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Research Papers

Resonance Oscillation of Third-Order Forced van der Pol System With Fractional-Order Derivative

[+] Author and Article Information
Nguyen Van Khang

Department of Applied Mechanics,
Hanoi University of Science and Technology,
Hanoi 100000, Vietnam
e-mail: khang.nguyenvan2@hust.edu.vn

Bui Thi Thuy

Department of Mechanics,
Hanoi University of Mining and Geology,
Hanoi 100000, Vietnam
e-mail: thuybt167ncs@gmail.com

Truong Quoc Chien

Department of Applied Mechanics,
Hanoi University of Science and Technology,
Hanoi 100000, Vietnam
e-mail: chienams@gmail.com

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received November 13, 2015; final manuscript received April 27, 2016; published online May 24, 2016. Assoc. Editor: Dumitru Baleanu.

J. Comput. Nonlinear Dynam 11(4), 041030 (May 24, 2016) (5 pages) Paper No: CND-15-1371; doi: 10.1115/1.4033555 History: Received November 13, 2015; Revised April 27, 2016

This study aims to investigate the harmonic resonance of third-order forced van der Pol oscillator with fractional-order derivative using the asymptotic method. The approximately analytical solution for the system is first determined, and the amplitude–frequency equation of the oscillator is established. The stability condition of the harmonic solution is then obtained by means of Lyapunov theory. A comparison between the traditional integer-order of forced van der Pol oscillator and the considered fractional-order one follows the numerical simulation. Finally, the numerical results are analyzed to show the influences of the parameters in the fractional-order derivative on the steady-state amplitude, the amplitude–frequency curves, and the system stability.

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Figures

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Fig. 1

The amplitude–frequency curve, where δp = 0

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Fig. 2

The amplitude–frequency curve, where δp = 1 and p = 0.25

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Fig. 3

The amplitude–frequency curve, where δp = 1 and p = 0.5

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Fig. 4

The amplitude–frequency curve, where δp = 1 and p = 0.75

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Fig. 5

The amplitude–frequency curve, where δp = 1 and p = 0.5 (the circlets denote the solution by numerical integration)

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Fig. 6

The amplitude–frequency curves corresponding to δp = 1 and different values of p

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Fig. 7

The amplitude–frequency curves corresponding to p = 0.5 and different values of δp

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Fig. 8

The amplitude–frequency curves corresponding to different values of E

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