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Bifurcations in a birhythmic biological system with time-delayed noise

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Abstract

We study the effects of recycled noise on the dynamics of a birhythmic biological system. This noise is generated by the superposition of a primary Gaussian white noise source with a second component (its replicas delayed of time τ). We find that under the influence of this kind of noise, the dynamics of the birhythmic biological system can be well characterized through the concept of stochastic bifurcation, consisting in a qualitative change of the stationary probability distribution. Analytical results are obtained following the quasiharmonic assumption through the Langevin and Fokker–Planck equations. Comparing the analytical and numerical results, we find good agreement when the frequencies of both attractors are equal, while the predictions of the analytic estimates deteriorate when the two frequencies depart. We also find that the increase of noise intensity leads to coherence resonance.

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References

  1. Springer, M., Paulsson, J.: Biological physics: harmonies from noise. Nature (London) 439, 27 (2006)

    Article  Google Scholar 

  2. Gardiner, C.W.: Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences. Springer, Berlin (2004)

    MATH  Google Scholar 

  3. Risken, H.: The Fokker-Planck Equation. Springer, Berlin (1996)

    MATH  Google Scholar 

  4. Andronov, A.A., Vitt, A.A., Khaikin, S.E.: The Theory of Oscillations. Nauka, Moscow (1981)

    MATH  Google Scholar 

  5. Yamapi, R., Filatrella, G., Aziz-Aloui, M.A.: Global stability analysis of birhythmicity in a self-sustained oscillator. Chaos 20, 013114 (2010)

    Article  MathSciNet  Google Scholar 

  6. Yamapi, R., Filatrella, G., Aziz-Alaoui, M.A., Hilda, C.: Effective Fokker-Planck equation for birhythmic modified van der Pol oscillators. Chaos 22, 043114 (2012)

    Article  Google Scholar 

  7. Yamapi, R., Enjieu Kadji, H.G., Filatrella, G.: Stability of the synchronization manifold in nearest neighbors non identical van der pol-like oscillators. Nonlinear Dyn. 61, 275–294 (2010)

    Article  MATH  Google Scholar 

  8. Zakharova, A., Vadivasova, T., Anishchenko, V., Koseska, A., Kurths, J.: Stochastic bifurcations and coherencelike resonance in a self-sustained bistable noisy oscillator. Phys. Rev. E 81, 011106 (2010)

    Article  Google Scholar 

  9. Goulding, D., Melnik, S., Curtin, D., Piwonski, T., Houlihan, J., Gleeson, J.P., Huyet, G.: Kramer’s law for a bistable system with time-delayed noise. Phys. Rev. E 76, 031128 (2007)

    Article  Google Scholar 

  10. Chéagé Chamgoué, A., Yamapi, R., Woafo, P.: Dynamics of a biological system with time-delayed noise. Eur. Phys. J. Plus 127(5), 59 (2012)

    Article  Google Scholar 

  11. Mukandavire, Z., Das, P., Chiyaka, C., Gazi, N.H., Das, K., Shiri, T.: HIV/AIDS model with delay and the effects of stochasticity. J. Math. Model. Algorithms 10(2), 181 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  12. Gazi, N.H., Das, K., Mukandavire, Z., Chiyaka, C., Das, P.: A study of cholera model with environmental fluctuations. Int. J. Math. Models Methods Appl. Sci. 4(3), 150 (2010)

    Google Scholar 

  13. Kaiser, F.: Coherent oscillations in biological systems, I, Bifurcation phenomena and phase transitions in an enzyme substrate reaction with ferroelectric behavior. Z. Naturforsch., A 33, 294 (1978)

    Google Scholar 

  14. Fröhlich, H.: Coherence and the action of enzymes. In: Welch, G.R. (ed.) The Fluctuating Enzyme, p. 421. Wiley, New York (1986)

    Google Scholar 

  15. Enjieu Kadji, H.G., Chabi Orou, J.B., Yamapi, R., Woafo, P.: Nonlinear dynamics and strange attractor in the biological system. Chaos Solitons Fractals 32, 862 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  16. Kaiser, F.: Coherent modes in biological systems. In: Illinger, K.H. (ed.) Biological Effects of Nonionizing Radiation. A.C.S Symp. Series, p. 157 (1981)

    Google Scholar 

  17. Kaiser, F., Eichwald, C.: Bifurcation structure of a driven multi-limit-cycle van der Pol oscillator (I). The superharmonic resonance structure. Int. J. Bifurc. Chaos 1, 485 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  18. Arnold, L.: Random Dynamical System. Springer, Berlin (2003)

    Google Scholar 

  19. Horsthemke, W., Lefever, R.: Noise Induced Transitions. Theory and Applications in Physics, Chemistry, and Biology. Springer Series in Synergetics, vol. 15. Springer, Berlin (1983)

    Google Scholar 

  20. Schimansky-Geier, L., Herzel, H.: Positive Lyapunov-exponents in the Kramers- oscillator. J. Stat. Phys. 70, 141 (1993)

    Article  MATH  Google Scholar 

  21. Eichwald, C., Kaiser, F.: Bifurcation structure of a driven multi-limit-cycle van der Pol oscillator (II). Int. J. Bifurc. Chaos 1, 711 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  22. Enjieu Kadji, H.G., Yamapi, R., Chabi Orou, J.B.: Synchronization of two coupled self-excited systems with multi-limit cycles. Chaos 17, 033113 (2007)

    Article  MathSciNet  Google Scholar 

  23. Yamapi, R., Nana Nbendjo, B.R., Enjieu Kadji, H.G.: Dynamics and active control of motion of a driven multi-limit-cycle van der Pol oscillator. Int. J. Bifurc. Chaos 17(4), 1343 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  24. Strogatz, S.: Nonlinear Dynamics and Chaos. Addison-Wesley, Reading (1994)

    Google Scholar 

  25. Enjieu Kadji, H.G.: Synchronization dynamics of nonlinear self-sustained oscillations with applications in physics, engineering and biology. Ph.D. Dissertation of Physics, Institut de Mathématiques et de Sciences Physiques (I.M.S.P.), Porto-Novo, Universite d’Abomey-Calavi, Benin (June 2006)

  26. Kaiser, F.: Coherent oscillations in biological systems: interaction with extremely low frequency fields. Radio Sci. 17, 17S (1981)

    Article  Google Scholar 

  27. Li, V.-X., Goldbeter, A.: Oscillatory isozymes as the simplest model for coupled biochemical oscillators. J. Theor. Biol. 138, 149 (1989)

    Article  MathSciNet  Google Scholar 

  28. Knuth, D.E.: The Art of Computer Programming vol. 2. Addison-Wesley, Reading (1969)

    MATH  Google Scholar 

  29. Fox, R.F., Gatland, I.R., Roy, R., Vemuri, G.: Fast, accurate algorithm for numerical simulation of exponentially correlated colored noise. Phys. Rev. A 38, 11 (1988)

    Article  Google Scholar 

  30. Stratonovich, R.L.: Selected Topics in the Theory of Random Noise, vols. 1 and 2. Gordon and Breach, New York (1963)

    Google Scholar 

  31. Freidlin, M.I., Wencel, A.D.: Random Perturbations in Dynamical Systems. Springer, New York (1984)

    Book  Google Scholar 

  32. Pikovsky, A.S., Kurths, J.: Coherence resonance in a noise-driven excitable system. Phys. Rev. Lett. 78, 775 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  33. Borromeo, M., Giusepponi, S., Marchesoni, F.: Stochastic synchronization via noise recycling. Phys. Rev. E 74, 031121 (2006)

    Article  Google Scholar 

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Authors and Affiliations

  1. Laboratory of Modeling and Simulation in Engineering, Biomimetics and Prototypes, Faculty of Science, University of Yaoundé I, Box 812, Yaoundé, Cameroon

    A. Chéagé Chamgoué, R. Yamapi & P. Woafo

  2. Department of Physics, Faculty of Science, University of Douala, Box 24 157, Douala, Cameroon

    R. Yamapi

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  1. A. Chéagé Chamgoué
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  2. R. Yamapi
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  3. P. Woafo
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Correspondence to R. Yamapi.

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Chéagé Chamgoué, A., Yamapi, R. & Woafo, P. Bifurcations in a birhythmic biological system with time-delayed noise. Nonlinear Dyn 73, 2157–2173 (2013). https://doi.org/10.1007/s11071-013-0931-7

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  • DOI: https://doi.org/10.1007/s11071-013-0931-7

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