It has been shown recently that border collision bifurcation in a piecewise smooth map can lead to a situation where a fixed point remains stable at both sides of the bifurcation point, and yet the orbit becomes unbounded at the point of bifurcation because the basin of attraction of the stable fixed point shrinks to zero size. Such bifurcations have been named "dangerous bifurcations." In this...

We consider the local bifurcations that can happen to a quasiperiodic orbit in a 3-dimensional map: (a) a torus doubling resulting in two disjoint loops, (b) a torus doubling resulting in a single closed curve with two loops, (c) the appearance of a third frequency, and (d) the birth of a stable torus and an unstable torus. We analyze these bifurcations in terms of the stability of the point at...

We investigate bifurcations of burst oscillations with rectangular waveform observed in a modified Bonhöffer-van der Pol equation, which is considered as a circuit model for neurons of a feeding rhythm generator. In particular, we clarify a mechanism of properties in a one-parameter graph on the period of oscillations, showing a staircase with hysteresis jumps, by studying a successive bifurcat...

We explore the multifractal, self-similar organization of heteroclinic and homoclinic bifurcations of saddle singularities in the parameter space of the Shimizu–Morioka model that exhibits the Lorenz chaotic attractor. We show that complex transformations that underlie the transitions from the Lorenz attractor to wildly chaotic dynamics are intensified by Shilnikov saddle-foci. These transforma...

Abstract One-dimensional maps with discontinuities are known to exhibit bifurcations somewhat different those of continuous maps. Freed from the constraints continuity, and hence balance stability that is maintained through fold, flip, other standard bifurcations, attractors discontinuous can appear as if nowhere, change period or almost arbitrarily. But in fact this misleading, one includes st...