Simultaneous border-collision and period-doubling bifurcations

We unfold the codimension-two simultaneous occurrence of a border-collision bifurcation and a period-doubling bifurcation for a general piecewise-smooth, continuous map. We find that with sufficient nondegeneracy conditions, a locus of period-doubling bifurcations emanates nontangentially from a loc...

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Bibliographic Details
Published in:Chaos (Woodbury, N.Y.) N.Y.), 2009-09, Vol.19 (3), p.033146-033146-11
Main Authors: Simpson, D. J. W., Meiss, J. D.
Format: Article
Language:English
Subjects:
Algorithms
Computer Simulation
Models, Statistical
Nonlinear Dynamics
Oscillometry - methods
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Summary:We unfold the codimension-two simultaneous occurrence of a border-collision bifurcation and a period-doubling bifurcation for a general piecewise-smooth, continuous map. We find that with sufficient nondegeneracy conditions, a locus of period-doubling bifurcations emanates nontangentially from a locus of border-collision bifurcations. The corresponding period-doubled solution undergoes a border-collision bifurcation along a curve emanating from the codimension-two point and tangent to the period-doubling locus here. In the case that the map is one-dimensional local dynamics is completely classified; in particular, we give conditions that ensure chaos.
ISSN:1054-1500
1089-7682
DOI:10.1063/1.3227645