Skip to main content
SpringerLink
Log in

Vibrational Resonance in Time-Delayed Nonlinear Systems

  • Chapter
  • First Online:
Nonlinear Dynamics and Complexity

Part of the book series: Nonlinear Systems and Complexity ((NSCH,volume 8))

Abstract

Time-delay is ubiquitous in many dynamical systems. The role of single and multiple time-delay on vibrational resonance in a single Duffing oscillator and in a system of n Duffing oscillators coupled unidirectionally and driven by both a low- and a high-frequency periodic force is presented. The investigation is performed through both theoretical approach and numerical simulation. Theoretically determined values of the amplitude of the high frequency force and the delay-time at which resonance occurs are in very good agreement with the numerical simulation. A major consequence of time-delay feedback is that it gives rise to a periodic or quasiperiodic pattern of vibrational resonance profile with respect to the time-delay parameter. For the system of n-coupled oscillators with a single time-delay coupling, the response amplitudes of the oscillators are shown to be independent of the time-delay. In the case of a multi time-delayed coupling, undamped signal propagation occurs for coupling strength (δ) above a certain critical value (denoted as δ u). Further, the response amplitude approaches a limiting value Q L with the oscillator number i. Analytical expressions for both δ u and Q L are determined.

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

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 109.99
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. Gammaitoni L, Hanggi P, Jung P, Marchesoni F (1998) Rev Mod Phys 70:223

    Article  Google Scholar 

  2. McDonnell MD, Stocks NG, Pearce CEM, Abbott D (2008) Stochastic resonance. Cambridge University Press, Cambridge

    Book  MATH  Google Scholar 

  3. Zambrano S, Casado JM, Sanjuan MAF (2007) Phys Lett A 366:428

    Article  Google Scholar 

  4. Pikovsky AS, Kurths J (1997) Phys Rev Lett 78:775

    Article  MathSciNet  MATH  Google Scholar 

  5. Landa PS, McClintock PVE (2000) J Phys Math Gen 33:L433

    Article  MathSciNet  MATH  Google Scholar 

  6. Blekhman II, Landa PS (2004) Int J Non Lin Mech 39:421

    Article  MATH  Google Scholar 

  7. Chizhevsky VN (2008) Int J Bifurcat Chaos 18:1767

    Article  Google Scholar 

  8. Jeyakumari S, Chinnathambi V, Rajasekar S, Sanjuan MAF (2009) Phys Rev E 80:046608

    Article  Google Scholar 

  9. Baltanas JP, Lopez L, Blekhman II, Landa PS, Zaikin A, Kurths J, Sanjuan MAF (2003) Phys Rev E 67:066119

    Article  Google Scholar 

  10. Rajasekar S, Jeyakumari S, Chinnathambi V, Sanjuan MAF (2010) J Phys Math Theor 43:465101

    Article  MathSciNet  Google Scholar 

  11. Jeyakumari S, Chinnathambi V, Rajasekar S, Sanjuan MAF (2009) Chaos 19:043128

    Article  Google Scholar 

  12. Rajasekar S, Abirami K, Sanjuan MAF (2011) Chaos 21:033106

    Article  Google Scholar 

  13. Ullner E, Zaikin A, Garcia-Ojalvo J, Bascones R, Kurths J (2003) Phys Lett A 312:348

    Article  MathSciNet  Google Scholar 

  14. Yang JH, Zhu H (2012) Chaos 22:013112

    Article  Google Scholar 

  15. Rajasekar S, Used J, Wagemakers A, Sanjuan MAF (2012) Comm Nonlinear Sci Numer Simulat 17:3435

    Article  MATH  Google Scholar 

  16. Deng B, Wang J, Wei X, Tsang KM, Chan WL (2010) Chaos 20:013113

    Article  Google Scholar 

  17. Jeevarathinam C, Rajasekar S, Sanjuan MAF (2013) Ecol Complex 15:33

    Article  Google Scholar 

  18. Yao C, Zhan M (2010) Phys Rev E 81:061129

    Article  Google Scholar 

  19. Yang JH, Liu XB (2011) Phys Scripta 83:065008

    Article  Google Scholar 

  20. Chizhevsky VN, Giacomelli G (2008) Phys Rev E 77:051126

    Article  Google Scholar 

  21. Atay FM (2010) Complex time-delay systems: theory and applications. Springer, Berlin

    Book  Google Scholar 

  22. Lakshmanan M, Senthilkumar DV (2010) Dynamics of nonlinear time-delay systems. Springer, Berlin

    MATH  Google Scholar 

  23. Choe CU, Dahms T, Hovel P, Scholl E (2010) Phys Rev E 81:025205(R)

    Google Scholar 

  24. Fischer I, Vicente R, Buldu JM, Peil M, Mirasso CR, Torrent MC, Garcia-Ojalvo J (2006) Phys Rev Lett 97:123902

    Article  Google Scholar 

  25. Vincente R, Gollo LL, Mirasso CR, Fischer I, Pipa G (2008) Proc Natl Acad Sci USA 105:17157

    Article  Google Scholar 

  26. Flunkert V, D’Huys O, Danckaert J, Fischer I, Scholl E (2009) Phys Rev E 79:065201(R)

    Google Scholar 

  27. Ramana Reddy DV, Sen A, Johnston GL (2000) Phys Rev Lett 85:3381

    Article  Google Scholar 

  28. Gassel M, Glatt E, Kaiser F (2007) Fluct Noise Lett 7:L225

    Article  Google Scholar 

  29. Bonnin M, Corinto F, Gilli M (2007) Int J Bifurcat Chaos 17:4033

    Article  MathSciNet  MATH  Google Scholar 

  30. Dahlem MA (2009) Phil Trans Roy Soc Lond 367:1079

    Article  MathSciNet  MATH  Google Scholar 

  31. Yang JH, Liu XB (2010) J Phys A Math Theor 43:122001

    Article  Google Scholar 

  32. Yang JH, Liu XB (2010) Chaos 20:033124

    Article  Google Scholar 

  33. Jeevarathinam C, Rajasekar S, Sanjuan MAF (2011) Phys Rev E 83:066205

    Article  Google Scholar 

  34. Hu D, Yang JH, Liu XB (2012) Comm Nonlinear Sci Numer Simulat 17:1031

    Article  MathSciNet  Google Scholar 

  35. Daza A, Wagemakers A, Rajasekar S, Sanjuan MAF (2013) Comm Nonlinear Sci Numer Simulat 18:411

    Article  MathSciNet  MATH  Google Scholar 

  36. Yang JH, Liu XB (2010) Phys Scripta 82:025006

    Article  Google Scholar 

  37. Ahlborn A, Parlitz U (2005) Phys Rev E 72:016206

    Article  MathSciNet  Google Scholar 

  38. Shahverdiev EM, Shore KA (2008) Phys Rev E 77:057201

    Article  Google Scholar 

  39. Gakkhar S, Negi K, Sahani SK (2009) Comm Nonlinear Sci Numer Simulat 14:850

    Article  MathSciNet  MATH  Google Scholar 

  40. Englert A, Kinzel W, Aviad Y, Butkovski M, Reidler I, Zigzag M, Kanter I, Rosenblum M (2010) Phys Rev Lett 104:114102

    Article  Google Scholar 

  41. Englert A, Heiligenthal S, Kinzel W, Kanter I (2011) Phys Rev E 83:046222

    Article  Google Scholar 

  42. Lee WS, Restrepo JG, Ott E, Antonsen TM (2011) Chaos 21:023122

    Article  MathSciNet  Google Scholar 

  43. Gakkhar S, Singh A (2012) Comm Nonlinear Sci Numer Simulat 17:914

    Article  MathSciNet  MATH  Google Scholar 

  44. Martinenghi R, Rybalko S, Jacquot M, Chembo YK, Larger L (2012) Phys Rev Lett 108:244101

    Article  Google Scholar 

  45. Cushing JM (1977) Integro-differential equations and delay models in population dynamics. Springer, Berlin

    Book  Google Scholar 

  46. Maza D, Mancini H, Boccaletti S, Genesio R, Arecchi FT (1998) Int J Bifurcat Chaos 8:1843

    Article  MATH  Google Scholar 

  47. Khrustova N, Veser G, Mikhailov A, Imbihl R (1995) Phys Rev Lett 75:3564

    Article  Google Scholar 

  48. Schanz M, Pelster A (2003) Phys Rev E 67:056205

    Article  Google Scholar 

  49. Larger L, Goldgebuer J, Erheux T (2004) Phys Rev E 69:036210

    Article  Google Scholar 

  50. Erneux T, Glorieux P (2010) Laser dynamics. Cambridge University Press, Cambridge

    Book  Google Scholar 

  51. Argyris A, Syvridis D, Larger L, Annovazzi-Lodi V, Colet P, Fischer I, Garcia-Ojalvo J, Mirasso CR, Pesquera L, Shore KA (2005) Nature 438:343

    Article  Google Scholar 

  52. Uchida A, Amano K, Inoue M, Hirano K, Naito S, Someya H, Oowada I, Kurashige T, Shiki M, Yoshimori S, Yoshimura K, Davis P (2008) Nat Photon 2:728

    Article  Google Scholar 

  53. MacDonald N (1989) Biological delay systems: linear stability theory. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  54. Hashemi M, Valizadeh A, Azizi Y (2012) Phys Rev E 85:021917

    Article  Google Scholar 

  55. Suel GM, Garcia-Ojalvo J, Liberman LM, Elowitz MB (2006) Nature 440:545

    Article  Google Scholar 

  56. Mergenthaler K, Engbert R (2007) Phys Rev Lett 98:138104

    Article  Google Scholar 

  57. Cabrera JL, Milton JG (2002) Phys Rev Lett 89:158702

    Article  Google Scholar 

  58. Bauer S, Brox O, Kreissl J, Sartorius B, Radziunas M, Sieber J, Wunsche HJ, Henneberger F (2004) Phys Rev E 69:016206

    Article  Google Scholar 

  59. Gerstner W, van Hemmen JL (1993) Phys Rev Lett 71:312

    Article  Google Scholar 

  60. Bacci A, Huguenard JR, Prince DA (2003) J Neurosci 23:859; (2006) Neuron 49:119

    Google Scholar 

  61. Diez-Martinez O, Segundo JP (1983) Biol Cybern 47:33

    Article  Google Scholar 

  62. Pakdaman K, Vibert JF, Boussard E, Azmy N (1996) Neural Network 9:797

    Article  Google Scholar 

  63. Kane DM, Shore KA (2005) Unlocking dynamical diversity: optical feedback effects on semiconductor lasers. Wiley, New York

    Book  Google Scholar 

  64. Shahverdiev EM, Shore KA (2008) Phys Rev E 77:057201

    Article  Google Scholar 

  65. Jeevarathinam C, Rajasekar S, Sanjuan MAF (2013) Chaos 23:013136

    Article  Google Scholar 

  66. Pipes LA, Harvill LR Applied mathematics for engineers and physicists, 3rd edn. McGraw-Hill, New York

    Google Scholar 

Download references

Acknowledgements

MAFS acknowledges the financial support from the Spanish Ministry of Science and Innovation under Project No. FIS2009-09898.

Author information

Authors and Affiliations

  1. Nonlinear Dynamics, Chaos and Complex Systems Group, Departamento de Física, Universidad Rey Juan Carlos, Tulipán s/n, 28933, Móstoles, Madrid, Spain

    S. Rajasekar & M. A. F. Sanjuán

Authors
  1. S. Rajasekar
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. A. F. Sanjuán
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. A. F. Sanjuán .

Editor information

Editors and Affiliations

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

    Valentin Afraimovich

  2. School of Engineering Mechanical and Industrial Engineering, Southern Illinois University Edwardsville, Edwardsville, Illinois, USA

    Albert C. J. Luo

  3. Shandong Normal University, Ji’nan, People's Republic of China

    Xilin Fu

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Rajasekar, S., Sanjuán, M.A.F. (2014). Vibrational Resonance in Time-Delayed Nonlinear Systems. In: Afraimovich, V., Luo, A., Fu, X. (eds) Nonlinear Dynamics and Complexity. Nonlinear Systems and Complexity, vol 8. Springer, Cham. https://doi.org/10.1007/978-3-319-02353-3_9

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-02353-3_9

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-02352-6

  • Online ISBN: 978-3-319-02353-3

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation