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Alternate pacing of border-collision period-doubling bifurcations

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Abstract

Unlike classical bifurcations, border-collision bifurcations occur when, for example, a fixed point of a continuous, piecewise C 1 map crosses a boundary in state space. Although classical bifurcations have been much studied, border-collision bifurcations are not well understood. This paper considers a particular class of border-collision bifurcations, i.e., border-collision period-doubling bifurcations. We apply a subharmonic perturbation to the bifurcation parameter, which is also known as alternate pacing, and we investigate the response under such pacing near the original bifurcation point. The resulting behavior is characterized quantitatively by a gain, which is the ratio of the response amplitude to the applied perturbation amplitude. The gain in a border-collision period-doubling bifurcation has a qualitatively different dependence on parameters from that of a classical period-doubling bifurcation. Perhaps surprisingly, the differences are more readily apparent if the gain is plotted versus the perturbation amplitude (with the bifurcation parameter fixed) than if plotted versus the bifurcation parameter (with the perturbation amplitude fixed). When this observation is exploited, the gain under alternate pacing provides a useful experimental tool to identify a border-collision period-doubling bifurcation.

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Authors and Affiliations

  1. Department of Biomedical Engineering, Duke University, Durham, NC, 27708, USA

    Xiaopeng Zhao

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

    David G. Schaeffer

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  1. Xiaopeng Zhao
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  2. David G. Schaeffer
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Correspondence to Xiaopeng Zhao.

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Zhao, X., Schaeffer, D.G. Alternate pacing of border-collision period-doubling bifurcations. Nonlinear Dyn 50, 733–742 (2007). https://doi.org/10.1007/s11071-006-9174-1

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  • DOI: https://doi.org/10.1007/s11071-006-9174-1

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