Dynamic perturbation analysis of fractional order differential quasiperiodic Mathieu equation

Abstract

The paper investigates the influence of parameters on the stability of fractional order differential quasiperiodic Mathieu equations. First, we use the perturbation method to obtain approximate expressions (i.e., transition curves) for the stability and unstable region boundaries of the equation. After obtaining the approximate expression of the transition curve, we use Lyapunov's first method to analyze the stability of the fractional order differential quasiperiodic Mathieu system, thereby obtaining the conditions for the stability of the fractional order differential quasiperiodic Mathieu equation system. Second, by comparing the approximate expressions of the transition curve of the steady-state periodic solution of the quasiperiodic Mathieu oscillator under different parameter conditions, we obtained the conclusion that the fractional order differential term exists in the form of equivalent stiffness and equivalent damping in the fractional order differential quasiperiodic Mathieu system. By comparison, we have summarized the general forms of equivalent linear damping and equivalent stiffness of the system. Through this general form, we can define an approximate expression for the thickness of unstable regions to better study the characteristics of fractional order differential quasiperiodic Mathieu systems. Finally, the influence of the parameters of the fractional order differential quasiperiodic Mathieu equation on the transition curve of the equation was intuitively analyzed through numerical simulation, to analyze the stability changes in the equation.