非线性问题中的动力学及其复杂性

In mathematical sciences and other subjects, we face many nonlinear problems, which is the important contents in nonlinear sciences. The mathematical models in nonlinear problems are often nonlinear differential equations and dynamical systems, whose solution or orbit represents the phenomenon in the nonlinear problems. The periodic and almost periodic motion and homoclinic orbit are the important nonlinear phenomenon and the significant research topic. This project will concern mathematical theory with nonlinear problem and study the mathematical problem which need be solved. In the project, we will study the coincidence degree and almost periodic problem, the frequency and module of almost periodic solution of impulsive differential equations, which reflect the oscillation and complexity of the motion, and the homoclinic solution of Hamiltonian systems with variable expoents. We will investigate the dynamics for age structure system and analyse the higher co-dimension bifurcation. We will also study the mathematical model in ecology and the spreading of the motion.

在数学及其它许多学科中, 常遇到非线性问题, 它是非线性科学的重要内容. 非线性问题的数学模型常用非线性微分方程和动力系统来描述,它的解或轨道反应了非线性问题的变化. 周期运动,概周期运动和同宿轨是重要的非线性现象,是重要的研究内容. 本项目联系非线性问题和数学理论, 研究在数学中需要解决的数学问题. 本项目将研究重合度理论与概周期问题; 研究概周期脉冲微分方程概周期解的频率和模,它反映了运动的复杂性; 研究变指数Hamilton系统的同宿解; 对年龄结构方程研究动力学性质并分析高余维分支情况; 研究生态学中的数学模型及运动的传播问题和速度.

许多科学中的问题是非线性问题，项目研究了一些非线性系统的动力学问题。项目研究了脉冲微分方程的概周期解，对概周期解的模进行了刻画；项目研究了一些年龄结构方程的稳定性和分支问题，研究了B-T分支和Hopf分支；项目研究了传染病动力学模型，研究了解的传播性质和行波解的存在性；项目研究了随机恒化器模型，研究了保持性、灭绝性、和随机吸引子；项目使用临界点理论研究了微分方程同宿解和基态解的存在性。