几类含参数微分方程的极限环个数研究

Differential equations and dynamical systems are important research fields of modern applied mathematics, which are widely used to describe in various studies of mathematical models of epidemiological, physics, biological, engineering, economical and many other phenomena. The main problems in the field focus on the dynamical behavior of periodic solutions, including the existence, location and stability. This project mainly studies limit cycles of dynamical systems with multiple parameters. These systems are one dimensional and two dimensional systems in smooth case or piecewise smooth case. By the Melnikov function method, multiply parameters method and changing the stability of homoclinic orbits or a focus, the main purpose of this project is to obtain new results on the number of limit cycles for polynomial systems or piecewise smooth Hamiltonian systems and on the existence and number of periodic solutions of periodic equations. Besides, we also develop new bifurcation methods for investigating limit cycles, studying the maximum number of periodic orbits for finite smooth system.

微分方程与动力系统是现代应用数学的一个重要研究方向，涉及流行病学、物理学、生物学、工程和经济等各个领域，主要研究微分方程解的动力学性态，包括周期解的存在性、位置和稳定性等问题。本项目主要研究含多参数的动力系统的极限环的个数，研究对象是一维周期方程与平面自治系统，包括光滑和分段光滑两种情况。主要目的是采用Melnikov函数方法、多参数方法、以及改变同宿环或者焦点的稳定性等方法，获得几类平面多项式系统或分段光滑近哈密顿系统极限环的个数，以及一些一维周期方程周期解的存在性和个数等。另外，将寻求极限环分支的新方法，深入探讨有限光滑系统的极限环个数问题。

本项目主要研究含参数的微分方程的极限环的个数，研究对象是平面自治系统，以及一维和高维的周期方程，包括光滑和分段光滑两种情况。采用Melnikov函数方法，以及改变同宿环的稳定性方法，获得了几类平面光滑的近哈密顿系统的极限环个数。采用多参数平均法，获得一维和高维周期方程周期解的存在性和个数，结果还被推广到了分段光滑的情形。