We extend a typical system that possesses a transcritical bifurcation to a fractional-order version. The bifurcation and the resonance phenomenon in the considered system are investigated by both analytical and numerical methods. In the absence of external excitations or simply considering only one low-frequency excitation, the system parameter induces a continuous transcritical bifurcation. When both low- and high-frequency forces are acting, the high-frequency force has a biasing effect and it makes the continuous transcritical bifurcation transit to a discontinuous saddle-node bifurcation. For this case, the system parameter, the high-frequency force, and the fractional-order have effects on the saddle-node bifurcation. The system parameter induces twice a saddle-node bifurcation. The amplitude of the high-frequency force and the fractional-order induce only once a saddle-node bifurcation in the subcritical and the supercritical case, respectively. The system presents a nonlinear response to the low-frequency force. The system parameter and the low-frequency can induce a resonance-like behavior, though the high-frequency force and the fractional-order cannot induce it. We believe that the results of this paper might contribute to a better understanding of the bifurcation and resonance in the excited fractional-order system.
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November 2015
Research-Article
Bifurcation Transition and Nonlinear Response in a Fractional-Order System
J. H. Yang,
J. H. Yang
1
School of Mechatronic Engineering,
e-mail: jianhuayang@cumt.edu.cn
China University of Mining and Technology
,Xuzhou 221116
, China
e-mail: jianhuayang@cumt.edu.cn
1Corresponding author.
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M. A. F. Sanjuán,
M. A. F. Sanjuán
Nonlinear Dynamics, Chaos, and Complex Systems Group,
Departamento de Física,
e-mail: miguel.sanjuan@urjc.es
Departamento de Física,
Universidad Rey Juan Carlos
,Tulipán s/n
,Móstoles, Madrid 28933
, Spain
e-mail: miguel.sanjuan@urjc.es
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H. G. Liu,
H. G. Liu
School of Mechatronic Engineering,
China University of Mining and Technology
,Xuzhou 221116
, China
Search for other works by this author on:
G. Cheng
G. Cheng
School of Mechatronic Engineering,
China University of Mining and Technology
,Xuzhou 221116
, China
Search for other works by this author on:
J. H. Yang
School of Mechatronic Engineering,
e-mail: jianhuayang@cumt.edu.cn
China University of Mining and Technology
,Xuzhou 221116
, China
e-mail: jianhuayang@cumt.edu.cn
M. A. F. Sanjuán
Nonlinear Dynamics, Chaos, and Complex Systems Group,
Departamento de Física,
e-mail: miguel.sanjuan@urjc.es
Departamento de Física,
Universidad Rey Juan Carlos
,Tulipán s/n
,Móstoles, Madrid 28933
, Spain
e-mail: miguel.sanjuan@urjc.es
H. G. Liu
School of Mechatronic Engineering,
China University of Mining and Technology
,Xuzhou 221116
, China
G. Cheng
School of Mechatronic Engineering,
China University of Mining and Technology
,Xuzhou 221116
, China
1Corresponding author.
Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received October 1, 2014; final manuscript received December 28, 2014; published online April 16, 2015. Assoc. Editor: J. A. Tenreiro Machado.
J. Comput. Nonlinear Dynam. Nov 2015, 10(6): 061017 (9 pages)
Published Online: November 1, 2015
Article history
Received:
October 1, 2014
Revision Received:
December 28, 2014
Online:
April 16, 2015
Citation
Yang, J. H., Sanjuán, M. A. F., Liu, H. G., and Cheng, G. (November 1, 2015). "Bifurcation Transition and Nonlinear Response in a Fractional-Order System." ASME. J. Comput. Nonlinear Dynam. November 2015; 10(6): 061017. https://doi.org/10.1115/1.4029512
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