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Research Papers

Bifurcation Transition and Nonlinear Response in a Fractional-Order System

[+] Author and Article Information
J. H. Yang

School of Mechatronic Engineering,
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,
Universidad Rey Juan Carlos,
Tulipán s/n,
Móstoles, Madrid 28933, Spain
e-mail: miguel.sanjuan@urjc.es

H. G. Liu, 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 10(6), 061017 (Nov 01, 2015) (9 pages) Paper No: CND-14-1229; doi: 10.1115/1.4029512 History: Received October 01, 2014; Revised December 28, 2014; Online April 16, 2015

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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Topics: Bifurcation
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Figures

Grahic Jump Location
Fig. 1

The system parameter μ induced transcritical bifurcation diagram for system (1) and system (2)

Grahic Jump Location
Fig. 2

Phase trajectories of system (3) under the excitation of the low-frequency force only. For numerical simulation, y = dx∕dt, F = 0, ω = 0.2, and f = 0.2, 0.5, 1.0, 1.5, and 2.0 from the inside to the outside in each subplot.

Grahic Jump Location
Fig. 3

Analytical prediction of the saddle-node bifurcation that induced by the system parameter μ for Ω = 10. The continuous thick lines are the stable branches and the dashed thin lines are the unstable branches.

Grahic Jump Location
Fig. 4

Phase trajectories of system (3) under the excitation of the two harmonic forces. For numerical simulation, y = dx∕dt, f = 0.1, ω = 1, Ω = 10, and F = 1, 5, 8, and 12 from the inside to the outside in each subplot.

Grahic Jump Location
Fig. 5

Numerical prediction of the transcritical and saddle-node bifurcations that induced by the system parameter μ. The simulation parameters are f = 0.01, ω = 1, and Ω = 10.

Grahic Jump Location
Fig. 6

Analytical prediction of the saddle-node bifurcation that induced by the force amplitude F for Ω = 10. The continuous thick lines are the stable branches and the dashed thin lines are the unstable branches.

Grahic Jump Location
Fig. 7

Phase trajectories of system (3) under the excitation of the two harmonic forces. For numerical simulation, y = dx∕dt, f = 0.1, ω = 1, Ω = 10, and F = 3, 10, 20, and 40 from the inside to the outside in each subplot. The phase trajectories are divergent when F = 20 and 40 in (a), F = 10, 20 and 40 in (b), F = 40 in (c) and (d).

Grahic Jump Location
Fig. 8

Numerical prediction of the saddle-node bifurcation that induced by the force amplitude F. The simulation parameters are f = 0.01, ω = 1, and Ω = 10.

Grahic Jump Location
Fig. 9

Analytical prediction of the saddle-node bifurcation that induced by the fractional-order α. The continuous thick lines are the stable branches and the dashed thin lines are the unstable branches. The simulation parameters are Ω = 10 and F = 10, 20, 40, 60 from the left curve to the right curve in each subplot.

Grahic Jump Location
Fig. 10

Numerical prediction of the saddle-node bifurcation that induced by the fractional-order α. The simulation parameters are f = 0.01, ω = 1, and Ω = 10.

Grahic Jump Location
Fig. 11

The response amplitude Q versus the system parameter μ presents the resonance-like behavior. The continuous lines are the analytical predictions and the discrete points are the numerical results. The simulation parameters are f = 0.1, ω = 1, F = 1, and Ω = 10.

Grahic Jump Location
Fig. 12

The response amplitude Q versus the force amplitude F does not present the resonance-like behavior. The continuous lines are the analytical predictions and the discrete points are the numerical results. The simulation parameters are f = 0.1, ω = 0.5, and Ω = 10.

Grahic Jump Location
Fig. 13

The response amplitude Q versus the fractional-order α does not present the resonance-like behavior. The continuous lines are the analytical predictions and the discrete points are the numerical results. The simulation parameters are f = 0.1, ω = 0.5, F = 10, and Ω = 10.

Grahic Jump Location
Fig. 14

The resonance behavior of Q versus the low-frequency ω depends on the fractional-order α. The continuous lines are the analytical predictions and the discrete points are the numerical results. The simulation parameters are f = 0.1, F = 6, and Ω = 20.

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