Bifurcation and stability analysis of fractional quintic oscillator system with power damping term

Abstract

In this study, the resonance, bifurcation and stability of a class of fractional quintic systems are discussed. The implicit function expressions of the amplitude–frequency relationship of the steady-state response of the system under Coulomb dry friction, linear viscous damping, square damping, and cubic damping are derived by using the average method. Lyapunov’s first method is used to study the stability of the steady-state response of the system under the action of Coulomb dry friction, linear viscous damping, cubic damping, and analyze the influence of system parameters on the steady-state response. The amplitude–frequency curve of the system is drawn based on the implicit function expression and compared with the numerical solution. The primary resonance behavior of the system under four damping conditions is discussed when the fractional differential order, cubic stiffness coefficient, quintic stiffness coefficient, and fractional differential term coefficient take different values. The fork bifurcation and saddle–node bifurcation caused by the change in damping coefficient and fractional differential term coefficient under different parameters are studied.