Simultaneous Border-Collision and Period-Doubling Bifurcations

Abstract

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 non-degeneracy conditions, a locus of period-doubling bifurcations emanates non-tangentially 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 are completely classified; in particular, we give conditions that ensure chaos.

22 pages; 5 figures

References in corpus (4)

• Border Collision Bifurcations in Two Dimensional Piecewise Smooth Maps
• Multiple Attractor Bifurcations: A Source of Unpredictability in Piecewise Smooth Systems
• Discontinuity induced bifurcations of non-hyperbolic cycles in nonsmooth systems
• Discontinuity Induced Bifurcations in a Model of Saccharomyces cerevisiae

Cited by in corpus (4)

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• Chaos and stability in a two-parameter family of convex billiard tables
• Local bifurcation structure of a bouncing ball system with a piecewise polynomial function for table displacement