本文通过分析具有脉冲效应(状态重置过程)的四次累积发放神经元模型的同宿分岔，研究了该系统周期解的存在性和稳定性。主要针对系统具有两个平衡点的情形，对系统鞍点附近的动力学行为进行了定性分析，研究了多种情况下系统的1-阶同宿环的存在性。以 为分岔参数，当系统发生同宿分岔后，利用脉冲动力系统理论及庞加莱映射的不动点理论，证明了不同情形下1-阶同宿环附近1-阶周期解的存在性和稳定性。最后，数值模拟了系统的周期解，验证了理论结果的正确性。本文所采用的方法提供了一种寻找脉冲动力系统周期解的策略。
Abstract
This paper studies the existence and stability of the periodic solution by analyzing the homoclinic bifurcation of the quartic integrate-and-fire(IF) neuron model with impulse effect (state reset process).We mainly aim at the system has two equilibrium points, qualitatively analyze the dynamic behavior near the saddle point of the system, and investigate the existence of the order-1 homoclinic cycle of the system in different cases. With as the bifurcation parameter, when the system occurs homoclinic bifurcation, the existence and stability of the order-1 periodic solutions near the order-1 homoclinic cycle in different cases are proved by using the theory of impulsive dynamic system and the fixed point theory of the Poincaré map. Finally, the periodic solutions of the system are simulated numerically to verify the theoretical results.The approach employed in this paper provides a strategy for finding periodic solutions to impulse dynamical systems.
关键词
四次累积发放神经元模型 /
脉冲效应 /
1-阶同宿环 /
同宿分岔 /
1-阶周期解
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Key words
quartic integrate-and-fire neuron model /
impulse effort /
order-1 homoclinic cycle /
homoclinic bifurcation /
order-1 periodic solution
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