Duffing方程调和解和次调和解的研究

The Duffing equation is a typical nonlinear vibration system, it has many important applications in mechanical, economic, mechanical and electronic engineering technology. It is a description of resonance phenomena, harmonic vibration, subharmonic vibration, quasiperiodic oscillation, which is a mathematical model of strange attraction and chaos phenomena. The project is concerned with the existence of harmonic and subharmonic solutions for several kinds of Duffing equations. By means of the Poincaré-Birkhoff twist theorem and phase-plan analysis methods, we wil study the existence of harmonic and subharmonic solutions for the quasilinear Duffing equation, the Duffing equation with singularity, the Duffing equation with delay; by using the twist theorem, we will deal with the existence of harmonic and subharmonic solutions of the damped Duffing equation. The work of the project will give a more complete theory system for the existence of harmonic and subharmonic solutions of the Duffing equations, which has important theoretical significance and application value.

Duffing方程是一个典型的非线性振动系统，它在机械、经济、材料力学和电子工程技术中有许多重要的应用。它是描述共振现象、调和振动、次调和振动、拟周期振动、奇异吸引子和混沌现象的数学模型。本项目将研究几类Duffing方程调和解和次调和解的存在性。利用Poincaré-Birkhoff 扭转定理和相平面分析，研究拟线性Duffing方程、奇性Duffing方程、具有时滞项的Duffing方程调和解和次调和解的存在性；利用满足耗散的扭转定理，研究带阻尼的Duffing方程调和解和次调和解的存在性。本项目的研究工作将给出一个较为完整的Duffing方程调和解和次调和解存在性理论体系，具有重要的理论意义和应用价值。

本项目，我们研究奇性Dufing方程、带阻尼的Duffing方程和拟线性Duffing方程调和解和次调和解的存在性。具体包括：1.对带阻尼Duffing方程的研究，由于Poincaré-Birkhoff扭转定理要求的条件必须是保面积的（保守系统），而带阻尼的Duffing方程是一个耗散系统，因此传统研究Duffing方程次调和解的方法Poincaré-Birkhoff扭转定理已不再适用。本项目中，我们引入满足耗散系统的扭转定理，然后通过该扭转定理和相平面分析，研究带阻尼的Duffing方程调和解和次调和解的存在性；2.对拟线性Duffing方程的研究，通过把拟线性Duffing方程转化为一阶二维系统，利用Poincaré-Birkhoff扭转定理和相平面分析，研究拟线性Duffing方程调和解和次调和解的存在性。3.对奇性Duffing方程的研究，为了便于使用相平面分析法研究奇性Duffing方程，我们首先把在原点的奇性，平移到-1，并通过把Duffing方程转化为二维一阶系统，利用高维的在Hamiltonian系统下的Poincaré-Birkhoff定理，得到一类奇性Duffing方程调和解和次调和解的存在性。