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Analytical routes of period-m motions to chaos in a parametric, quadratic nonlinear oscillator

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Abstract

In this paper, analytical routes of periodic motions to chaos in a parametric, quadratic oscillator are investigated. The analytical solutions of period-m motions in such a parametric oscillator are presented from the finite term Fourier series, and the stability and bifurcation of the period-m motions are discussed through the eigenvalue analysis. The routes of periodic motions to chaos in such parametric oscillator are illustrated through harmonic amplitudes varying with excitation amplitude in the finite term Fourier series solution. From the routes of period-m motion to chaos, numerical illustrations of periodic motions are given through trajectories and analytical harmonic amplitude spectrum.

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References

  1. Luo ACJ, Yu B (2014) Bifurcation trees of periodic motion to chaos in a parametric, quadratic nonlinear oscillator. Int J Bifurcation Chaos (in press)

  2. Luo ACJ, Yu B (2014) On effects of excitation amplitude on bifurcation trees of periodic motions in parametric, quadratic nonlinear oscillator. Nonlinear Dyn Mobile Robot 1(2):231–263

  3. Mathieu E (1868) Memoire sur le mouvement vibratoire d’une membrance deforme elliptique. J Matheematique 2(13):137–203

    Google Scholar 

  4. Mathieu E (1873) Cours de physique methematique. Gauthier-Villars, Paris

    Google Scholar 

  5. Hill GW (1886) On the part of the motion of the lunar perigee. Acta Math 8:1–36

    Article  MathSciNet  MATH  Google Scholar 

  6. Whittaker ET (1913) General solution of Mathieu’s equation. Proc Edinb Math Soc 32:75–80

    Article  Google Scholar 

  7. Whittaker ET, Watson GN (1935) A course of modern analysis. Cambridge University Press, London

    Google Scholar 

  8. McLachlan NW (1947) Theory and applications of Mathieu equations. Oxford University Press, London

    Google Scholar 

  9. Sevin E (1961) On the parametric excitation of pendulum-type vibration absorber. ASME J Appl Mech 28:330–334

    Article  MATH  Google Scholar 

  10. Hsu CS (1963) On the parametric excitation of a dynamics system having multiple degrees of freedom. ASME J Appl Mech 30:369–372

    Article  Google Scholar 

  11. Hsu CS (1965) Further results on parametric excitation of a dynamics system. ASME J Appl Mech 32:373–377

  12. Hayashi C (1964) Nonlinear oscillations in physical systems. Mcgraw-Hill, New York

    MATH  Google Scholar 

  13. Tso WK, Caughey TK (1965) Parametric excitation of a nonlinear system. ASME J Appl Mech 32:899–902

    Article  MathSciNet  MATH  Google Scholar 

  14. Mond M, Cederbaum G, Khan PB, Zarmi Y (1993) Stability analysis of non-linear Mathieu equation. J Sound Vib 167(1):77–89

    Article  MathSciNet  MATH  Google Scholar 

  15. Zounes RS, Rand RH (2000) Transition curves for the quasi-periodic Mathieu equations. SIAM J Appl Math 58(4):1094–1115

    Article  MathSciNet  Google Scholar 

  16. Luo ACJ (2004) Chaotic motion in the resonant separatrix bands of a Mathieu–Duffing oscillator with twin-well potential. J Sound Vib 273:653–666

    Article  Google Scholar 

  17. Shen JH, Lin KC, Chen SH, Sze KY (2008) Bifurcation and route-to-chaos analysis for Mathieu–Duffing oscillator by the incremental harmonic balance method. Nonlinear Dyn 52:403–414

    Article  MATH  Google Scholar 

  18. Luo ACJ (2012) Continuous dynamical systems. Higher Education Press/L&H scientific, Beijing/Glen Carbon

  19. Luo ACJ, Huang JZ (2012) Analytical dynamics of period-m flows and chaos in nonlinear systems. Int J Bifurcation Chaos 22: Article No: 1250093 (29 pages)

  20. Luo ACJ, Huang JZ (2012) Analytical routines of period-1 motions to chaos in a periodically forced Duffing oscillator with twin-well potential. J Appl Nonlinear Dyn 1:73–108

    Article  MATH  Google Scholar 

  21. Luo ACJ, Huang J (2012) Unstable and stable period-m motions in a twin-well potential Duffing oscillator. Discontinuity Nonlinearity Complex 1:113–145

    Article  MATH  Google Scholar 

  22. Luo ACJ, O’Connor D (2013) On periodic motions in parametric hardening Duffing oscillator. Int J Bifurcation Chaos 24(1): Article No: 1430004 (17 pages)

  23. Luo ACJ, Yu B (2013) Analytical solutions for stable and unstable period-1 motions in a periodically forced oscillator with quadratic nonlinearity. ASME J Vib Acoust 135: Article No: 034505 (5 pages)

  24. Luo ACJ, Yu B (2013) Complex period-1 motions in a periodically forced, quadratic nonlinear oscillator. J Vib Control. doi:10.1177/1077546313490525

  25. Luo ACJ, Yu B (2013) Period-m motions in a periodically forced oscillator with quadratic nonlinearity. Discontinuity Nonlinearity Complex 2:265–288

    Google Scholar 

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Authors and Affiliations

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

    Albert C. J. Luo & Bo Yu

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  1. Albert C. J. Luo
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  2. Bo Yu
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Correspondence to Albert C. J. Luo.

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Luo, A.C.J., Yu, B. Analytical routes of period-m motions to chaos in a parametric, quadratic nonlinear oscillator. Int. J. Dynam. Control 4, 1–22 (2016). https://doi.org/10.1007/s40435-014-0112-7

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  • DOI: https://doi.org/10.1007/s40435-014-0112-7

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