On chaotic behavior, stability analysis, and vibration control of the van der Pol–Mathieu–Duffing oscillator under parametric force and resonance

Abstract

The van der Pol–Mathieu–Duffing oscillator (VMDO) finds applications across diverse fields due to its ability to model complex dynamic behaviors. Oscillators are versatile tools used in fields such as mechanics, biology, and electrical engineering. Its ability to model complex behaviors makes it an important subject of study for researchers and engineers studying nonlinear dynamic systems. A parametric forcing excited the VMDO under feedback control is investigated and discussed in the most severe resonance scenario. The multiple-scales-strategy (MSS) is utilized to obtain the approximate solution (AS). Furthermore, the AS is validated against the numerical solution (NS) obtained using the Runge–Kutta of fourth-order (RK-4) method. A negative-velocity-feedback (NVF) and negative-cubic-velocity-feedback (NCVF) controllers are integrated into the primary system to mitigate unwanted vibrations, which can significantly impact the system’s efficiency, particularly under resonance conditions. The stability analysis is thoroughly examined, and optimal feedback gains are selected to suppress amplitude peaks. A range of response curves is presented to illustrate and compare the effectiveness of the controllers. Bifurcation diagrams and Poincaré maps (PM) are utilized to investigate the system’s diverse motions, providing valuable insights into its complex behavior and the variations it undergoes under different conditions. The results are explained through the displayed curves and provide insights into the system’s dynamics. The VMDO is a versatile model in various scientific and engineering disciplines. Its ability to embody nonlinear dynamics makes it invaluable for studying the stability, resonance, and chaotic behavior of complex systems. As research continues, its applications may expand into new areas of technology.

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