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

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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.

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Authors and Affiliations

  1. Department of Biomedical Engineering Center for Nonlinear and Complex Systems, Duke University, Durham, NC, 27708, USA

    Xiaopeng Zhao & Daniel J. Gauthier

  2. Department of Mathematics Center for Nonlinear and Complex Systems, Duke University, Durham, NC, 27708, USA

    David G. Schaeffer

  3. Department of Physics and Center for Nonlinear and Complex Systems, Duke University, Durham, NC, 27708, USA

    Carolyn M. Berger & Daniel J. Gauthier

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  1. Xiaopeng Zhao
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  2. David G. Schaeffer
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  3. Carolyn M. Berger
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  4. Daniel J. Gauthier
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Correspondence to Xiaopeng Zhao.

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Zhao, X., Schaeffer, D.G., Berger, C.M. et al. Small-signal amplification of period-doubling bifurcations in smooth iterated maps. Nonlinear Dyn 48, 381–389 (2007). https://doi.org/10.1007/s11071-006-9092-2

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  • DOI: https://doi.org/10.1007/s11071-006-9092-2

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