Skip to main content
SpringerLink
Log in

Analytical Period-m Motions in a Parametric, Quadratic Nonlinear Oscillator

  • Chapter
  • First Online:
Complex Motions and Chaos in Nonlinear Systems

Part of the book series: Nonlinear Systems and Complexity ((NSCH,volume 15))

  • 973 Accesses

Abstract

Analytical solutions of period-m motions in a parametric quadratic nonlinear oscillator are obtained through the finite Fourier series, and the corresponding stability and bifurcation analysis for periodic motions are discussed. The bifurcation trees of periodic motions to chaos in a parametric oscillator with quadratic nonlinearity are presented. Numerical illustration shows good agreement between the analytical and numerical results.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 54.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

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

    Google Scholar 

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

    MATH  Google Scholar 

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

    MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

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

    MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  7. 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 

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

    Article  Google Scholar 

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

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  11. 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 

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

    Article  MathSciNet  MATH  Google Scholar 

  13. 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 

  14. 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 Dynamics 52:403–414

    Article  MATH  Google Scholar 

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

    MATH  Google Scholar 

  16. 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)

    Google Scholar 

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

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  19. Luo ACJ, O’Connor D (2014) On periodic motions in Mathieu-Duffing oscillator. Int J Bifurcation Chaos 24:Article no:1430004 (17 pages)

    Google Scholar 

  20. 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 Acous 135:Article no:034505 (5 p)

    Google Scholar 

  21. Luo ACJ, Yu B (2015) Complex period-1 motions in a periodically forced, quadratic nonlinear oscillator. J Vib Control. 21(1):896–906

    Google Scholar 

  22. Luo ACJ, Yu B (2013) Period-m motions in a periodically forced oscillator with quadratic nonlinearity. Discontin Nonlinearity Complex 2(3):265–288

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Southern Illinois University Edwardsville, Edwardsville, IL, 62026-1805, USA

    Albert C. J. Luo & Bo Yu

Authors
  1. Albert C. J. Luo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Bo Yu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Albert C. J. Luo .

Editor information

Editors and Affiliations

  1. San Luis Potosi University, San Luis Potosi, Mexico

    Valentin Afraimovich

  2. Institute of Engineering, Polytechnic of Porto, Porto, Portugal

    José António Tenreiro Machado

  3. School of Energy&PowerEngineering, Xi’an Jiaotong University, Shaanxi Province, Shaanxi, China

    Jiazhong Zhang

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Luo, A.C.J., Yu, B. (2016). Analytical Period-m Motions in a Parametric, Quadratic Nonlinear Oscillator. In: Afraimovich, V., Machado, J., Zhang, J. (eds) Complex Motions and Chaos in Nonlinear Systems. Nonlinear Systems and Complexity, vol 15. Springer, Cham. https://doi.org/10.1007/978-3-319-28764-5_9

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-28764-5_9

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-28762-1

  • Online ISBN: 978-3-319-28764-5

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation