2014
DOI: 10.1016/j.physd.2013.12.011
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Discontinuity-induced bifurcation cascades in flows and maps with application to models of the yeast cell cycle

Abstract: This paper applies methods of numerical continuation analysis to document characteristic bifurcation cascades of limit cycles in piecewise-smooth, hybrid-dynamical-system models of the eukaryotic cell cycle, and associated period-adding cascades in piecewise-defined maps with gaps. A general theory is formulated for the occurrence of such cascades, for example given the existence of a period-two orbit with one point on the system discontinuity and with appropriate constraints on the forward trajectory for near… Show more

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Cited by 9 publications
(9 citation statements)
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“…These typically involve gaps arising due to a flow that grazes a surfaces of discontinuity, often an impact or frictional sticking surface, an electronic control surface, see e.g. [5,19,27], or a switching thresholds in a sleep-wake cycle [3] or in cell mitosis [18], as sketched in fig. 2.…”
Section: Cherry Flowmentioning
confidence: 99%
See 1 more Smart Citation
“…These typically involve gaps arising due to a flow that grazes a surfaces of discontinuity, often an impact or frictional sticking surface, an electronic control surface, see e.g. [5,19,27], or a switching thresholds in a sleep-wake cycle [3] or in cell mitosis [18], as sketched in fig. 2.…”
Section: Cherry Flowmentioning
confidence: 99%
“…therefore if a periodic orbit of type or q lr p exists, the condition (18) implies that periodic orbits of type or p lr b exist for all b = 1, . .…”
Section: (I)mentioning
confidence: 99%
“…Theory of piecewise-smooth maps has attracted the interest of many researchers lately to analyze the non-linear dynamics of various types of systems. These systems are from different fields like power electronics and drives [1][2][3][4][5] bio sciences [6][7][8][9] finance [10][11][12], analysis of relaxation oscillators in planar fast-slow systems [13] to name the few. Analysis of 1-D linear piecewisesmooth discontinuous (LPSD) map has been the focus point and given special attention by the researchers in recent years due to the inherent rich dynamics in it even though the underlying map equation is very simple [14].…”
Section: Introductionmentioning
confidence: 99%
“…This analysis is complicated by the fact that one of the constitutive laws has a discontinuity in its derivative. Such discontinuities can arise in both hybrid discrete-continuous systems and piecewise-smooth models, which capture a wide range of biological phenomena including yeast cell cycles (Jeffrey and Dankowicz 2014), yeast culture population dynamics (Simpson et al 2009), gene regulatory networks (Casey et al 2005), neuron firing (Coombes and Doole 2010), as well as a range of models from ecology (Dercole et al 2007). In our work here, we document both traditional (smooth) saddle node and Hopf bifurcations, as well as their discontinuity-induced (non-smooth) counterparts.…”
Section: Introductionmentioning
confidence: 99%