平面与高维系统周期轨的分支方法

Differential equations and dynamical systems are important research fields of modern mathematics and nonlinear sciences. The main problems in the field focus on global behavior of solutions, including periodic orbits, invariant sets and attractors. The project mainly studies limit cycles of planar systems and periodic solutions of higher dimensional systems with multiple parameters. These systems can be smooth or non-smooth. The problem studied appears in two types: local and non-local. Then the main purpose of the project is to develop new bifurcation methods for investigating limit cycles and periodic solutions and to obtain new results on the number of limit cycles for polynomial or piecewise polynomial systems and on the uniqueness and existence of periodic solutions with their number. We also presents the application of our theorems to some delay differential equations and differential systems with practice background, studying conditions for the existence of multiple periodic orbits and maximum number and uniqueness of periodic orbits in detail.

微分方程与动力系统是现代数学与非线性科学的一个重要研究领域，研究的主要问题是微分方程解的全局性态，包括周期轨、不变集与吸引子等。本项目主要研究含多参数的平面系统的极限环问题与高维系统的周期轨问题，这些系统可以是光滑的，也可以是非光滑的；这些问题可以是局部的，也可以是非局部的。主要目的是建立极限环与周期轨的新的分支技术， 获得一些平面多项式系统或分段多项式系统极限环的个数，以及一些三维系统或更高维系统周期轨的存在性及其个数与唯一性等。我们还将研究这些技术方法对时滞微分方程和有实际背景的微分方程的应用，深入探讨周期轨道的个数以及存在性与唯一性。

本项目研究的微分系统包括一维周期方程、二维自治系统和一些高维自治系统，它们既有光滑微分方程，又有分段光滑的微分方程。本项目对这几类系统的周期解或极限环的分支理论进行了一系列深入研究，建立了判定极限环个数的新方法。我们还研究这些技术方法对时滞微分方程和有实际背景的微分方程的应用，深入探讨周期轨道的个数以及存在性与唯一性。