右端不连续复值微分方程的动力学行为及其应用研究

Complex-valued differential equation with discontinuous right-handed sides has wide applications in many fields, such as rotating fluid, images processing and information security. In this project, we will construct the system of Filippov complex-valued differential inclusion for discontinuous complex-valued differential equation. Combined with non-smooth analysis theory, set-valued analysis, differential inclusion and non-smooth critical point theory, we will investigate the dynamic behavior for complex-valued differential equation, including the initial value problem and some basic properties, such as existence and uniqueness, different kinds of stability, convergence behavior (convergence in finite time) for equilibrium (periodic solution) . We will further study the stability theory of complex-valued differential inclusion, and apply these theories and methods to study different kinds of mathematical models in neural networks and complex networks, which described by complex-valued differential equation with discontinuous right-hand side. This research not only enriches and develops some basic theory of discontinuous differential equation, but also provides effective method and theoretical basis to solve many practical problems with discontinuous character.

右端不连续复值微分方程在流体力学、图像处理以及信息科学等领域有着广泛的应用. 本项目拟构建不连续复值微分方程所对应的微分包含系统,结合集值映射理论、微分包含理论、非光滑临界点理论等现代数学工具, 研究不连续复值微分方程的动力学行为, 主要包括初值问题和解的基本性质、平衡点(周期解)的存在唯一性、稳定性和收敛性(有限时间收敛性)等动力学行为. 完善和发展复值微分包含的稳定性理论, 研究神经网络和复杂网络同步领域中与右端不连续复值微分方程相关的模型. 本项目的开展, 将进一步丰富和发展不连续微分方程的基本理论, 同时为分析和解决众多受不连续因素影响的实际问题提供有效方法和理论依据.

右端不连续复值微分方程在流体力学、图像处理以及信息科学等领域有着广泛的应用. 我们构建了不连续复值微分方程所对应的微分包含系统,利用集值映射理论、微分包含理论、非光滑临界点理论研究不连续复值微分方程的动力学行为, 主要包括解的存在性和延拓性、平衡点(周期解)的存在唯一性、稳定性和收敛性(有限时间收敛性)等动力学行为. 完善和发展了复值微分包含的稳定性理论, 改进了复值微分包含的不动点定理, 研究不连续复值神经网络平衡点、周期解的存在性唯一性以及稳定性和稳定化问题.设计了切换控制器实现不连续复值复杂网络领域的同步.我们的研究成果进一步丰富和发展了不连续微分方程的基本理论, 同时为分析和解决众多受不连续因素影响的实际问题提供有效方法和理论依据.