Gluing and grazing bifurcations in periodically forced 2dimensional integrateandfire models
Abstract
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 squarewave pulse, the system possesses a periodic orbit which may undergo smooth and nonsmooth grazing bifurcations. We perform a semirigorous study of the existence of periodic orbits for a particular model consisting of a leaky integrateandfire model with a dynamic threshold. We use the stroboscopic map, which in this context is a $2$dimensional piecewisesmooth discontinuous map. For some parameter values we are able to show that the map is a quasicontraction 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.
 Publication:

arXiv eprints
 Pub Date:
 October 2016
 arXiv:
 arXiv:1610.02930
 Bibcode:
 2016arXiv161002930G
 Keywords:

 Mathematics  Dynamical Systems