Simple Coupled System And Synchronization Of Coupled Metronome

As a common phenomenon in the nature, synchronization always show that at least two oscillating system’s phase behave in a same tone. The earliest research about synchronization was the observation of coupling pendulums by Huygens in1673. As a common equipment for the research of synchronization, metronome are widely used in the experiment. We put a number of pendulum clocks hanging from an elastically fixed horizontal beam. It has been shown that after a transient, we can observe numerous dynamical phenomenon. In this thesis, we study the synchronization phenomenon of several number of coupling metronomes, include complete synchronization and phase synchronization. The main contents are as follows:In section one, we introduce rudiments about nonlinear dynamics, include the definition of synchronization, bifurcation phenomenon, and so on. Then we introduce the research history and present situation of coupling metronome.Section two is the study about two metronomes hanging on a horizontal beam. In the part of theory analysis, we find that after a transient, we can observe two types of synchronization:(1) complete synchronization.(2) anti-phase synchronization. furthermore, we study the relationship between the rolling friction and the synchronization styles. The impact of rolling friction on the synchronization (or anti-synchronization) time are discovered. Finally, several complex behaviors of coupled metronomes such as in-phase and anti-phase synchronization conditions of non-identical systems are studied. In addition, we make experiment tests and find the in-phase and anti-phase synchronization corresponded to the theory analysis. We change the experiment material to change the rolling friction. The experiment result have a good coincidence with the theory analysis.Section three and four are the study about three metronomes. First, we do numerical simulations, then get four stable result with different initial condition:(1) complete synchronization.(2) symmetrical synchronization.(3) four-periods behavior.(4) quasi-periodic behavior. Through plotting domain of basin, we get the initial condition for corresponded result. Then we study the phase synchronization of a number of different pendulum clocks hanging from a horizontal beam. Pendulum have different period of oscillations but have the same masses. As the difference is small, we can observe phase synchronization, phase locking and non-phase locking. As for experiment, we observed the corresponded phenomenon, by CCD and software we get good experiment result.Section five is the summary and outlook about this thesis.