几类高维非线性动力系统多周期解分岔的研究及应用

The research and application on bifurcation theory of multi-periodic solutions for high dimensional nonlinear dynamical systems is the international frontier subject in the field of nonlinear dynamics. The purpose of this project is to study the theories and applications on bifurcation of multi-periodic solutions for high dimensional nonlinear dynamical systems closely related to many complex nonlinear phenomena. (1) The project will study the bifurcation of multi-periodic solutions and the upper bound for the number of multi-periodic solutions for high-dimensional polynomial system with degree N and nonautonomous multiple time scale system. Develop the subharmonic Melnikov method to study the bifurcation of multi-periodic solutions for 2n dimensional nonautonomous quasi-Hamiltonian system. (2) The project will study the hypernormal form of high-dimensional nonlinear nilponent singularities, Hopf singularities and smooth maps. Develop and complete the new technology to deal with the complex computation in large size block matrix. (3) The project will also investigate the reduction and bifurcation of multi-periodic solutions of respiratory vibration and respiratory torsion coupling vibration model related to the circular truss antenna, reveal the generation mechanism of large amplitude vibration caused by the multi-periodic solutions under the excitation of the system parameters such as thermal load, structural damping, and present parameter control methods for the design of anti-vibrations. We will strive to make meaningful and breakthrough innovation work in the field of bifurcation theory of multi-periodic solutions and engineering applications for high dimensional nonlinear dynamical systems.

高维非线性动力系统多周期解分岔的研究及应用是国际非线性动力学领域的前沿课题。本项目旨在研究与诸多复杂非线性现象密切相关的高维非线性动力系统多周期分岔理论及应用中的若干问题。(1)研究高维N次多项式系统及非自治多时间尺度系统的多周期解分岔及个数上界；发展适用于研究2n维非自治拟Hamilton系统多周期解分岔的次谐Melnikov方法。(2)研究高维幂零与Hopf型非线性动力系统及光滑映射的超规范形；发展并完善处理大尺寸分块矩阵中相关大规模计算的新技术研究。(3)研究具有实际应用背景的环形桁架天线呼吸振动、扭转与呼吸耦合振动模型的简化与多周期解分岔问题，理论揭示在交变热载荷、结构阻尼等系统参数激励下由于多周期解的存在导致系统大幅振动现象的产生机理，并提出减振设计的参数控制方法。本项目力争在高维非线性动力系统多周期解分岔理论及工程应用领域做出有意义乃至突破性的创新工作。

高维非线性动力系统多周期解分岔的研究及应用是非线性科学的重要分支，也是国际动力学领域的前沿课题和科研难题。本项目深入研究了与诸多复杂非线性现象密切相关的高维非线性系统的多周期解分岔理论及应用中的若干问题，并取得了重要成果。.(1) 发展平均理论与Melnikov方法研究了高维N次多项式系统及非自治多时间尺度系统多周期解的存在性、个数与几何构型，并成功将次谐Melnikov方法推广至2n维自治与非自治拟Hamilton系统的多周期解分岔理论。.(2) 进一步发展新次数函数、多重李括号与分块矩阵新记号表示相结合的方法，研究了几类高维幂零系统、Hopf系统与光滑映射的超规范形理论，获得超规范形计算中简化超大尺寸分块矩阵相关大规模计算的新方法。.(3) 研究了环形桁架天线动力学模型的简化与多周期振动等复杂动力学行为，探讨了热激励等参数对系统动力学行为的影响规律，获得多周期轨道的存在性判据及相对位置，揭示了由于分岔导致系统失稳的产生机理，为相应工程系统的减振控制与动态优化设计提供有价值的理论指导。.本项目的研究工作按照预定计划圆满完成，取得了较丰硕的成果。共完成及发表论文32篇，其中SCI论文16篇、EI论文7篇、核心期刊论文5篇，出版专著2部。基于项目影响力，牵头成立北京交叉科学学会。