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 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.
Original language | English |
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Journal | Communications in Nonlinear Science and Numerical Simulation |
Volume | 70 |
Pages (from-to) | 48-73 |
ISSN | 1007-5704 |
DOIs | |
Publication status | Published - 2019 |
Keywords
- Integrate-and-fire
- Hybrid systems
- Piecewise smooth 2d maps
- Quasi-contractions