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TECHNICAL PAPERS

An Investigation of Coupled van der Pol Oscillators

[+] Author and Article Information
Lesley Ann Low, Per G. Reinhall, Duane W. Storti

Department of Mechanical Engineering, University of Washington, Seattle, WA 98195

J. Vib. Acoust 125(2), 162-169 (Apr 01, 2003) (8 pages) doi:10.1115/1.1553469 History: Received September 01, 2000; Revised September 01, 2002; Online April 01, 2003
Copyright © 2003 by ASME
Topics: Stability , Motion , Equations
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References

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Murray, J. D., 1993, Mathematical Biology, Springer-Verlag, New York.
Storti,  D. W. and Reinhall,  P. G., 1995, Stability of in-phase and out-of-phase modes for a pair of linearly coupled van der Pol oscillators. Stability, Vibration and Control of Systems, 2(B), 1–23.
Blevins, R. D., 1977, Flow Induced Vibrations. Van Nostrand, New York.
Cohen,  A. H., Holmes,  P. J., and Rand,  R. H., 1982, The nature of the coupling between segmental oscillators of the lamprey spinal generator for locomotion: A mathematical model. J. Math. Biol., 13, 345–369.
Rand,  R. H., and Holmes,  P. J., 1980, Bifurcation of periodic motions in two weakly coupled van der Pol oscillators. Int. J. Non-Linear Mech., 15, 387–399.
Reinhall, P. G., and Storti, D. W., 1995, A numerical investigation of phase-locked and chaotic behavior of coupled van der Pol oscillators. Proceedings of the 15th Biennial ASME Conference on Mechanical Vibration and Noise.
Storti,  D. W., and Reinhall,  P. G., 2000, Phase-locked mode stability for coupled van der Pol oscillators. ASME J. Vibr. Acoust., 122(3), 318–323.
Chiu,  C. H., Lin,  W. W., and Wang,  C. S., 1998, Synchronization in a lattice of coupled van der Pol systems. Int. J. Bifurcation Chaos Appl. Sci. Eng., 8(12), 2353–2373.
Storti,  D. W., and Rand,  R. H., 1982, Dynamics of two strongly coupled van der Pol oscillators. Int. J. Non-Linear Mech., 17(3), 143–152.
Yoshinaga,  T., Kawakami,  H., and Yoshikawa,  K., 1991, Mode bifurcations in diffusively coupled van der Pol equations. IEICE Trans. Commun., 74(6), 1420–1427.
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Storti, D. W., Nevrinceanu, C., and Reinhall, P. G., 1993, Perturbation solution of an oscillator with periodic van der Pol damping. Dynamics and Vibration of Time-Varying Systems and Structures, ASME DE, 56:397–402.
Andersen,  C. M., and Geer,  J. F., 1982, Power series expansions for the frequency and period of the limit cycle of the van der Pol equation. SIAM (Soc. Ind. Appl. Math.) J. Appl. Math., 42, 678–693.
Dadfar,  M. B., Greer,  J., and Andersen,  C. M., 1984, Perturbation analysis of the limit cycle of the van der Pol equation. SIAM (Soc. Ind. Appl. Math.) J. Appl. Math., 44, 881–895.
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Figures

Grahic Jump Location
Transition curve, ε=2.5. (Response at Pts. A,B and C shown in Figs. 3, 4 and 10)
Grahic Jump Location
Transition curves, ε=0.5, 2.5, 5, 7.5 and 10. (ε=0.5 on top and ε=10 on bottom)
Grahic Jump Location
In-phase response (Pt. A on Fig. 1), ε=2.5,εA=3.0,εB=−0.4
Grahic Jump Location
Out-of-phase response (Pt. B on Fig. 1), ε=2.5,εA=2.0,εB=−1.4
Grahic Jump Location
Shifted asymmetric trajectory (ε=10,εB=−2.5,εA=1.765)
Grahic Jump Location
Shifted symmetric trajectory (ε=10,εB=−2.5,εA=2.75)
Grahic Jump Location
Transition curve, ε=10, with shifted symmetric (rightmost region with thick outline), and shifted asymmetric regions (leftmost region with thick outline)
Grahic Jump Location
Transition curve, ε=10.0 (upper left), 7.5 (upper right), 5.0 (lower left), 2.5 (lower right) with shifted symmetric (rightmost region with thick outline), and shifted asymmetric regions (leftmost region with thick outline)
Grahic Jump Location
Phase lag (ϕ) versus εB(ε=10,εA=0.3)
Grahic Jump Location
Time history of chaotic regime (ε=10,εB=−2.5,εA=2.033)
Grahic Jump Location
Bifurcation plot, x,y amplitude per cycle versus εA(ε=10 and εB=−2.5)
Grahic Jump Location
Correlation dimension (ε=10,B=−2.5,A=2.033)
Grahic Jump Location
Poincare map (ε=10,B=−2.5,A=2.033)
Grahic Jump Location
Transition curves, ε=1(εΔ=0.5,εΔ=0.25,εΔ=0)
Grahic Jump Location
Transition curve, ε=1(εΔ=0.25 and εΔ=0)
Grahic Jump Location
Transition curve, ε=1(εΔ=0.5 and εΔ=0)
Grahic Jump Location
Time history, beating (T=50 sec), no phase-locking region with detuning (εΔ=0.25,ε=1,εA=0.4912,εB=0.0004)
Grahic Jump Location
Transition curves (analytical results) for the network of 2, 3, 4, 5 and 6 coupled van der Pol oscillators, ε=1.0
Grahic Jump Location
Analytical (line) and numerical (points) transition curves of 3 and 4 coupled van der Pol oscillators, ε=1.0

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