Shrinking Point Bifurcations of Resonance Tongues for Piecewise-Smooth, Continuous Maps

Abstract

Resonance tongues are mode-locking regions of parameter space in which stable periodic solutions occur; they commonly occur, for example, near Neimark-Sacker bifurcations. For piecewise-smooth, continuous maps these tongues typically have a distinctive lens-chain (or sausage) shape in two-parameter bifurcation diagrams. We give a symbolic description of a class of "rotational" periodic solutions that display lens-chain structures for a general $N$-dimensional map. We then unfold the codimension-two, shrinking point bifurcation, where the tongues have zero width. A number of codimension-one bifurcation curves emanate from shrinking points and we determine those that form tongue boundaries.

27 pages, 6 figures

References in corpus (1)

• Border Collision Bifurcations in Two Dimensional Piecewise Smooth Maps

Cited by in corpus (11)

• Aspects of Bifurcation Theory for Piecewise-Smooth, Continuous Systems
• Discontinuity induced bifurcations of non-hyperbolic cycles in nonsmooth systems
• Sequences of Periodic Solutions and Infinitely Many Coexisting Attractors in the Border-Collision Normal Form
• Resonance near Border-Collision Bifurcations in Piecewise-Smooth, Continuous Maps
• Scaling laws for large numbers of coexisting attracting periodic solutions in the border-collision normal form
• Subsumed homoclinic connections and infinitely many coexisting attractors in piecewise-linear maps
• Unfolding homoclinic connections formed by corner intersections in piecewise-smooth maps
• Unfolding codimension-two subsumed homoclinic connections in two-dimensional piecewise-linear maps
• Border-collision bifurcations in a driven time-delay system
• A synopsis of the non-invertible, two-dimensional, border-collision normal form with applications to power converters
• Critical curves of a piecewise linear map