具脉冲影响的Van der Pol方程的复杂动力学行为研究

The Van der Pol Equation which derived from oscillation circuit has been widely used in the filed of natural science. However, the effect of instantaneous mutations on the system exist objectively in LC oscillation circuit. The Van der Pol equation with such impulsive effects becomes a special kind of discontinuous system due to the emergence of pulse, and it owns more practical significances to research on its complex dynamical behaviors in the field of mechanical engineering, finance, economics, space science, etc. To overcome the limitations of traditional methods of impulsive differential equation, this project plans to adopt the flow theory of discontinuous dynamical system to research on its complex dynamical behaviors, such as switchability, chatter, grazing motion, and stick motion, by the structure of time-varying impulsive surface boundary. Based on such analysis, equilibrium point and periodic motion will be studied, as well as the attractor and bifurcation problems, through the method of local singularity theory in discontinuous dynamical system. Finally, the above complex dynamical analysis will be applied on the Fermi acceleration problem, which could perfect the particle acceleration theory in space science. To research on the Van der Pol Equation with impulsive effects and get analytical conditions of complex dynamical behaviors, as well as bifurcation results, the project may supply a new method based on the theory of discontinuous dynamical system for impulsive differential equation’s qualitative research and expand the theoretical research of Van der Pol Equation. What’s more, it can supply a method of solving the instantaneous effect on practical problems and a theoretical direction.

源自振荡电路问题的Van der Pol方程已广泛应用于自然科学各领域，由于LC振荡电路中瞬时突变的影响是客观存在的，描述这一问题的具脉冲影响的Van der Pol方程因脉冲的出现而成为一种特殊的不连续系统，其相应复杂动力学行为的研究在机械工程、金融经济、空间科学等实践领域更具实际意义。为克服传统方法对脉冲问题研究的局限，本项目拟通过构造动态脉冲面边界，利用不连续动力系统流转换理论研究流的穿越、chatter、擦边、粘合等复杂行为；并基于此，采用不连续动力系统局部奇异性理论研究系统平衡点及周期吸引子性态和分支问题；最后，将所得复杂动力学结果应用于Fermi加速模型，完善空间粒子加速理论。通过对具脉冲影响的Van der Pol方程的研究，得到复杂动力学行为的解析条件和分支结果，可望发展和完善Van der Pol方程的理论研究，并为系统的应用研究提供一种解决瞬时突变影响的方法和理论指导。

源自振荡电路问题的Van der Pol方程因脉冲的影响而成为一种特殊的不连续系统，本项目利用不连续动力系统的最新理论，克服了传统方法对脉冲问题研究的局限。首先，本项目围绕脉冲Van der Pol方程的复杂动力学行为展开研究，借助流转换理论、局部奇异性理论、G函数、L函数、离散映射等工具，得到了擦边及粘合运动的解析条件，同时得到了系统周期吸引子性态、稳定性及分支结果，为今后借助不连续动力系统理论研究脉冲Van der Pol方程提供了必要的基础性工作；其次，将所得复杂动力学结果应用到了Fermi模型中，得到了极限环式运动对Fermi粒子加速的影响，为脉冲Van der Pol方程的进一步应用展示了良好的景象。