一类一维混沌映射的拓扑条件  

Topological Conditions on a Class of One-dimensional Chaotic Maps

作　　者：裴森[1] 孙野[1] 赵珍[1] 王海涛[1] 佘志坤[1] 

机构地区：[1]北京航空航天大学数学与系统科学学院,北京100191

出　　处：《数学的实践与认识》2009年第19期213-227,共15页Mathematics in Practice and Theory

摘　　要：20世纪中期以来,人们在物理、天文、气象等领域中发现了大量的混沌现象.这些新发现引发了近几十年来对混沌现象的研究.由于它的困难程度和在解决实际问题中的巨大价值,对混沌现象的研究成为动力系统乃至数学中的一个长期的前沿和热点研究方向.混沌现象最本质的特征是初值敏感性,保证有初值敏感性的一个充分条件是系统具有正Lyapunov指数.因此研究系统是否具有正Lyapunov指数成为研究系统是否出现混沌的重要方法.从拓扑角度给出了一类一维映射出现混沌现象的充分条件.从拓扑的角度来研究,将加深对此类映射出现混沌的机理的认识.研究此类映射,最重要的是研究临界点、临界点轨道及它们的相互关系.我们采用临界点的逆像建立拓扑工具,使用这一拓扑工具分析临界点轨道与临界点的复杂关系,研究临界点逆轨道的运动形态、相应开集的拓扑特征,进而导出系统出现混沌的拓扑特征及它与Lyapunov指数之间的关系.Since the middle of the 20th century, a large number of chaotic phenomena have emerged in the research of physits, astronomy, meteorology, etc. These new discoveries have aroused the research of chaos in last decades. Because of its difficulty and great value in solving practical problems,the research of chaos has been a front-line and important field in dynamic systems and even in mathematics. The essential character of chaos is the sensitive dependence on initial conditions. A sufficient condition for a system to have sensitive dependence on initial conditions is to have a positive Lyapunov exponent. Therefore, to investigate whether a system has a positive Lyapunov exponent is an important way in the research of chaotic systems. In this text. we give a sufficient condition for a class of one-dimensional chaotic maps from a topological point of view for the first time. Researching this problem in a topological method will cement our understanding of the chaotic structure of such maps. The most important thing in the research of these maps is to study critical points, critical points＇ orbits and their correlation. We construct topological tool with pre-images of critical points. Then we analyze the complex relation between critical points and their orbits, research the dynamics of critical points＇pre-orbits and the topological characteristic of the corresponding open set, from which we get the topological condition on chaotic maps and its relation with Lyapunov exponent.

关 键 词：混沌 拓扑 非双曲 LYAPUNOV指数 

分 类 号：O415.5[理学—理论物理] Q516[理学—物理]