高维非线性动力系统次谐Melnikov函数及多周期解分岔的研究及应用

Research on bifurcation theory of multi-periodic solutions for high dimensional nonlinear dynamical systems, closely related to many complex nonlinear phenomena in practical engineering, has become the international frontier subject in the field of nonlinear dynamics. This project focus on the bifurcation theory and geometry of multi-periodic solutions for high dimensional nonlinear dynamical systems and nonlinear dynamical analysis of iced cable model by developing and improving the high dimensional subhamonic Melnikov theory. (1) Based on the dynamic curvilinear coordinate frame and Poincare mapping, the project will improve the high dimensional harmonic Melnikov theory to study the bifurcation of multi-periodic solutions for more general systems. (2) The project will study the exact number, parameter control and geometry of the periodic solution for 4-dimensional Qusi-Hamiltonian system by the improved subhamonic Melnikov theory. (3) The project will also investigate the bifurcation of multi-periodic solutions for iced cable model with practical engineering background, reveal the mechanism of losing stability and physical phenomena caused by bifurcation of periodic solutions, and present the parameter control for anti-vibration design. We will strive to achieve innovative results in the field of bifurcation theory of multi-periodic solutions and 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同实际工程中诸多复杂非线性现象密切相关，已成为国际非线性动力学领域的前沿课题。本项目致力于发展和改进高维次谐Melnikov理论，研究高维非线性动力系统的多周期解分岔理论与几何结构，并应用于覆冰悬索模型的非线性动力学分析。(1)基于动态曲线坐标标架与Poincare映射相结合的方法，发展和改进高维次谐Melnikov理论，使其适用于更为广泛系统多周期解分岔的研究。(2)探寻利用改进的高维次谐Melnikov理论获得4维拟Hamilton系统周期解的精确个数、参数控制与几何构型。(3)研究具有实际工程背景的覆冰悬索模型的多周期解分岔问题，理论揭示由于多周期解分岔导致系统失稳的产生机理与其相关的物理现象，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应用领域取得创新性成果。

高维非线性动力系统多周期解分岔的研究是非线性动力学与控制领域的热点和难点问题之一，对研究实际工程模型中诸多复杂非线性现象有重要意义和价值。本项目主要完成的研究工作：. (1)基于动态曲线坐标标架与Poincaré映射相结合的方法，得到了2n维Hamilton系统和2n维具有孤立正交不变环面系统分别在自治和非自治扰动下的次谐Melnikov函数，进而研究了四类系统周期解分岔的判定条件。. (2)基于周期变换与Poincaré映射相结合的方法，将平面系统周期解的相关理论推广至高维，得到了4维共振等时中心系统分别在自治和非自治扰动下的广义Melnikov函数，进而研究两类系统周期解分岔的判定条件。. (3)利用改进的高维次谐Melnikov理论获得4维Hamilton系统分别在自治和非自治扰动下周期解的精确个数、参数控制与几何构型，并将理论结果应用于覆冰悬索结构、 负泊松比超材料板、复合材料层合圆柱壳等系统多周期解分岔的研究中，为系统的减震设计和参数控制提供有效的论指导。. 本项目研究工作按照预定计划顺利完成，达到预期研究目标。