Gluing and grazing bifurcations in periodically forced 2-dimensional integrate-and-fire models
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In this work we consider a general class of 2-dimensional hybrid systems. Assuming that the system possesses an attracting equilibrium point, we show that, when periodically driven with a square-wave pulse, the system possesses a periodic orbit which may undergo smooth and nonsmooth grazing bifurcations. We perform a semi-rigorous study of the existence of periodic orbits for a particular model consisting of a leaky integrate-and-fire model with a dynamic threshold. We use the stroboscopic map, which in this context is a 2-dimensional piecewise-smooth discontinuous map. For some parameter values we are able to show that the map is a quasi-contraction possessing a (locally) unique maximin periodic orbit. We complement our analysis using advanced numerical techniques to provide a complete portrait of the dynamics as parameters are varied. We find that for some regions of the parameter space the model undergoes a cascade of gluing bifurcations, while for others the model shows multistability between orbits of different periods
CitationGranados, A., Huguet, G. Gluing and grazing bifurcations in periodically forced 2-dimensional integrate-and-fire models. "Communications in nonlinear science and numerical simulation", 1 Maig 2019, vol. 70, p. 48-73.