| Synchronization is a widely observed in nonlinear science. There are all kinds of synchronous phenomenon whether in nature and artificial theory or environment. As early as1673, Huygens found the synchronization phenomenon in the pendulum experiment. From then on, people began research synchronization phenomenon in all kinds of system widely. And development to the modern times, the mechanical metronome, relative to early pendulum, has been improved and developed greatly. Due to small volume, accurate timing, frequency easily to adjust, and many other advantages, mechanical metronome were widely used in the experiment of synchronous. Several metronomes which are fixed on boards of different material, with the metronome random vibration as the initial values, will appear very rich dynamic phenomena after a brief coupling.In this paper, we made a coupled mechanical metronomes system and studied the phenomenon of a variety of synchronization of the two or three metronome under the condition of strong coupling, including in-phase synchronization, anti-phase synchronization, phase-locked synchronization, quasi-periodic phenomenon, chaos, and so on. Then we firstly observed the phenomenon of irrational phase synchronization (IPS) in experimental studies. We also analyzed the dynamic behavior in theoretical. Finally, we researched the pivot angle’s time series of the pendulum by adjusting the frequency of the metronome and using the triplet synchronous theory. We confirmed the triplet synchronization phenomenon in the three coupled metronomes’ system for the first time. Further, we simulated the theory model of the metronomes’system and exhibited the parameter space of the triplet synchronization in the metronomes’ system. The main content of this paper is as follows:The first chapter is introduction. It introduces the basic knowledge of the nonlinear dynamics and basic theories, including the knowledge of dynamics equation, Lyapunov index, definition of synchronization and so no. These knowledge will be used in the article. And it also introduced the research history and development of the metronome system.The second chapter introduced the device and the working principle of the metronome itself. Experiment system were introduced in this article, the theoretical model is derived. We also analyzed the related theory of metronome system. This chapter is the experimental and theoretical basis for the full text. All the work will be revolved around the system of this chapter.The third chapter studies synchronous behavior of the metronomes. Two non-identical metronomes (adaptive parameter is very small) will be stable in the in-phase synchronization or anti-phase synchronization state from theory and experiment. And with the increase of system damping, the ratio of anti-phase synchronization will increase. For three identical metronomes, there will four states:in-phase synchronization, anti-phase synchronization, four periodic synchronization and quasi-periodic state. And the distribution of the four states will be different with the change of the system damping. For three non-identical metronome with little frequency mismatch, there is also a similar phenomenon, but only the phase-locked synchronization and quasi-periodic synchronization.The fourth chapter studies the irrational synchronization phenomenon. In a chaotic system, if the coupling system of two oscillators can keep the irrational proportion of phase synchronization state, we say the system is in irrational synchronization state. This phenomenon also exists in other nonlinear system: We found the chaos phenomena in the metronome system. And on this basis, we find the irrational synchronization phenomenon, and prove it indirectly through the experiment.The fifth chapter studies the integer ratio of synchronization with three metronomes. We analysis the system of three metronomes by the synchronous index which is represent the synchronization state of the system. We obtained the corresponding integer ratio synchronization results with the experimental results.The sixth chapter is the summary and outlook about this thesis. |
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