高维非线性系统双Hopf分叉理论研究及应用

The piezoelectric composite laminates selected in this project can be described to be high-dimensional nonlinear systems in the engineering system. When the system is subjected to the combined effect of in-plane and transverse incentives, it will appear complex nonlinear dynamical behavior. It is very important to research the double Hopf bifurcation of high-dimensional nonlinear dynamical systems in the practical problems. Research objectives of this project are: Promotion to study the singularity theory which is used to study the local bifurcation of nonlinear systems, and making it can be applied to high-dimensional nonlinear dynamical systems in different resonance situations. Used the promoted singularity theory we will study the double Hopf bifurcation of piezoelectric composite laminates under the combined effects of in-plane and transverse incentives from both theoretical and numerical modes. Reveal the different generating mechanism of Hopf bifurcation, double Hopf bifurcation and periodic motion of high-dimensional nonlinear dynamical systems, and get a different classification and open fold existing in periodic motion of the systems, to provide valuable theoretical basis of the design and control of piezoelectric composite laminate.

本项目选取的压电复合材料层合板在工程系统中，可以用高维非线性系统来描述，当系统受到面内和横向激励联合作用下，将会出现复杂的非线性动力学行为。研究工程实际问题中高维非线性动力系统的双Hopf分叉是非常重要的课题。本项目的研究目标为：推广用来研究非线性系统局部分叉的奇异性理论，使其可以应用到不同共振情形下高维非线性动力系统中。利用推广后的奇异性理论，从理论和数值两方面研究受面内和横向激励联合作用下压电复合材料层合板的双Hopf分叉问题。揭示高维非线性动力系统的Hopf分叉，双Hopf分叉以及不同周期运动的产生机理，并得到非线性动力系统存在不同周期运动的分类和开折，为压电复合材料层合板的优化设计和控制提供有价值的理论依据。

本项目选取的压电复合材料层合板在工程系统中，可以用高维非线性系统来描述，当系统受到面内和横向激励联合作用下，将会出现复杂的非线性动力学行为。研究实际工程问题中高维非线性动力系统的双Hopf分叉是非常重要的课题。本项目主要做了如下研究工作：推广用来研究非线性系统局部分叉的奇异性理论，使其可以应用到不同共振情形下高维非线性动力系统中；利用推广后的奇异性理论，从理论和数值两方面研究受面内和横向激励联合作用下压电复合材料层合板的双Hopf分叉问题。揭示了高维非线性动力系统的Hopf分叉，双Hopf分叉以及不同周期运动的产生机理，并得到了非线性动力系统存在不同周期运动的分类和开折，为压电复合材料层合板的优化设计和控制提供了有价值的理论依据.