右端不连续时滞微分方程的多稳定性及其应用研究

In recent years, delay differential equations with discontinuous right-hand sides are research hotspots in the field of differential equations and dynamical systems at home and broad. Among them, the study of mutistability and related dynamics is one of the most challenging issues. This program will construct the differential inclusion systems corresponding to the delay differential equations with discontinuous right-hand sides, in the sense of Filippov solutions, by using modern mathematical theories and methods such as differential inclusions, set-valued mappings and nonsmooth analysis, combining with the geometric character of discontinuity for the right-hand side function on state variables and the state space decomposition techniques, we will study the coexistence、(in-)stability、estimation of attraction basins of multiple attractors(equilibrium points、periodic solutions and almost periodic solutions) of the discontinuous delay differential equations, we will also explore the relationship between the stable manifolds of unstable equilibrium points and the boundaries of attraction basins of stable equilibrium points, and clarify the influence mechanism of discontinuities and delay effects on multistable system dynamics, then apply the theoretical results to the study of dynamics of neural network and biomathematics models with discontinuous phenomena. This research not only enriches and develops the basic theory of discontinuous delay differential equations, but also provides theoretical basis and guarantee for the neural network design and biological population control, so it has certain theoretical significance and application value.

右端不连续时滞微分方程是近年来国内外微分方程与动力系统领域的研究热点,其中多稳定性及相关动力学的研究是具有挑战性的前沿课题之一。本项目拟构建右端不连续时滞微分方程所对应的微分包含系统，在Filippov解意义下，运用微分包含、集值映射、非光滑分析等现代数学理论和方法，结合右端函数关于状态变量不连续的几何特征和状态空间分解技术，研究不连续时滞微分方程多吸引子(平衡点、周期解和概周期解)的共存性、(不)稳定性、吸引域估计等，探索不稳定平衡点的稳定流形与稳定平衡点吸引域边界之间的关系，阐明不连续因素和时滞效应对多稳系统动力学的影响机理，并把理论成果应用于神经网络和生物数学领域中含有不连续现象的动力学研究。本项目的研究将丰富和发展不连续时滞微分方程的基本理论，同时为神经网络设计和生物种群控制提供理论依据和保证，具有一定的理论意义和应用价值。

右端不连续时滞微分方程大量出现在控制工程、生物学、物理学和经济学等领域的动力学建模中。由于不连续多稳系统右端函数向量场连续性的缺失和自身结构的复杂性导致现有的动力系统理论及方法都不能直接应用，从而给理论分析带来很大困难。本项目主要研究内容包括：(1) 不连续时滞微分方程解(周期解、概周期解)的存在性和稳定性研究；(2) 不连续时滞微分方程的同步性研究； (3) 不连续时滞扩散微分方程的多同步性研究； (4) 若干来源于工程领域和具有现实生物学背景的时滞微分方程的(多)解的定性与稳定性问题研究。通过本项目的研究，建立和发展了若干具有创新性的研究右端不连续时滞微分方程多稳定性的理论和方法，揭示不连续因素和时滞效应对不连续微分方程多稳定性及相关动力学的具体影响，丰富和发展不连续微分方程的稳定性理论和研究方法，为其应用研究提供理论基础。