On the Definition of Chaos in Discrete Dynamical System

• Main Authors: Wang Wef-Fu, 王文甫
• Other Authors: Chun-Kong Law
• Format: Others
• Language: zh-TW
• Published: 1998
• Online Access: http://ndltd.ncl.edu.tw/handle/07805213083166224634

碩士 === 國立中山大學 === 應用數學研究所 === 86 === Let $X$ be a metric space. A continuous function $f : X \rightarrow X$ is said to be ( topologically ) transitive if for any pair of non-empty open sets $U, V \subseteq X$, there exists a positive integer $k$ such that $f^k(U) \cap V \neq \ep$. In his popular book, Devaney defined a function $f$ to be chaotic if $f$ is transitive, has dense periodic points and is sensitive upon initial conditions. Recently, there are a number of results on the inter-relationship among these three components. The most surprising result is that when the space is an interval, then transitivity implies the other two components, and hence chaos.\parOn the other hand, Block and Coppel defined a function $f$ to be chaotic on an intervalif $f$ has a periodic point whose period is not $2^n$. This condition is equivalent to the existence of a homoclinic orbit, that is, there is some periodic point $x_0$ ( $f^n(x_0)=x_0$ ) and $y$ which lies in a local unstable set of $x_0$ such that$f^{km}(y)=x_0$ for some $k>0$.\parIn this thesis, we study the properties of transitivity and show that if $x_0$ is arepelling fixed point and $I$ is a compact interval, then chaos in Devaney's sensewould imply chaos in Block-Coppel's sense.\parCrannell, in a study on transitivity, introduced the concepts of blending and the setof eventual preimages of a fixed point $x_0$,$$E[x_0] = \{x\in I \mid f^k(x) = x_0, \mbox{ for some } k \in {\bf N} \cup \{0\}\}\ .$$We find that the last concept is the strongest of all three. Finally we shall usesome examples to illustrate the different concepts.