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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 os- cillator is discretized to obtain implicit discrete mappings. From the implicit mapping structures, periodic motions varying with excita- tion 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 pre- sented. Thus, from the analytical prediction, numerical simulations of periodic motions are performed for comparison of numerical and an- alytical results. The harmonic amplitude spectrums of periodic mo- tions are also presented for harmonic effects on the periodic motions. Through this study, the order of symmetric period-1 to chaotic mo- tions (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 indepen- dent 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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