Skip to main content
SpringerLink
Log in

Nonlinear oscillators with state variable damping and elastic coefficients

  • Published:
Pramana Aims and scope Submit manuscript

Abstract

In the present work, we investigate the chaotic and limit-cycle behaviour of the new models of damped nonlinear oscillators with elastic coefficients. The study concerning the stability of these models in their autonomous state presents periodical regions of stability and instability, a very rich dynamical behaviour. The analytical investigations on the existence of limit cycles show that a periodic solution (and therefore a limit cycle) in the (x, y)-plane encompasses the origin. These models can be used to describe with some approximations, the artificial pacemaker. The chaos analysis shows the effect of state variable damping and elastic coefficient on the appearance of chaotic dynamics. Due to the complexity of the analogical functions, an experimental study was made by real implementation of an Arduino Card based on the Runge–Kutta algorithm of order 4. The results obtained show a good correlation with numerical results.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13

Similar content being viewed by others

References

  1. R E Mickens, Circuits, Syst. Signal Process. 8, 187 (1989)

    Article  Google Scholar 

  2. S K Joshi, Int. J. Dynam. Control. 9, 602 (2021)

    Article  MathSciNet  Google Scholar 

  3. S Dashkovskiy and S Pavlichkov, Automatica 112, 108643 (2020)

    Article  Google Scholar 

  4. S S Vwalker and J A Connelly, Circuits, Syst. Signal Process. 2, 213 (1983)

    Google Scholar 

  5. J Cheng and Y Zhan, Appl. Math. Comput. 365, 124714 (2020)

    MathSciNet  Google Scholar 

  6. Y Tang, Z Wu, S Peng and F Qian, Automatica 113, 108766 (2020)

    Article  Google Scholar 

  7. L Enrique, B Gonzalez, R Q Bermudez and R Q Torres, Circuits, Syst. Signal Process. 39, 4775 (2020)

    Google Scholar 

  8. M Xiao, G Jiang and J Cao, Circuits, Syst. Signal Process. 35, 2041 (2016)

    Article  Google Scholar 

  9. Y Han, Opt. Commun. 445, 262 (2019)

    Article  ADS  Google Scholar 

  10. J P Ramirez, E Garcia and J Alvarez, Commun. Nonlinear Sci. Numer. Simul. 80, 104977 (2020)

    Article  MathSciNet  Google Scholar 

  11. C R Goncalves, B Stefan, V Tholakanahalli, A Römer, I Hofmann, M Reinartz, G Sameer, K Sievert, S Nalan, I Grunwald and H Sievert, Cardiovasc. Revasc. Med. 21, 726 (2020)

    Article  Google Scholar 

  12. J J Zebrowski, K Grudziński, T Buchner, P Kuklik and J Gac, Chaos 17, 015121 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  13. K Grudziński, J J Żebrowski and R Baranowski, Biomed. Tech. Eng. 51, 210 (2006)

    Article  Google Scholar 

  14. K Grudziński, J J Żebrowski and R Baranowski, Complex dynamics in physiological systems: From heart to brain (Springer, 2009) Vol. 2, pp. 127–136

  15. J J Zebrowski, IEEE Eng. Med. Biol. Mag. 28, 24 (2009)

    Google Scholar 

  16. G C Cardarilli, Appl. Sci. 9, 3653 (2019)

    Article  Google Scholar 

  17. S Behnia and J Ziaei, Chaotic Model. Simul. 3, 281 (2016)

    Google Scholar 

  18. R FitzHugh, Biophys J. 1, 445 (1961)

    Article  ADS  Google Scholar 

  19. J Nagumo, S Arimoto and S Yoshizawat, Proc. IRE 50, 2061 (1962)

    Article  Google Scholar 

  20. W H Steeb, Lett. Math. Phys. 2, 171 (1977)

    Article  ADS  MathSciNet  Google Scholar 

  21. W H Steeb and A Kunick, Int. J. Nonlinear Mech. 17, 41 (1982)

    Article  ADS  Google Scholar 

  22. W H Steeb and A Kunick, Phys. Lett. 95A, 269 (1983)

    Article  ADS  Google Scholar 

  23. W H Steeb, Int J. Non-Linear Mech. 22, 349 (1987)

    Article  ADS  Google Scholar 

  24. W H Steeb, W Erig and A Kunick, Phys. Lett.\(A\)93, 267 (1983)

    Article  ADS  MathSciNet  Google Scholar 

  25. H Hochstadt and B H Stephan, Arch. Rt. Mech. Anal. 23, 368 (1967)

    Google Scholar 

  26. R N D’Heedene, Diff. Eq. 5, 564 (1996)

  27. D W Storti and P G Reinhall, Nonlinear Dyn. 2, 1 (1997)

    Article  Google Scholar 

  28. D A Linkens, Bull. Math. Biol. 39, 359 (1977)

    Google Scholar 

  29. T Kai and K Tomita, Prog. Theor. Phys. 61, 54 (1979)

    Article  ADS  Google Scholar 

  30. T Kai and K Tomita, J. Stat. Phys. 21, 65 (1979)

    Article  ADS  Google Scholar 

  31. J Dreitlein and M Smoes, J. Theor. Eiol. 46, 559 (1974)

    ADS  Google Scholar 

  32. B Van der Pol, London, Edinburgh Dublin Philos. Mag. J. Sci. Ser. 7–2, 978 (1926)

    Google Scholar 

  33. G Bub and L Glass, Int. J. Bifurc. Chaos 5, 359 (1995)

    Article  Google Scholar 

  34. S Behnia, J Ziaei, M Ghiassi and A Akhshani, Chin. J. Phys. 53, 120702 (2015)

    Google Scholar 

  35. S Behnia, J Ziaei and M Ghiassi, Iranian Conference on Electrical Engineering (ICEE), https://doi.org/10.1109/IranianCEE.2013.6599874 (2013)

  36. L Makouo and PWoafo, Chaos Solitons Fractals 94, 95 (2017)

    Article  ADS  Google Scholar 

  37. H Simo and P Woafo, Int. J. Bifurc. Chaos 22, 1 1250003 (2012)

    Article  Google Scholar 

  38. K Grudzinski and J J Zebrowski, Physica A 336, 153 (2004)

    Article  ADS  MathSciNet  Google Scholar 

  39. B Van Der Pol and J Van Der Mark, Philos. Mag. J. Sci. Ser. 7, 763 (1928)

    Google Scholar 

  40. B Van Der Pol and J Van Der Mark, Philos. Mag. 2, 978 (1926)

    Article  Google Scholar 

  41. J C Chedjou, H B Fotsin and P Woafo, Phys. Scr. 55, 390 (1997)

    Article  ADS  Google Scholar 

  42. G Xu, Y Shekofteh, A Akgül, C Li and S Panahi, Entropy 20, 86 (2018)

    Article  ADS  Google Scholar 

  43. C Li and J C Sprott, Phys. Lett. A 378, 178 (2014)

    Article  ADS  MathSciNet  Google Scholar 

  44. C Li, J C Sprott, Z Yuan and H Li, Int. J. Bifurc. Chaos 25, 1530025 (2015)

    Article  Google Scholar 

  45. C Li, J C Sprott and H Xing, Nonlinear Dyn. 87, 1351 (2017)

    Article  Google Scholar 

  46. C Li, J C Sprott, A Akgul, Herbert H C Iu and Y Zhao, Chaos 27, 083101 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  47. C Li, W Joo-Chen Thio, H Ho-Ching Iu and T Lu, IEEE Access. 6, 12945 (2018)

    Article  Google Scholar 

  48. H Chaté, Nonlinearity 7, 185 (1994)

    Article  ADS  MathSciNet  Google Scholar 

  49. V V Castets, E Dulos, J Boissonade and P D Kepper, Phys. Rev. Lett. 64(24), 2953 (1990)

    Article  ADS  Google Scholar 

  50. P Coullet and C Tresser, J. de Phys. Colloque 39, C5-25 (1978)

    Google Scholar 

  51. I Bendixson, Acta Math. 24, 1 (1901)

    Article  MathSciNet  Google Scholar 

  52. R Thepi Siewe, U Simo Domguia and P Woafo, Commun. Nonlinear Sci. Numer. Simul. 69, 343 (2019)

    Article  ADS  Google Scholar 

  53. R Thepi Siewe, U Simo Domguia and P Woafo, IJNSNS 19(2), 153 (2018)

  54. E M Tekougoum, U G Ngouabo, S Noubissie, H B Fotsin and P Woafo, Commun. Nonlinear Sci. Numer. Simul. 62, 454 (2018)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Research Unit Condensed Matter, Electronics and Signal Processing, Université de Dschang, P.O. Box 67, Dschang, Cameroon

    R F Fonkou & Patrick Louodop

  2. UR de Mécanique et de Modélisation des Systèmes Physiques (UR-2MSP), UFR/DSST, Université de Dschang, BP 67, Dschang, Cameroun

    R F Fonkou & P K Talla

Authors
  1. R F Fonkou
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Patrick Louodop
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. P K Talla
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Fonkou, R.F., Louodop, P. & Talla, P.K. Nonlinear oscillators with state variable damping and elastic coefficients. Pramana - J Phys 95, 210 (2021). https://doi.org/10.1007/s12043-021-02230-w

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s12043-021-02230-w

Keywords

PACS Nos

Search

Navigation