small constant. Analyzing complex synchronization patterns in multi-frequency systems have been deeply applied in various fields, especially widely in biological systems such as the interaction of respiratory, cardiac and brain activities 13 . When considering a large number of interacting oscillators with complex interacting network structure and random natural frequency distribution, there are rich dynamics and patterns in the processes that the coupled system transits from incoherent states to full phase locking states with increasing coupling interactions. Before reaching full phase locking state (any pair of coupling units satisfy the phase locking condition), the system may become a partial phase synchronization regime 7 , i.e. some pairs or groups of oscillators are phase locked while others are not which forms several synchronous clusters. To reveal all synchronous states of a network efficiently, Kralemann et al. 14 defined a synchronous index to detect a triplet synchronization which is realized when triplets of interacting oscillators adjust their phases and frequencies so that the following conditions are fulfilled for t is larger than a transient time T, ϕ ϕ ϕ ( ) + ( ) + ( ) < , ( ) n t m t l t C 2 1 2 3 where the integers n, m, l can be both positive and negative and C is a small constant. However in parallel, the conditions of the pair-wise synchrony equation (1) may not be satisfied for any pair of units. Although, triplet synchronization is theoretically predicted and detected in oscillator networks from observed data, it is expected to reveal various pattern formations of coupled oscillators and to contribute to research in neuroscience based on the binding by-synchrony hypothesis 15 . To our best knowledge, no experimental observation on triplet synchronization has been observed so far. Experimental discussion on the triplet synchronization is important for various applications, such as the interaction of different brain regions, where oscillations with a hierarchy of frequencies are ubiquitous. Coupled pendulums are deemed as a paradigmatic model of exploring the dynamics of coupled systems since the pioneering work of Huygens. Recently, many scientific teams carried out a variety of experimental research work, such as Wu Ye et al. 16 showed a relationship theoretically and experimentally between the initial values and the friction damping force and the stable synchronous states of coupled metronomes system. Oliveira et al. 17 experimentally explored Huygens synchronization in two clocks hanging from an aluminum rail fixed to a masonry wall. Hu et al. 18 studied the synchronous behavior of three coupled metronomes, discovering a variety of synchronous states and the envelope synchronization phenomenon. Martens et al. found Chimera states in coupled metronome systems 19 , etc 20,21 . Therefore, coupled pendulums are a promising candidate to observe triplet synchronization experimentally. We set up here a coupled system with three globally coupled pendulums and apply the synchronous index defined in Eq. (2) to observe the triplet synchronization.
In this article, we try to experimentally observe the triplet synchrony with the aid of the synchronous index. A model based on our experimental setup is built and analyzed to verify our experimental results. Experimental setup Figure 1 shows the experiment platform which consists of three metronome units supported by a piece of folded A4 paper on two aluminum pipes, a CCD (Charge Coupled Device) acquisition system connected to a computer and software of LABVIEW. The metronomes in our experimental work are all the Taktell Piccolino (Series 890) manufactured by Wittner GmbH & Co.KG in Germany. In order to improve the accuracy and simplicity, the latest experimental system are ameliorated based on the previous system [18][19][20][21][22][23] . An organic glass base of hollow cuboids shape is used to ensure that the system will not produce deformation because of its own weight. Two aluminum pipes (with 39 mm inside diameter, 41 mm outside diameter, and 100 mm length) are put on and perpendicular to the base. The aluminum pipes have a lot of advantages, such as that their rolling friction and shape hardly change and they have lighter mass.
Since the total energy supplied by the metronome units is limited (last about 20 minutes), it is difficult to realize synchronization if the coupling strength is not sufficient large. In order to enhance the effect of coupling, the crux of the problem is to provide more energy or reduce unnecessary loss. Without changing the structure of metronome, a paper-made platform was applied to substitute the coupling board used in the previous works 16,18 . A few pieces of A4 paper are folded as undulating shape so that it is strong enough to support the metronomes. Thus, with the lightness of the coupling board, the energy of the system will not waste too much of the kinetic energy of the coupling board. As a result, the coupling strength is guaranteed strong enough to realize synchronization between metronomes on the coupling board.
Three metronomes are put on the coupling board and a red wafer is pasted at the end of each pendulum and on the coupling board to improve the accuracy of recognition for the CCD camera. Then the motion of the pendulums and the coupling board can be conveniently recorded by tracing the center of the red wafers. In order to get accurate data, a camera with a high frame rate is set up by which one can record videos with a resolution of 720p (1280*720 pixels) and a frame rate of 30 frames per second. The time series of the pendulum of each metronome are recorded by handling the videos.
In our experiments, the metronomes are numbered as 1 to 3 from left to right. By adjusting the equivalent lengths of pendulums, we may change the frequency of the metronome slightly. (Noted that there are some slight differences between the actual frequencies and the set values which are caused by the instrumental errors (about 0.4%) in the mechanical structure of the metronomes. Therefore, we use all measured values of frequencies other than nominal ones of the metronome).
To provide enough energy necessary for the coupling, we set a relatively high value of initial frequency of metronomes as f 1 = 160 beats per minute (BPM) and f 1 = 176 BPM, while adjusting the value of the initial frequency f 3 of the 3 rd metronome from 120 BPM to 200 BPM and so 107 values of f 3 are obtained. The time series for each initial frequency constellation are collected by the CCD camera, and the corresponding phases are calculated with the aid of a computer.

Analysis Methods
The synchronous ratio and order parameter are both effective indicators verifying the synchronous behavior of system. In the work of Kralemann et al. 14 , the two indicators were combined and a special synchronous index was proposed as shown in Eq. (3) and (4). The pair-wise synchronization indices can be described as follows when two oscillators i and j are coupled: are the phases of the oscillators i and j respectively, n and m are integers, and < > is average on the time t. The oscillators i and j are considered as being n:m synchronized when the value of the pair-wise synchronization index γ , ( , ) n m i j is equal to 1. Accordingly, when all three oscillators are coupled, the triplet synchronization indices can be calculated by equation (4):

Experimental Results
According to the recorded data of the swing angle ϕ ( ) γ , , max n m l is approaching 1, while the pair-wise indices of the corresponding intervals of f 3 are small (approaching zero). Therefore, the system reaches the triplet synchrony in those intervals of parameter f 3 . That is to say, any pair of metronomes is asynchronous but the whole system of the three metronomes is in a triplet synchronous state when the parameter f 3 is in one of those four intervals. The corresponding integer parameters n, m, l are marked on the peaks of corresponding frequencies f 3 . for (i, j) = (2, 3) distribute centrally on the circle, while those of (i, j) = (1, 3) and (1, 2) distribute uniformly on the circle. Therefore, pair-wise synchronization is only built between metronomes 2 and 3, which coincide well with the results presented in Fig. 2(a). However, the dots of  Fig. 3 (g-j) which are corresponding to that in Fig. 3 (c-f). Obviously, there is no pair-wise synchronization but a triplet synchronization with = − n m l : : 1: 2: 1. What should be mentioned is that there are small amount of scatter dots in Fig. 3(e,j) which is caused by the sampling error of the CCD. As a result, the amount of data shown on the circle is limited (here we present data of 8 seconds) to observe the collective dynamics clear. To exhibit the collective dynamics more efficiently and without being influenced by the scatter dots caused by the sampling errors, the phase angle θ of the dot on the circle is . If the coupled system is in phase locking then the curve of corresponding θ ( ) P has a peak otherwise it will be uniformly distributed. Obviously, there is a peak of θ ( ) P around θ π = for γ , 2 3 and a uniform distribution of θ ( ) P for γ ,− , 1 3 as shown in Fig. 4 (a) when the coupled system is in pair-wise synchronization as Fig. 3(a). However, there is uniform distribution of θ ( ) P for γ , 1 3 , and peaks of θ ( ) P around θ π = , 0 for γ ,− , 1 2 1 as shown in Fig. 4 (b) when the coupled system is in triplet synchronization as Fig. 3(b).

Theoretical model and numerical results.
To reveal the observed triplet synchronization in the experimental setup, a theoretical model derived from the experimental devices is analyzed 24,25 . The experimental devices are abstracted from that in refs 26-29 as shown in Fig. 5 where several pendulums with the same mass are coupled through a board, which can move horizontally. All pendulums are swinging around the fixed point above and in a common upright plane. The length and swinging angles of pendulums are denoted by l i and φ i . Two aluminum pipes are set parallel under the coupling board and thus the board can move horizontally. The displacement of the board is denoted by x; c x and k x are damping and linear force respectively.
Without the damping and driving force, the Lagrange equation of the system is as follows: Scientific RepoRts | 5:17008 | DOI: 10.1038/srep17008 and can be simplified into, where M is the mass of the board, m i = 1 is the mass of the pendulum, x is the displacement of the board with ( ) = x 0 0. l i and φ i are the length and angle of the ith pendulum, g is the gravity. We define the right direction as the positive direction. In Huygens experiments, the coupling board was limited by magnetic substance, and therefore the parameter k x is remained to repeat the earlier work and it is set as = .
k 0 5 x . With the effects of damping and driving force, which is caused by the escapement mechanism of the metronomes, the dynamic equation of the coupled system can be solved:   t 0 0001 is helpful to solve the problems about Lyapunov exponent. At the time = t 0, the initial velocities of all objects and displacement of the coupling board are set as zero, and the initial values of the swinging angles are set randomly. The coupled system will go into stable states usually after 300 seconds of transient time.
While simulating equations (7) and (8) numerically, we fix the parameters as follows, = . which is less than 0.8. Even larger range of n, m, for example, [− 300, 300], the maximal pairwise synchronous indices are not larger than those when m and n are in the range of [− 100, 100]. Therefore, we may deduced that triplet synchronization is stable for given range of m and n. However, it is a time-consuming work for even larger range of n, m.
The largest three Lyapunov exponents are an effectively indicator of phase synchronization between the coupled system. Since the metronome is a mechanical system which is driven by discontinuous force from a spring, the standard algorithm does not work for the discontinuous dynamical equation. Thus the largest three of them, λ , ,  Fig. 6 (c), the largest three Lyapunov exponents are presented for different values of f 3 . Phase synchronization is characterized by a negative value of the third largest Lyapunov exponent λ 3 . In the intervals of triplet or pair-wise synchrony, two of the largest three exponents are zero and the other one is negative and it shows that the system is quasi-periodic at these moments. In other frequencies, none of the three exponents are negative and at least two of them are approaching to zero, so the system is in high dimensional quasi-periodic or chaotic state.
To compare the numerical results with the experimental ones, we recorded the time series of the coupled metronomes' angle for f 3 = 192 BPM and f 3 = 1752 BPM, respectively. We also cannot determine whether the coupled system is in triplet or pair-wise synchrony only from the time series. However, the phase space diagram of γ ( ( )) , , t Im n m l and γ ( ( )) , , t Re n m l (or γ ( BPM, there is no pair-wise synchronization between the metronomes 1,2, 1,3, and 2,3, since the dots distribute uniformly on the circle as shown in Fig. 7(g-i). However, triplet synchronization with n:m:l = 1:−2:1 is built between them as shown in Fig. 7(j).
Another approach to verify the motion states is via Poincare maps which are got by recording the values of φ 1 and φ 2 when φ 3 = 0 as shown in Fig. 8. The Poincare map (in Fig. 8(a), = f 175 3 BPM) is a horizontal line with small variation in the parameter space of φ 1 versus φ 2 when φ 3 = 0. Hence, it is confirmed this way that the coupled system built pair-wise synchronization only between metronomes 2 and 3 while there is no pair-wise synchronous between φ 1 and φ 2 . As a result, φ 1 influences the pair-wise synchronization between φ 2 and φ 3 which leads to a small modulation or perturbation. However, the Poincare map (in Fig. 8 BPM) shows that both phases φ 1 and φ 2 vary from zero to π 2 and remain in a functional relationship φ φ ( ) + ( ) < n t m t C 1 2 with = − n m : 1 : 2 when φ 3 = 0. Hence, there is no pair-wise synchronization between φ 2 and φ 3 , φ 1 and φ 3 according to the fact that φ 1 and φ 2 vary from zero to π 2 when φ 3 = 0. If the coupled metronome system is triplet synchronous with n:m:l = 1:−2:1, 1: 2 as shown the Poincare map in Fig. 8(b). Comparing Figs 2 and 6, it is obvious that the numerical analyses are consistent with our experimental results and the theoretical model is effective to describe the experimental process. However, it should be mentioned that there are some differences between them. Firstly, there are 5 intervals of triplet synchrony, in numerical results but only 4 of them are observed in our experiments. By checking our experimental results carefully, we find that there is a smaller peak at the frequency of = f 125 3 BPM, and its maximum value is less than 0.8. Next, the sections of pair-wise synchrony, which are observed in our experiments, are smaller than the simulated ones, especially at the frequency of ≈ f 160 Scientific RepoRts | 5:17008 | DOI: 10.1038/srep17008 Since the mass of the coupling board affects the strength of the coupling, it is necessary to investigate the effect of the mass of the coupling board on the synchronous index. As shown in Fig. 9(a), the ordinate denotes M, the mass of the board, and the abscissa denote f 3 , the frequency of the 3rd metronome. We find that the system shows extremely complex phenomena when the mass is small. With the increasing of the mass, only a few sections are maintained. When the mass is about 30, system will work as our theoretical analysis predicts. The frequency difference ∆f is fixed as ∆ = − = f f f 16 2 1 BPM, as shown in Fig. 9(b) (the ordinate denotes f 2 ), and third one will influence the synchronous index. In addition, if the frequency of the 1st metronome is fixed as f 1 = 160 BPM, we observe a relationship between f 2 and f 3 in Fig. 9(c).
Therefore, the distribution of the synchronous index is related to the frequencies of all metronomes. When the frequency differences are large enough, the behavior of the system depends on the coupling strength. We set the parameters in order to make the phenomena more clear and intuitive.
Let us consider the synchronization indices based on the parameter spaces of f 2 and f 3 by fixing the value of f 1 = 160 BPM. The results indicate that there are rich dynamics as the complete synchronization (white area), the triplet synchrony (black area), the pairwise synchrony (wine area) and unlocked states (light gray area) as shown in Fig. 10. Obviously, the probability for triplet synchrony is the smallest in the parameter spaces of f 2 and f 3 .
To sum up, it is obviously that a very tiny difference in the value of parameters in the model system may lead to a deviation from the synchronization states. In consequence, taking inevitable instrumental error into account, the numerical analyses can be considered to be consistent with our experimental results as well as in the theoretical model. Based on the method of synchronous index, the triplet synchrony is observed in the experimental system of coupled metronomes.

Discussion
In this paper, according to the theory of synchronization, an oscillation system of coupled metronomes was set up and triplet synchrony is discovered in a real experiment. By establishing the theoretical model and simulating the system numerically, we obtain consistent findings in experiments as well as theoretical models. By expanding the parameter spaces, we uncover more abundant nonlinear dynamic behaviors of coupled metronomes. It has been validated that the method of synchronous index is a useful tool to study experimental systems. Moreover, we propose a more direct method to determine triplet synchronization and pair-wise synchronization by a phase space representation of γ ( ( )) , , t Im n m l and γ ( ( )) , , t Re n m l (or γ ( ). This approach can be used as a basis for applications and to detect such synchronization in various fields such as engineering, neuroscience, biology, etc. The discovery of triplet synchrony in our experimental system will help us to explore the physical mechanism of complex synchronization patterns in other real systems as well. It is hoped to play a role of guidance and construction in complicated synchronization behaviors in future.

Methods
Experiments. For the convenience of CCD acquisition, the bobs of the pendulums are pasted with red wafers respectively. The model of the metronome is Series 890 by DE Taktell with a mass (94 g). Its energy is supplied by a hand wound spring. The frequency of the metronome can be adjusted by changing the position of the mass on the pendulum bob, which denotes that the equivalent pendulum length is changed. The standard settings of the metronome's frequency range from 40 ticks per minute (largo) to 208 ticks per minute (prestissimo), but not limited to the scale. The supporting folded A4 paper is light (4.366 g), since the pendulum bob's swing direction is perpendicular to the aluminum pipes axes, a bidirectional coupling between the metronomes is generated via the folded paper, which has a tuning impact on the metronomes. Two parallel aluminum pipes (with inner (external) diameter 39 mm (41 mm)) support the folded A4 paper. Under the aluminum pipes there is a horizontal adjustment equipment. (1) and (2)