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Experimental study of the irrational phase synchronization of coupled nonidentical mechanical metronomes.

Song Z, Wu Y, Liu W, Xiao J - PLoS ONE (2015)

Bottom Line: Here, using two coupled nonidentical periodic mechanical metronomes, we revisit this interesting phenomenon through experimental studies.It is demonstrated that under suitable couplings, the phases of the metronomes indeed can become locked into irrational ratios.Our studies provide a solid step toward further studies of IPS.

View Article: PubMed Central - PubMed

Affiliation: School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, China.

ABSTRACT
It has recently been observed in numerical simulations that the phases of two coupled nonlinear oscillators can become locked into an irrational ratio, exhibiting the phenomenon of irrational phase synchronization (IPS) [Phys. Rev. E 69, 056228 (2004)]. Here, using two coupled nonidentical periodic mechanical metronomes, we revisit this interesting phenomenon through experimental studies. It is demonstrated that under suitable couplings, the phases of the metronomes indeed can become locked into irrational ratios. Numerical simulations confirm the experimental observations and also reveal that in the IPS state, the system dynamics are chaotic. Our studies provide a solid step toward further studies of IPS.

No MeSH data available.


Related in: MedlinePlus

The experimental results.For f10 = 158.47 BPM and f20 = 156.52 BPM, we present (a) the time evolutions of ϕ1,2, where envelopes of ϕ1,2 exhibit anti-phase synchronization; (b) the phase trajectory in the  space; and (c) the Poincaré maps. (d) The variation of the phase-locking ratio, r, as a function of the natural frequency of the second metronome, f20. The arrow denotes the point at which r = π / 3.1.
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pone.0118986.g002: The experimental results.For f10 = 158.47 BPM and f20 = 156.52 BPM, we present (a) the time evolutions of ϕ1,2, where envelopes of ϕ1,2 exhibit anti-phase synchronization; (b) the phase trajectory in the space; and (c) the Poincaré maps. (d) The variation of the phase-locking ratio, r, as a function of the natural frequency of the second metronome, f20. The arrow denotes the point at which r = π / 3.1.

Mentions: In our experiments, we fix the frequency of the first metronome to 160 BPM while varying the frequency of the second metronome gradually from 152 to 168 BPM. Note that the attachment of the wafer shifts the natural frequency of the first metronome to f10 = 158.47 BPM. With the natural frequency of the second metronome chosen to be f20 = 156.52 BPM, we plot in Fig. 2A the time evolution of the two phases, ϕ1(t) and ϕ2(t). The envelopes of ϕ1 and ϕ2 exhibit the phenomenon of anti-phase synchronization [20]. To provide further details on the dynamics of the phases, in Fig. 2B, we present the system trajectory in the spaces. It is observed that the trajectory is irregular in both spaces. Using the Poincaré surface-of-section method (i.e., by recording ϕ2 at the moments when and ), we replot in Fig. 2C the discrete evolution of the system trajectory in the space, which reveals that the system dynamics are chaotic. A careful evaluation of the time evolution of ϕ1,2 also indicates that /Δrϕ − c/ < π, with r = π / 3.1 (the details of how to determine this irrational ratio are presented in the next section). That is, the two metronomes achieve the IPS state. To validate the experimental findings, we gradually vary the natural frequency of the second metronome from 152 to 168 BPM and investigate the synchronization behaviors of the two phases. The experimental results are presented in Fig. 2D. It is observed that as f20 increases, the phase-locking ratio, r, varies continuously. In particular, near the testing frequency f20 = 156.52, r continuously decreases as f20 increases. According to Ref. [16], given that r is continuously varying over some interval of the parameter space, IPS can always be achieved through careful tuning of the parameter within this interval.


Experimental study of the irrational phase synchronization of coupled nonidentical mechanical metronomes.

Song Z, Wu Y, Liu W, Xiao J - PLoS ONE (2015)

The experimental results.For f10 = 158.47 BPM and f20 = 156.52 BPM, we present (a) the time evolutions of ϕ1,2, where envelopes of ϕ1,2 exhibit anti-phase synchronization; (b) the phase trajectory in the  space; and (c) the Poincaré maps. (d) The variation of the phase-locking ratio, r, as a function of the natural frequency of the second metronome, f20. The arrow denotes the point at which r = π / 3.1.
© Copyright Policy
Related In: Results  -  Collection

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getmorefigures.php?uid=PMC4364733&req=5

pone.0118986.g002: The experimental results.For f10 = 158.47 BPM and f20 = 156.52 BPM, we present (a) the time evolutions of ϕ1,2, where envelopes of ϕ1,2 exhibit anti-phase synchronization; (b) the phase trajectory in the space; and (c) the Poincaré maps. (d) The variation of the phase-locking ratio, r, as a function of the natural frequency of the second metronome, f20. The arrow denotes the point at which r = π / 3.1.
Mentions: In our experiments, we fix the frequency of the first metronome to 160 BPM while varying the frequency of the second metronome gradually from 152 to 168 BPM. Note that the attachment of the wafer shifts the natural frequency of the first metronome to f10 = 158.47 BPM. With the natural frequency of the second metronome chosen to be f20 = 156.52 BPM, we plot in Fig. 2A the time evolution of the two phases, ϕ1(t) and ϕ2(t). The envelopes of ϕ1 and ϕ2 exhibit the phenomenon of anti-phase synchronization [20]. To provide further details on the dynamics of the phases, in Fig. 2B, we present the system trajectory in the spaces. It is observed that the trajectory is irregular in both spaces. Using the Poincaré surface-of-section method (i.e., by recording ϕ2 at the moments when and ), we replot in Fig. 2C the discrete evolution of the system trajectory in the space, which reveals that the system dynamics are chaotic. A careful evaluation of the time evolution of ϕ1,2 also indicates that /Δrϕ − c/ < π, with r = π / 3.1 (the details of how to determine this irrational ratio are presented in the next section). That is, the two metronomes achieve the IPS state. To validate the experimental findings, we gradually vary the natural frequency of the second metronome from 152 to 168 BPM and investigate the synchronization behaviors of the two phases. The experimental results are presented in Fig. 2D. It is observed that as f20 increases, the phase-locking ratio, r, varies continuously. In particular, near the testing frequency f20 = 156.52, r continuously decreases as f20 increases. According to Ref. [16], given that r is continuously varying over some interval of the parameter space, IPS can always be achieved through careful tuning of the parameter within this interval.

Bottom Line: Here, using two coupled nonidentical periodic mechanical metronomes, we revisit this interesting phenomenon through experimental studies.It is demonstrated that under suitable couplings, the phases of the metronomes indeed can become locked into irrational ratios.Our studies provide a solid step toward further studies of IPS.

View Article: PubMed Central - PubMed

Affiliation: School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, China.

ABSTRACT
It has recently been observed in numerical simulations that the phases of two coupled nonlinear oscillators can become locked into an irrational ratio, exhibiting the phenomenon of irrational phase synchronization (IPS) [Phys. Rev. E 69, 056228 (2004)]. Here, using two coupled nonidentical periodic mechanical metronomes, we revisit this interesting phenomenon through experimental studies. It is demonstrated that under suitable couplings, the phases of the metronomes indeed can become locked into irrational ratios. Numerical simulations confirm the experimental observations and also reveal that in the IPS state, the system dynamics are chaotic. Our studies provide a solid step toward further studies of IPS.

No MeSH data available.


Related in: MedlinePlus