Dynamic Response Analysis of a Van der Pol-Mathieu-Duffing Type System

The Van der Pol-Mathieu-Duffing (VPMD) oscillator is studied in this paper, which possesses three kinds of nonlinearity and has been extensively used in different fields such as micro-electro-mechanical system devices for energy harvesting, the fluid–structure interaction problem. The dynamic behaviors of the kind of system have important effect on safety and stability of the engineering devices. Herein, the incremental harmonic balance (IHB) method is used to study the first-order approximate solution, and the results are verified by the direct numerical simulation. Then the control effects of the system parameters are studied numerically. The phase portraits, time histories, Poincaré sections, power spectrum, bifurcation diagrams and Lyapunov exponents are derived under various parameter conditions, and the results show the complex dynamics such as the quasi-periodic and chaotic solutions as well as co-existing of chaotic attractors can occur under certain parameter conditions. Moreover, the system can display different nonlinear states via parameter control.