Analysis of synchronization of Van der Pol oscillators (2018-2019)
Synchronization is very commonly observed but a complex phenomenon. In this study, the effective methods of measurement and visualization of synchronization are investigated. Systems of mechanical metronomes modeled by coupled Van der Pol oscillators were analyzed both theoretically and experimentally. The equation of motion of the chaotic systems was solved numerically, changing several parameters including the initial conditions. Signal processing including Hilbert transformation was used to compute the phase difference and hence evaluate the rate of synchronization. Hilbert transform intuitively showed the decrease of phase difference as time passes and distinguish whether synchronization happens or not. By numerically sampling possible initial conditions and drawing “initial condition – resulting condition” diagram, the stability of synchronization was investigated. A new type of fractal was observed in “initial condition - resulting condition” diagram. The shape of fractal and area of synchronizing region changed rapidly as coupling constant and distribution of natural frequencies are changed. The same method of evaluating the rate and stability of synchronization is expected to be applied for various chaotic systems such as electrical, chemical and biological oscillators.