Small-signal amplification of period-doubling bifurcations in smooth iterated maps |
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Authors: | Xiaopeng Zhao David G Schaeffer Carolyn M Berger Daniel J Gauthier |
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Institution: | (1) Department of Biomedical Engineering Center for Nonlinear and Complex Systems, Duke University, Durham, NC 27708, USA;(2) Department of Mathematics Center for Nonlinear and Complex Systems, Duke University, Durham, NC 27708, USA;(3) Department of Physics and Center for Nonlinear and Complex Systems, Duke University, Durham, NC 27708, USA |
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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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Keywords: | Prebifurcation amplification Period-doubling bifurcation Cardiac dynamics |
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