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Synchronization of weakly nonlinear oscillators with Huygens' coupling
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10.1063/1.4816360
/content/aip/journal/chaos/23/3/10.1063/1.4816360
http://aip.metastore.ingenta.com/content/aip/journal/chaos/23/3/10.1063/1.4816360

Figures

Image of FIG. 1.
FIG. 1.

Huygens' original system of pendulum clocks and its generalized (and modern) version.

Image of FIG. 2.
FIG. 2.

Conditions (37) for in-phase synchronization plotted as a function of the mass of the coupling bar, i.e., . Figures 2(a) to 2(d) : . Figure 2(e) : .

Image of FIG. 3.
FIG. 3.

For a small coupling strength ( large) and initial conditions close to in-phase, system (19) , (20) synchronizes in-phase as depicted in Figures 3(a) and 3(b) . For the same coupling strength and initial conditions close to anti-phase, the oscillators synchronize in anti-phase, as shown in Figures 3(c) and 3(d) , where black line: and grey line: .

Image of FIG. 4.
FIG. 4.

When the coupling strength is increased (by decreasing ) the only synchronous solution in system (19) , (20) is anti-phase synchronization. In Fig. 4(b) , black line: and grey line: . In Fig. 4(c) , black line: transient behaviour and grey: long term behaviour.

Image of FIG. 5.
FIG. 5.

Photo of the experimental setup.

Image of FIG. 6.
FIG. 6.

Experimental results. For relatively large coupling strength, the oscillators practically synchronize in anti-phase. In Figures 6(a) and 6(b) , black line: and grey line: . Figure 6(d) does not contain transient behaviour.

Image of FIG. 7.
FIG. 7.

In this experiment, the mass of the coupling bar is increased to (coupling strength is decreased). As a consequence, the oscillators practically synchronize in-phase. In Figures 7(a) and 7(b) , black line: and grey line: . Figure 7(d) does not contain transient behaviour.

Image of FIG. 8.
FIG. 8.

Experimental results. For a light coupling bar (relatively large coupling strength μ), practical anti-phase synchronization occurs. In Figures 8(a) and 8(b) , black line: and grey line: . In Figure 8(d) , transient behaviour has been omitted.

Image of FIG. 9.
FIG. 9.

In this experiment, the oscillators practically synchronize in-phase. The oscillators have been released from initial conditions close to in-phase and the mass of the coupling bar has been increased to . InFigures 9(a) and 9(b) , black line: and grey line: . In Figure 9(d) , transient behaviour has been omitted.

Image of FIG. 10.
FIG. 10.

For initial conditions far from in-phase, the oscillators synchronize in anti-phase. Black line: and grey line: .

Tables

Generic image for table
Table I.

Parameter values for the energy-dependent escapement experiments.

Generic image for table
Table II.

Parameter values for the van der Pol escapement experiments.

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/content/aip/journal/chaos/23/3/10.1063/1.4816360
2013-07-30
2014-04-25
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Synchronization of weakly nonlinear oscillators with Huygens' coupling
http://aip.metastore.ingenta.com/content/aip/journal/chaos/23/3/10.1063/1.4816360
10.1063/1.4816360
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