Simulation of subharmonic frequency lock-in and phase synchronization using Van der Pol equation

This study examines subharmonic frequency lock-in and phase synchronization of vortex shedding behind a partially immersed circular cylinder placed in a steady water current and subjected to harmonic surface waves. A forced Van der Pol wake-oscillator model was developed to describe the nonlinear coupling between the wake dynamics and external periodic forcing, with parameters validated against wave-flume experiments. Both numerical and experimental results reveal that synchronization occurs when the forcing frequency is approximately twice the natural shedding frequency, forming Arnold-tongue-shaped regions in the amplitude-frequency plane. At the edges of these regions, chaotic and intermittent behaviors emerge, highlighting the nonlinear nature of the wake-forcing interaction. The strong agreement between model predictions and laboratory observations confirms the relevance of the Van der Pol framework for predicting vortex-induced vibration in oscillatory environments and provides insight into the transition between unsynchronized and fully phase-locked regimes. In addition, a machine learning framework based on a multilayer perceptron (MLP) was implemented to classify synchronization states directly from the frequency-forcing parameter space. The MLP successfully reproduced the experimentally observed synchronization boundaries and achieved prediction accuracies exceeding 93% for experimental data and 98% for theoretical data, demonstrating its capability to identify complex nonlinear lock-in dynamics. These results highlight the potential of combining physics-based reduced-order models with data-driven approaches for improved prediction and analysis of flow synchronization phenomena.