This survey article is concerned with the study of bifurcations of piecewise-smooth maps. We review the literature in circle maps and quasicontractions and provide paths through this literature to prove sufficient conditions for the occurrence of two types of bifurcation scenarios involving rich dynamics. The first scenario consists of the appearance of periodic orbits whose symbolic sequences and "rotation" numbers follow a Farey tree structure; the periods of the periodic orbits are given by consecutive addition. This is called the period adding bifurcation, and its proof relies on results for maps on the circle. In the second scenario, symbolic sequences are obtained by consecutive attachment of a given symbolic block and the periods of periodic orbits are incremented by a constant term. It is called the period incrementing bifurcation, in its proof relies on results for maps on the interval. We also discuss the expanding cases, as some of the partial results found in the literature also hold when these maps lose contractiveness. The higher dimensional case is also discussed by means of quasi-contractions. We also provide applied examples in control theory, power electronics and neuroscience where these results can be applied to obtain precise descriptions of their dynamics.

A technique is presented, based on the differential geometry of planar curves, to evaluate the excitability threshold of neuronal models. The aim is to determine regions of the phase plane where solutions to the model equations have zero local curvature, thereby defining a zero-curvature (inflection) set that discerns between sub-threshold and spiking electrical activity. This transition can arise through a Hopf bifurcation, via the so-called canard explosion that happens in an exponentially small parameter variation, and this is typical for a large class of planar neuronal models (FitzHugh-Nagumo, reduced Hodgkin-Huxley), namely, type II neurons (resonators). This transition can also correspond to the crossing of the stable manifold of a saddle equilibrium, in the case of type I neurons (integrators). We compute inflection sets and study how well they approximate the excitability threshold of these neuron models, that is, both in the canard and in the non-canard regime, using tools from invariant manifold theory and singularity theory. With the latter, we investigate the topological changes that inflection sets undergo upon parameter variation. Finally, we show that the concept of inflection set gives a good approximation of the threshold in both the so-called resonator and integrator neuronal cases.

In this work we consider a general non-autonomous hybrid system based on the integrate-and-fire model, widely used as simplified version of neuronal models and other types of excitable systems. Our unique assumption is that the system is monotonic, possesses an attracting subthreshold equilibrium point and is forced by means of periodic pulsatile (square wave) function. In contrast to classical methods, in our approach we use the stroboscopic map (time-T return map) instead of the so-called firing-map. It becomes a discontinuous map potentially defined in an infinite number of partitions. By applying theory for piecewise-smooth systems, we avoid relying on particular computations and we develop a novel approach that can be easily extended to systems with other topologies (expansive dynamics) and higher dimensions. More precisely, we rigorously study the bifurcation structure in the twodimensional parameter space formed by the amplitude and the duty cycle of the pulse. We show that it is covered by regions of existence of periodic orbits given by period adding structures. They do not only completely describe all the possible spiking asymptotic dynamics but also the behavior of the firing rate, which is a devil's staircase as a function of the parameters.

Circumnutation in Helianthus annuus L. was investigated by measurements lasting 4-7 weeks using a picture analysis system. The rhythmicity of circumnutation vigour (intensity) with regard to the trajectory length and period of individual circumnutations were examined. Three photoperiod conditions were applied [light/dark (LD), continuous light (LL) and LD followed by LL]. Data were processed by the Fourier analysis. Statistical analysis included the examination of circumnutation mean frequencies and correlation tests. Both parameters, trajectory length and period, revealed a daily (24 h) modulation in LD with a weak correlation between them, whereas in LL no daily modulation of the parameters was observed. After LD-LL transition, the parameters were gradually losing their daily modulation. Despite a very strong modulation of the trajectory length in LD, the period was quite stable in all groups tested, but only in LD were there no statistical differences in the number of circumnutations per 24 h among the plants studied. LD was concluded to be the strong synchronizer, making the plants circumnutate regularly. Regardless of the presence or absence of daily modulation, the infradian (several and more days long) harmonics of the trajectory length were the same in each group. These findings strongly support the view that circumnutation in sunflower, widely known as an ultradian rhythm, also possesses daily and infradian modulations of its intensity. To the authors' knowledge, this is the first report of circumnutation that was obtained by a picture analysis system in such a large timescale.

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