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Gluing and grazing bifurcations in periodically forced 2-dimensional integrate-and-fire models

机译:周期性强制二维积分并发射模型中的胶合和放牧分叉

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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-andfire 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. (C) 2018 Elsevier B.V. All rights reserved.
机译:在这项工作中,我们考虑一类普通的二维混合系统。假设系统拥有一个吸引性的平衡点,我们表明,当用方波脉冲周期性地驱动时,系统具有一个周期性的轨道,该轨道可能会经历光滑和非光滑的掠射分叉。我们对特定模型的周期性轨道的存在进行了半严格的研究,该模型包括具有动态阈值的漏泄积分和发射模型。我们使用频闪观测图,在这种情况下,它是二维的分段平滑不连续图。对于某些参数值,我们能够证明该图是具有(局部)唯一的极大值周期轨道的准收缩。我们使用高级数值技术对分析进行补充,以提供参数变化时动力学的完整描述。我们发现,对于参数空间的某些区域,模型经历了一系列的胶合分叉,而对于其他参数,模型则表现出不同周期轨道之间的多重稳定性。 (C)2018 Elsevier B.V.保留所有权利。

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