基于不连续阈值控制的蚊媒传染病模型的动力学研究

近年来，不连续系统在工程、生物及物理等领域得到广泛的应用。针对状态脉冲微分动力系统，本项目主要从两个方面进行研究。1）基于脉冲微分系统的理论，拟改进和推广拟庞加莱映射法寻找状态依赖的脉冲微分方程的阶K周期解的证明方法。尤其阶-2周期解的证明方法；寻找微分方程的轨迹稳定的判定方法和有限时间稳定的方法，寻找状态脉冲微分方程的同宿分岔条件。2）将上述的理论方法应用到具有状态脉冲控制的蚊媒生物系统中，研究系统的复杂动力学，并给出其生物意义。预期研究结果将可促进不连续微分方程理论的进一步发展，也可为解决生物工程领域中的实际问题提供理论依据和有效方法。

In recent years, discontinuous systems have been widely used in engineering, biology and physics. Aiming at the state impulsive differential dynamic system, this project mainly studies from two aspects. 1) Based on the theory of impulsive differential system, we propose to improve and extend the quasi-Poincare mapping method to find the order-K periodic solution of the state-dependent impulsive differential equation. Especially the proof method of th order-2 periodic solution; The method of determining trajectory stability and finite-time stability of differential equation is discussed. The condition of homoclinic bifurcation of state impulsive differential equation is investigated. 2) Apply the above theoretical methods to a mosquito-population biological system with state impulsive control, study the complex dynamics of the system, and give its biological significance. It is expected that the research results will promote the further development of discontinuous differential equation theory and provide theoretical basis and effective method for solving practical problems in the field of bioengineering.