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ISSN: 2475-4811 (print)
ISSN: 2475-482X (online)
Journal of Vibration Testing and System Dynamics

C. Steve Suh (editor), Pawel Olejnik (editor),

Xianguo Tuo (editor)

Pawel Olejnik (editor)

Lodz University of Technology, Poland

Email: pawel.olejnik@p.lodz.pl

C. Steve Suh (editor)

Texas A&M University, USA

Email: ssuh@tamu.edu

Xiangguo Tuo (editor)

Sichuan University of Science and Engineering, China

Email: tuoxianguo@suse.edu.cn

A Series of Symmetric Period-1 Motions to Chaos in a Two-degree-of-freedom van der Pol-Duffing Oscillator

Journal of Vcibration Testing and System Dynamics 2(2) (2018) 119--153 | DOI:10.5890/JVTSD.2018.06.003

Yeyin Xu; Albert C.J. Luo

Department of Mechanical and Industrial Engineering, Southern Illinois University Edwardsville, IL 62026-1805, USA

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Abstract

In this paper, independent periodic motions in the two-degree-of-freedom (2-DOF) van der Pol-Duffing oscillator are investigated. From the semi-analytical method, the 2-DOF van der Pol-Duffing oscillator is discretized to obtain implicit discrete mappings. From the implicit mapping structures, periodic motions varying with excitation frequency are obtained semi-analytically, and the corresponding stability and bifurcation are obtained by eigenvalue analysis. The frequency-amplitude characteristics of periodic motions are also presented. Thus, from the analytical prediction, numerical simulations of periodic motions are performed for comparison of numerical and analytical results. The harmonic amplitude spectrums of periodic motions are also presented for harmonic effects on the periodic motions. Through this study, the order of symmetric period-1 to chaotic motions (i.e., 1(S)⊳1(A)⊳3(S)⊳2(A)⊳· · ·⊳m(A)⊳(2m+1)(S)⊳· · · ) (m→ꝏ) is discovered. Chaotic motions or catastrophe jumping phenomena between the two independent periodic motions exist. The independent periodic motions can be used for specific applications in phase locking, and such results can be useful to develop series of the van der Pol-Duffing circuits for applications.

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