Eur. Phys. J. B (2015) 88: 310
https://doi.org/10.1140/epjb/e2015-60315-y

Regular Article

Bifurcation and resonance in a fractional Mathieu-Duffing oscillator

¹ School of Mechatronic Engineering, China University of Mining and Technology, Xuzhou 221116, P.R. China
² Jiangsu Key Laboratory of Mine Mechanical and Electrical Equipment, China University of Mining and Technology, Xuzhou 221116, P.R. China
³ Nonlinear Dynamics, Chaos and Complex Systems Group, Departamento de Física, Universidad Rey Juan Carlos, Tulipán s/n, 28933 Móstoles, Madrid, Spain

^a e-mail: jianhuayang@cumt.edu.cn

Received: 22 April 2015
Received in final form: 30 July 2015
Published online: 26 November 2015

Abstract

The bifurcation and resonance phenomena are investigated in a fractional Mathieu-Duffing oscillator which contains a fast parametric excitation and a slow external excitation. We extend the method of direct partition of motions to evaluate the response for the parametrically excited system. Besides, we propose a numerical method to simulate different types of local bifurcation of the equilibria. For the nonlinear dynamical behaviors of the considered system, the linear stiffness coefficient is a key factor which influences the resonance phenomenon directly. Moreover, the fractional-order damping brings some new results that are different from the corresponding results in the ordinary Mathieu-Duffing oscillator. Especially, the resonance pattern, the resonance frequency and the resonance magnitude depend on the value of the fractional-order closely.