不连续动力系统理论及应用

This project focuses on the theoretical exploration and application research of the brand-new field of discontinuous dynamical systems. The selected topic comes from practical problems in various modern fields of science and technology, such as automatic control, engineering technology, comlpex networks, biology science and information technology. Considering the discontinuous dynamical systems appearing in such different fields, the project will explore related essential dynamic characteristics, develop new theories and methods, and research on the application models, which has both scientific and practical significance. This project commits to investigate the basic theories and application issues of discontinuous dynamical systems. By utilizing the new idea derived from the energy layer measurement in physics and the new method of flow theory in time-varying domains, we try to set up new dynamic theories of several typical discontinuous dynamical systems. In addition, some innovative achievements about Van der Pol oscillator model with impulsive effect, metal cutting model, impact oscillator model and oscillator synchronization model would also be obtained based on the new idea and method, with simulation experiments. Especially, we will furtherly develop the dynamic theory of discontinuous systems and try to establish related flow theory for non-connectable domains and when obstacles appear in the vector field on the discontinuous boundary. We try to improve and perfect the theories and application research of discontinuous dynamical systems and promote the investigation in such field.

本项目对于不连续动力系统这一崭新研究领域进行理论探索和应用研究。本项目选题由实际问题驱动，针对自动控制、工程技术、复杂网络、生命科学、信息技术等诸多领域中出现的不连续动力系统，探索其动力学本质特征，发展新的理论和方法，开展其应用模型研究，具有重要的科学意义和实用价值。本项目对不连续动力系统若干亟待解决的具根本性的理论与应用课题开展研究，运用源于物理的能量层度量的新思想和动态域流转换的新方法，力争对几类典型的不连续动力系统建立动力学新理论；还将在脉冲Van der Pol振子模型、金属切割模型、振荡撞击模型、振子同步模型等的应用研究方面取得基于新思想、新方法的创新成果并进行数值仿真试验。特别是进一步开拓、发展不连续动力系统的动力学理论，力争在不连通域及不连续边界上向量场出现障碍的情形下建立相应的流转换理论。通过本项目研究力争填补不连续动力系统理论与应用研究的某些空白，促进对该领域研究的发展。

本项目对不连续动力系统若干亟待解决的具根本性的理论与应用课题开展研究。本项目选题由实际问题驱动，针对自动控制、工程技术、复杂网络、生命科学、信息技术等诸多领域中出现的不连续动力系统，探索其动力学本质特征，发展新的理论和方法，开展其应用模型研究，具有重要的科学意义和实用价值。首先,本项目致力于探索若干典型不连续动力系统的动力学特征与规律，运用研究动态域不连续动力系统的新思想新方法，给出了具有干摩擦的水平碰撞振动系统、单自由度斜碰撞振动系统、以及双自由度碰撞振动系统的动力学创新结果；得到了以悬架系统为代表的具有非对称阻尼性质的不连续动力系统的复杂动力学创新结果。其次，本项目开展关于不连续动力系统实际问题模型的应用研究及仿真实验，得到了关于由VdP振荡器激励的Fermi加速器模型的复杂加速机理的创新结果，还得到了人口动力学模型等的应用创新结果。另外，本项目进一步开拓、发展不连续动力系统的动力学理论，給出了在状态边界向量场转换的切换系统复杂动力学的重要创新成果。