三维自治微分系统的极限环、等时中心与临界周期分支

常微分方程定性理论已在无线电、航空工程、自动控制等众多领域中得到应用。在定性理论中微分系统极限环、等时中心与临界周期是相互关联的重要问题，对它们的研究有重要的理论和应用价值。.本项目研究三维自治微分系统在不变曲面上的极限环、等时中心与临界周期分支问题。由于技术上的困难，对于三维分片光滑系统上述问题的研究结果甚少。极限环方面，探索研究三维光滑微分系统和三维分片光滑系统在不变曲面上极限环的方法和技术。等时中心问题，建立三维光滑系统在中心流形上奇点周期常数计算的递推算法，探索三维分片光滑系统中心流形的存在性和其上周期常数的计算方法。扩展三维光滑系统临界周期分支的研究，探索三维分片光滑系统临界周期的存在性和数目。作为应用，本项目将解决一些电路系统和控制系统的Hopf分支和多重Hopf分支问题，研究机翼气动弹性系统的动力学行为,应用等时中心和临界周期分支方法研究和获得非线性波方程的动力学性质。

The qualitative theory of ordinary differential equation has been applied in many fields such as radio, aeronautical engineering and automatic control. It has important theoretical and applied value to study the limit cycle, the isochronous center and the bifurcation of critical periods which are classic and correlative problems in the qualitative theory of the differential system.. In this project, we study the problems on the limit cycles, isochronous center and critical periods of three-dimensional autonomous differential systems. For three dimensional piecewise smooth differential systems, the results on these problems are very few up to now. On limit cycle problem, we explore new methods and techniques to study the existence and number of limit cycles for both three-dimensional smooth differential systems and three-dimensional piecewise smooth differential systems in invariant surface. On isochronous center problems of three-dimensional differential systems, we set up recursive algorithms of the period constant on the center manifold for three-dimensional smooth differential systems，and also explore the existence of the center manifold and algorithm of the period constant for three dimensional piecewise smooth systems. On critical periods of three-dimensional differential systems, we extend its investigation in three-dimensional smooth systems, and explore the the existence and the number of critical periods for the three-dimensional piecewise smooth system. As applications, we solve the Hopf bifurcation and multiple Hopf bifurcation for some circuit systems and control systems and obtian the dynamic behavior of wing aeroelasticity systems. We also apply the methods of isochronous center and the critical period to study the dynamic properties of nonlinear wave equations.