Nonlinear Dynamics

, Volume 48, Issue 4, pp 381–389

Small-signal amplification of period-doubling bifurcations in smooth iterated maps

  • Xiaopeng Zhao
  • David G. Schaeffer
  • Carolyn M. Berger
  • Daniel J. Gauthier
Original Article

Abstract

Various authors have shown that, near the onset of a period-doubling bifurcation, small perturbations in the control parameter may result in much larger disturbances in the response of the dynamical system. Such amplification of small signals can be measured by a gain defined as the magnitude of the disturbance in the response divided by the perturbation amplitude. In this paper, the perturbed response is studied using normal forms based on the most general assumptions of iterated maps. Such an analysis provides a theoretical footing for previous experimental and numerical observations, such as the failure of linear analysis and the saturation of the gain. Qualitative as well as quantitative features of the gain are exhibited using selected models of cardiac dynamics.

Keywords

Prebifurcation amplification Period-doubling bifurcation Cardiac dynamics 

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References

  1. 1.
    Heldstab, J., Thomas, H., Geisel, T., Randons, G.: Linear and nonlinear response of discrete dynamical systems I. Periodic attractors. Z. Phys. B 50, 141–150 (1983)CrossRefGoogle Scholar
  2. 2.
    Arneodo, A.: Scaling for a periodic forcing of a period-doubling system. Phys. Rev. Lett. 53, 1240; 54, 86 (1984)Google Scholar
  3. 3.
    Argoul, F., Arneodo, A., Richetti, O., Roux, J.C., Swinney, H.L.: Transitions to chaos in the presence of an external periodic field – crossover effect in the measure of critical exponents. Europhys. Lett. 3, 643 (1987)Google Scholar
  4. 4.
    Kuznetsov, S.P.: Effect of a periodic external perturbation on a system which exhibits an order-chaos transition through period-doubling bifurcation. JETP Lett. 39, 133 (1984)Google Scholar
  5. 5.
    Kuznetsov, S.P., Pikovsky, A.S.: Renormalization group for the response function and spectrum of the period-doubling system. Phys. Lett. A 94, 1 (1989)MathSciNetGoogle Scholar
  6. 6.
    Ivan’Kov, N.Yu., Kuznetsov, S.P.: Different types of scaling in the dynamics of period-doubling maps under external periodic driving. Discrete Dyn. Nat. Soc. 5, 223 (2000)MATHCrossRefGoogle Scholar
  7. 7.
    Wiesenfeld, K.: Virtual Hopf phenomenon: a new precursor of period-doubling bifurcations. Phys. Rev. A 32, 1744 (1985)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Wiesenfeld, K., McNamara, B.: Small-signal amplification in bifurcating dynamical systems. Phys. Rev. A 33, 629–642 (1986); erratum: ibid 33, 3578 (1986)Google Scholar
  9. 9.
    Bryant, P., Wiesenfeld, K.: Suppression of period-doubling and nonlinear parametric effects in periodically perturbed systems. Phys. Rev. A 33, 2525 (1986)CrossRefGoogle Scholar
  10. 10.
    Svensmark, H., Wiesenfeld, K.: Scaling law for the idler near a bifurcation. Phys. Rev. A 46, 787 (1992)CrossRefGoogle Scholar
  11. 11.
    Vohra, S.T., Bucholtz, F., Koo, K.P., Dagenais, D.M.: Experimental observation of period-doubling suppression in the strain dynamic of a magnetostrictive ribbons. Phys. Rev. Lett. 66, 212 (1991)CrossRefGoogle Scholar
  12. 12.
    Vohra, S.T., Wiesenfeld, K.: Experimental test of the normal form for period doubling bifurcations. Physica D 86, 27 (1995)MATHCrossRefGoogle Scholar
  13. 13.
    Kravtsov, Yu.A., Surovyatkina, E.D.: Nonlinear saturation of prebifurcation noise amplification. Phys. Lett. A 319, 348 (2003)MATHCrossRefGoogle Scholar
  14. 14.
    Surovyatkina, E.D.: Rise and saturation of the correlation time near bifurcation threshold. Phys. Lett. A 329, 169 (2004)CrossRefMathSciNetGoogle Scholar
  15. 15.
    Chialvo, D.R., Michaels, D.C., Jalife, J.: Supernormal excitability as a mechanism of chaotic dynamics of activation in cardiac Purkinje fibers. Circ. Res. 66, 525–545 (1990)Google Scholar
  16. 16.
    Hall, G.M., Gauthier, D.J.: Experimental control of cardiac muscle alternans. Phys. Rev. Lett. 88, 198102 (2002)CrossRefGoogle Scholar
  17. 17.
    Fox, J.J., Bodenschatz, E., Gilmour, Jr. R.F.: Period-doubling instability and memory in cardiac tissue. Phys. Rev. Lett. 89, 138101 (2002)CrossRefGoogle Scholar
  18. 18.
    Berger, C.M., Dobrovolny, H., Zhao, X., Schaeffer, D.G., Krassowska, W., Gauthier, D.J.: Investigating a period-doubling bifurcation in cardiac tissue using alternate pacing. In: Dynamics Days, Bethesda, MD, January 4–7, 2006Google Scholar
  19. 19.
    American Heart Association: Heart disease and stroke statistics – 2004 update. American Heart Association, Dallas, TX, 2004Google Scholar

Copyright information

© Springer Science + Business Media B.V. 2006

Authors and Affiliations

  • Xiaopeng Zhao
    • 1
  • David G. Schaeffer
    • 2
  • Carolyn M. Berger
    • 3
  • Daniel J. Gauthier
    • 1
    • 3
  1. 1.Department of Biomedical Engineering Center for Nonlinear and Complex SystemsDuke UniversityDurhamUSA
  2. 2.Department of Mathematics Center for Nonlinear and Complex SystemsDuke UniversityDurhamUSA
  3. 3.Department of Physics and Center for Nonlinear and Complex SystemsDuke UniversityDurhamUSA

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