In this study, we propose a new systematic approximation method for theoretically analyzing synchronization dynamics exhibited by a general class of self-excited oscillators. In applied mathematics, the phase-reduction method, which enables us to systematically reduce high-dimensional oscillator dynamics to a one-dimensional reduced equation, has been widely used to analyze synchronization dynamics. However, this method has a critical drawback that it can be used only when the forcing applied to the oscillator is sufficiently weak, which hinders many potential applications of this method in mechanical engineering. To overcome this drawback, we propose an extended phase-reduction method that is applicable to self-excited oscillators driven by strong forcing. The method works even if the orbit of the oscillator is deformed largely, as long as the forcing can be decomposed into a slowly varying component and remaining weak fluctuations. Using the proposed method, we theoretically analyze a synchronization phenomenon of self-excited oscillators induced by strong periodic forcing. We also verify the validity and robustness of our method through numerical experiments.
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