Parametric Frequency Analysis of Mathieu–Duffing Equation

Abstract

The classic linear Mathieu equation is one of the archetypical differential equations which has been studied frequently by employing different analytical and numerical methods. The Mathieu equation with cubic nonlinear term, also known as Mathieu–Duffing equation, is one of the many extensions of the classic Mathieu equation. Nonlinear characteristics of such equation have been investigated in many papers. Specifically, the method of multiple scale has been used to demonstrate the pitchfork bifurcation associated with stability change around the first unstable tongue and Lie transform has been used to demonstrate the subharmonic bifurcation for relatively small values of the undamped natural frequency. In these works, the resulting bifurcation diagram is represented in the parameter space of the undamped natural frequency where a constant value is allocated to the parametric frequency. Alternatively, this paper demonstrates how the Poincaré–Lindstedt method can be used to formulate pitchfork bifurcation around the first unstable tongue. Further, it is shown how higher order terms can be included in the perturbation analysis to formulate pitchfork bifurcation around the second tongue, and also subharmonic bifurcations for relatively high values of parametric frequency. This approach enables us to demonstrate the resulting global bifurcation diagram in the parameter space of parametric frequency, which is beneficial in the bifurcation analysis of systems with constant undamped natural frequency, when the frequency of the parametric force can vary. Finally, the analytical approximations are verified by employing the numerical integration along with Poincaré map and phase portraits.

A modified perturbation method for global dynamic analysis of generalized mixed Rayleigh–Liénard oscillator with cubic and quintic nonlinearities

“…Many fundamental nonlinear oscillation models, such as Duffing, Van der Pol, Rayleigh, Liénard, and Mathieu, have been extensively studied to describe nonlinear phenomena, such as nonlinear resonance, chaos, bifurcation, limit cycle, hysteresis and jump, analytic solutions, plasma oscillations, and noise effects [2]. Other than the studies on fundamental oscillation models mentioned above, many composition forms of fundamental oscillators also attract the attention of researchers, such as Duffing-Van der Pol oscillator [3], Mathieu-Duffing oscillator [4], Rayleigh-Duffing oscillator [5], Rayleigh-van der Pol-Duffing oscillator [6], and Mathieu-van der Pol-Duffing oscillator [7]. Among the composition forms of oscillators, we, in this paper, focus on generalized mixed RayleighLiénard oscillator as follows [8]:…”