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Multistability and hidden attractors in an impulsive Goodwin oscillator with time delay

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Abstract

The release of luteinizing hormone (LH) is driven by intermittent bursts of activity in the hypothalamic nerve centers of the brain. Luteinizing hormone again stimulates release of the male sex hormone testosterone (Te) and, via the circulating concentration of Te, the hypothalamic nerve centers are subject to a negative feedback regulation that is capable of modifying the intermittent bursts into more regular pulse trains. Bifurcation analysis of a hybrid model that attempts to integrate the intermittent bursting activity with a continuous hormone secretion has recently demonstrated a number of interesting nonlinear dynamic phenomena, including bistability and deterministic chaos. The present paper focuses on the additional complexity that arises when the time delay in the continuous part of the model exceeds the typical bursting interval of the feedback. Under these conditions, the hybrid model is capable of displaying quasiperiodicity and border collisions as well as multistability and hidden attractors.

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References

  1. G.A. Leonov, N.V. Kuznetsov, V.I. Vagaitsev, Phys. Lett. A 375, 2230 (2011)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  2. G.A. Leonov, N.V. Kuznetsov, V.I. Vagaitsev, Physica D 241, 1482 (2012)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  3. V.O. Bragin, V.I. Vagaitsev, N.V. Kuznetsov, G.A. Leonov, J. Comp. Sys. Sci. Int. 50, 511 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  4. G.A. Leonov, N.V. Kuznetsov, Int. J. Bifurcat. Chaos 23, 1330002 (2013)

    Article  MathSciNet  Google Scholar 

  5. U. Chaudhuri, A. Prasad, Phys. Lett. A 378, 713 (2014)

    Article  MathSciNet  ADS  Google Scholar 

  6. Z. Wei, Q. Yang, Nonlin. Anal.: Real World Appl. 12, 106 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  7. Z. Wei, R. Wang, A. Liu, Math. Comp. Sim. 100, 13 (2014)

    Article  MathSciNet  Google Scholar 

  8. A.P. Kuznetsov, S.P. Kuznetsov, E. Mosekilde, N.V. Stankevich, Eur. Phys. J. Special Topics 222, 2391 (2013)

    Article  ADS  Google Scholar 

  9. A.P. Kuznetsov, S.P. Kuznetsov, E. Mosekilde, N.V. Stankevich, J. Phys. A: Math. Theor. 48, 125101 (2015)

    Article  MathSciNet  ADS  Google Scholar 

  10. S. Jafari, J.C. Sprott, S.M.R.H. Golpayegani, Phys. Lett. A 377, 699 (2013)

    Article  MathSciNet  ADS  Google Scholar 

  11. M. Molaie, S. Jafari, J.C. Sprott, S.M.R.H. Golpayegani, Int. J. Bifurcat. Chaos 23, 1350188 (2013)

    Article  Google Scholar 

  12. X. Wang, G.R. Chen, Nonlin. Dyn. 71, 429 (2013)

    Article  Google Scholar 

  13. Zh.T. Zhusubaliyev, E.A. Soukhoterin, E. Mosekilde, Math. Comp. Sim. 73, 364 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. Zh.T. Zhusubaliyev, E. Mosekilde, IEEE Trans. Circ. Syst. I 53, 1839 (2006)

    Article  MathSciNet  Google Scholar 

  15. Zh.T. Zhusubaliyev, E. Mosekilde, S.M. Maity, S. Mohanan, S. Banerjee, Chaos 16, 023122 (2006)

    Article  MathSciNet  ADS  Google Scholar 

  16. Zh.T. Zhusubaliyev, E. Mosekilde, O.O. Yanochkina, IEEE Trans. Power Electron. 26, 1270 (2011)

    Article  Google Scholar 

  17. Zh.T. Zhusubaliyev, E. Mosekilde, O.O. Yanochkina, Int. J. Bifurcat. Chaos 21, 2343 (2011)

    Article  MATH  Google Scholar 

  18. Zh.T. Zhusubaliyev, E.A. Soukhoterin, E. Mosekilde, Chaos, Solitons Fractals 13, 1889 (2002)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  19. Zh.T. Zhusubaliyev, E. Mosekilde, E.V. Pavlova, IEEE Trans. Industrial Inform. 9, 1937 (2013)

    Article  Google Scholar 

  20. Zh.T. Zhusubaliyev, E. Mosekilde, Math. Comp. Sim. 109, 32 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  21. Zh.T. Zhusubaliyev, E. Mosekilde, Bifurcations and Chaos in Piecewise-Smooth Dynamical Systems (World Scientific, Singapore, 2003)

  22. S. Banerjee, G.C. Verghese (eds.), Nonlinear Phenomena in Power Electronics (IEEE Press, New York, USA, 2001)

  23. C.K. Tse, Complex Behavior of Switching Power Converters (CRC Press, Boca Raton, USA, 2003)

  24. M. di Bernardo, C.J. Budd, A.R. Champneys, P. Kowalczyk, Piecewise-Smooth Dynamical Systems: Theory and Applications (Springer-Verlag, London, 2008)

  25. B.C. Goodwin, in Advances of Enzime Regulation, edited by G. Weber, Vol. 3 (Pergamon, Oxford, 1965), p. 425

  26. W.R. Smith, Bull. Math. Biol. 42, 57 (1980)

    ADS  MATH  Google Scholar 

  27. W.R. Smith, Amer. J. Physiol. 245, R473 (1983)

    Google Scholar 

  28. G. Enciso, E. Sontag, J. Math. Biol. 49, 627 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  29. W.S. Evans, L.S. Farhy, M.L. Johnson, in Methods in Enzymology: Computer methods, Volume A, edited by M.L. Johnson, L. Brand, Vol. 454 (2009), p. 345

  30. A.N. Churilov, A. Medvedev, A.I. Shepeljavyi, Automatica 45, 78 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  31. F.C.W. Wu, D.C. Irby, I.J. Clarke, J.T. Cummins, D.M. de Kretser, Biol. Reprod. 37, 501 (1987)

    Article  Google Scholar 

  32. N. Rasgon, L. Pumphrey, P. Prolo, S. Elman, A. Negrao, J. Licinio, A. Garfinkel, CNS Spectrums 8, 805 (2003)

    Google Scholar 

  33. J.J. Walker, J.R. Terry, K. Tsaneva-Atanasova, S.P. Armstrong, C.A. McArdle, S.L. Lightman, J. Neuroendocrinology 22, 1226 (2010)

    Article  Google Scholar 

  34. V. Lakshmikantham, D.D. Bainov, P.S. Simeonov, Theory of Impulsive Differential Equations (World Scientific, Singapore, 1989)

  35. A.M. Samoilenko, N.A. Perestyuk, Impulsive Differential Equations (World Scientific, Singapore, 1995)

  36. A.K. Gelig, A.N. Churilov, Stability and Oscillations of Nonlinear Pulse-modulated Systems (Birkhäuser, Boston, 1998)

  37. T. Yang, Impulsive Control Theory (Springer, Berlin, 2001)

  38. W.M. Haddad, V. Chellaboina, S.G. Nersesov, Impulsive and Hybrid Dynamical Systems: Stability, Dissipativity, and Control (Princeton Univ. Press, Princeton, 2006)

  39. Zh.T. Zhusubaliyev, A.N. Churilov, A. Medvedev, Chaos 22, 013121 (2012)

    Article  ADS  Google Scholar 

  40. A.N. Churilov, A. Medvedev, P. Mattsson, in Proceedings of the 51st IEEE Conference on Decision and Control (Maui, Hawaii, 2012), p. 362

  41. A. Churilov, A. Medvedev, P. Mattsson, in Proceedings of the 52nd IEEE Conference on Decision and Control (Florence, Italy, 2013), p. 2078

  42. A.N. Churilov, A. Medvedev, P. Mattsson, IEEE Trans. Automatic Control 59, 728 (2014)

    Article  MathSciNet  Google Scholar 

  43. A.N. Churilov, A. Medvedev, Automatica 50, 2187 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  44. A. Churilov, A. Medvedev, P. Mattsson, in Proceedings of the 53rd IEEE Conference on Decision and Control (Los Angeles, California, 2014), p. 765

  45. P. Mattsson, A. Medvedev, in Advances in Experimental Medicine and Biology, Vol. 823 (Springer, 2015), p. 23

  46. G. Lang, Pulsatility of Neuroendocrine Systems (CRC Press, Boca Raton, 1988)

  47. R.A. Steiner, W.J. Bremner, D.K. Clifton, Endocrinology 111, 2055 (1982)

    Article  Google Scholar 

  48. A.M. Matsumoto, W.J. Bremner, J. Clin. Endocrinol. Metab. 58, 609 (1984)

    Article  Google Scholar 

  49. S.L. Lightman, B.L. Conway-Campbel, Nat. Rev. Neurosci. 11, 710 (2010)

    Article  Google Scholar 

  50. M. Bergendahl, W.S. Evans, J.D. Veldhuis, Human Reproduc. Update 6, 507 (1996)

    Article  Google Scholar 

  51. P.E. Belchetz, T.M. Plant, Y. Nakai, E.J. Keogh, E. Knobil, Science 202, 631 (1978)

    Article  ADS  Google Scholar 

  52. W.R. Smith, Bull. Math. Biol. 42, 57 (1980)

    ADS  MATH  Google Scholar 

  53. T.M. Plant, Endocrinol. Rev. 7, 75 (1986)

    Article  Google Scholar 

  54. D.M. Keenan, J.D. Veldhuis, Am. J. Physiol. 38, E157 (1998)

    Google Scholar 

  55. G.M. de la Escalera, A.L.H. Choi, R.I. Weiner, Proc. Natl. Acad. Sci. 89, 1852 (1992)

    Article  ADS  Google Scholar 

  56. R. Vazquez-Martinez, S.L. Shorte, F.R. Boockfor, L.S. Frawley, Endocrinology 142, 2095 (2001)

    Google Scholar 

  57. K. Khadra, Y.X. Li, Biophys. J. 91, 74 (2006)

    Article  ADS  Google Scholar 

  58. A.N. Churilov, A. Medvedev, Zh.T. Zhusubaliyev, in Proceedings of the 19th World Congress The International Federation of Automatic Control, Cape Town, South Africa, 2014), p. 3340

  59. Zh.T. Zhusubaliyev, A.N. Churilov, A. Medvedev, in Proceedings of the 13th European Control Conference (Strasbourg, France, 2014), p. 3340

  60. H.E. Nusse, J.A. Yorke, Physica D 57, 39 (1992)

    Article  MathSciNet  ADS  Google Scholar 

  61. M. di Bernardo, M.I. Feigin, S.J. Hogan, M.E. Homer, Chaos, Solitons Fractals 10, 1881 (1999)

    Article  MATH  Google Scholar 

  62. O.V. Sosnovtseva, E. Mosekilde, Int. J. Bifurcat. Chaos 7, 1225 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  63. T. Kapitaniak, Yu.L. Maistrenko, Phys. Rev. E 58, 5161 (1998)

    Article  MathSciNet  ADS  Google Scholar 

  64. Y.A. Kuznetsov, Elements of Applied Bifurcation Theory (Springer–Verlag, New York, 2004)

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Zhusubaliyev, Z.T., Mosekilde, E., Churilov, A.N. et al. Multistability and hidden attractors in an impulsive Goodwin oscillator with time delay. Eur. Phys. J. Spec. Top. 224, 1519–1539 (2015). https://doi.org/10.1140/epjst/e2015-02477-8

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  • DOI: https://doi.org/10.1140/epjst/e2015-02477-8

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