On the relative coexistence of fixed points and period-two solutions near border-collision bifurcations

David J. W. Simpson

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arXiv · 1405.7083

On the relative coexistence of fixed points and period-two solutions near border-collision bifurcations

David J. W. Simpson

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Abstract

At a border-collision bifurcation a fixed point of a piecewise-smooth map intersects a surface where the functional form of the map changes. Near a generic border-collision bifurcation there are two fixed points, each of which exists on one side of the bifurcation. A simple eigenvalue condition indicates whether the fixed points exist on different sides of the bifurcation (this case can be interpreted as the persistence of a single fixed point), or on the same side of the bifurcation (in which case the bifurcation is akin to a saddle-node bifurcation). A similar eigenvalue condition indicates whether or not there exists a period-two solution on one side of the bifurcation. Previously these conditions have been combined to obtain five distinct scenarios for the existence and relative coexistence of fixed points and period-two solutions near border-collision bifurcations. In this Letter, it is shown that one of these scenarios, namely that two fixed points exist on one side of the bifurcation and a period-two solution exists on the other side of the bifurcation, cannot occur. The remaining four scenarios are feasible. Therefore there are exactly four distinct scenarios for fixed points and period-two solutions near border-collision bifurcations.

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David J. W. Simpson. 2014-05-27. On the relative coexistence of fixed points and period-two solutions near border-collision bifurcations. https://arxiv.org/abs/1405.7083

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