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`Mathematical Problems in EngineeringVolume 2014, Article ID 738087, 17 pageshttp://dx.doi.org/10.1155/2014/738087`
Research Article

## Subharmonic Resonance of Van Der Pol Oscillator with Fractional-Order Derivative

1Department of Mechanical Engineering, Shijiazhuang Tiedao University, No. 17 Bei Erhuan Dong Road, Shijiazhuang, Hebei 050043, China
2Key Laboratory of Transportation Tunnel Engineering, Ministry of Education, Southwest Jiaotong University, No. 111 North 2nd Ring Street, Chengdu 610031, China

Received 4 November 2013; Revised 16 January 2014; Accepted 17 January 2014; Published 25 February 2014

Copyright © 2014 Yongjun Shen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The subharmonic resonance of van der Pol (VDP) oscillator with fractional-order derivative is studied by the averaging method. At first, the first-order approximate solutions are obtained by the averaging method. Then the definitions of equivalent linear damping coefficient (ELDC) and equivalent linear stiffness coefficient (ELSC) for subharmonic resonance are established, and the effects of the fractional-order parameters on the ELDC, the ELSC, and the dynamical characteristics of system are also analysed. Moreover, the amplitude-frequency equation and phase-frequency equation of steady-state solution for subharmonic resonance are established. The corresponding stability condition is presented based on Lyapunov theory, and the existence condition for subharmonic resonance (ECSR) is also obtained. At last, the comparisons of the fractional-order and the traditional integer-order VDP oscillator are fulfilled by the numerical simulation. The effects of the parameters in fractional-order derivative on the steady-state amplitude, the amplitude-frequency curves, and the system stability are also studied.