Original Article

Nonlinear Dynamics

, Volume 48, Issue 4, pp 381-389

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

  • Xiaopeng ZhaoAffiliated withDepartment of Biomedical Engineering Center for Nonlinear and Complex Systems, Duke University Email author 
  • , David G. SchaefferAffiliated withDepartment of Mathematics Center for Nonlinear and Complex Systems, Duke University
  • , Carolyn M. BergerAffiliated withDepartment of Physics and Center for Nonlinear and Complex Systems, Duke University
  • , Daniel J. GauthierAffiliated withDepartment of Biomedical Engineering Center for Nonlinear and Complex Systems, Duke UniversityDepartment of Physics and Center for Nonlinear and Complex Systems, Duke University

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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