Math @ Duke

Publications [#244120] of David G. Schaeffer
Papers Published
 Zhao, X; Schaeffer, DG, Alternate Pacing of BorderCollision PeriodDoubling Bifurcations.,
Nonlinear Dynamics, vol. 50 no. 3
(November, 2007),
pp. 733742, ISSN 0924090X [19132134], [doi]
(last updated on 2019/07/18)
Abstract: Unlike classical bifurcations, bordercollision bifurcations occur when, for example, a fixed point of a continuous, piecewise C1 map crosses a boundary in state space. Although classical bifurcations have been much studied, bordercollision bifurcations are not well understood. This paper considers a particular class of bordercollision bifurcations, i.e., bordercollision perioddoubling 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 bordercollision perioddoubling bifurcation has a qualitatively different dependence on parameters from that of a classical perioddoubling bifurcation. Perhaps surprisingly, the differences are more readily apparent if the gain is plotted vs. the perturbation amplitude (with the bifurcation parameter fixed) than if plotted vs. 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 bordercollision perioddoubling bifurcation.


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