【作者】 杨柳；

【导师】 盛中平；

【作者基本信息】 东北师范大学 ， 计算数学， 2013， 硕士

【摘要】 本文讨论了线段上连续自映射构成的离散动力系统问题，综述了Sarkovskii定理的证明方法。在定理的证明过程中，涉及到映射f和迭代q次后f~q的周期点的周期变化关系，同时给出了几个统一结论及其具体实用的推论。讨论了周期点的几何个数与代数个数，给出了p周期点的一般几何个数公式和代数个数的上下界估计。分析了周期点的度量结构，提出了重周期轨的概念，就二重二周期轨给出了具体分析，证明了此时周期点的重数一定相等，由此可以定义为二周期轨的重数。证明了重周期轨的充要条件就是轨道特征为1；给出了周期轨二阶导特征表达式，进而给出三重周期轨的判别方法。分析了周期点的拓扑结构，引入了周期轨的拓扑图表示与矩阵表示，获得了在周期轨道的拓扑分类，亦即p周期轨共有(p-1)!类。并具体绘制了2-周期轨，3-周期轨，4-周期轨和5-周期轨的拓扑图谱。更多还原

【Abstract】 This paper discusses the discrete dynamical system problems for a continuousmap on a line, summarized the proving method of Sarkovskii theorem. In the prooffor Sarkovskii theorem, there are some periodic variation relations for a periodicpoints between original map and its multiple composite map. It gives several unifiedconclusions and specific practical inference for the periodic variation relations. Itdiscusses the geometric number and algebraic number of periodic points for period p,gives the geometric number of periodic points, and estimates the upper and lowerbounds of the algebraic number. It analyses the metric structure of periodic orbits,puts forward the concept of multiple periodic orbit, gives a detailed analysis for thedouble periodic orbit with period2, proves that at this time the mutiples of periodicpoints in the period orbit must be equal, hence the same multiple can be defined as themultiple for the periodic orbit. Obtains the result, the necessary and sufficientcondition for double periodic orbit is that the orbit characteristic is1. And it gives theorbit characteristic by the second derivative, and gives a sufficient condition for thetriple periodic orbits. At last, it analyses the topology structure of period orbits. Itintroduces the topological graph representation and the matrix representation forperiodic orbits, obtains the classification topologically for period orbits, and provesthe class number is (p-1)! for the period orbit with period p, specifically draws thetopology graph for the period orbit with period2,3,4and5.更多还原