数理解析研究所講究録 1246 巻 2002 年 66-71

66

On Definitions of Chaos
城西大学大学院理学研究科 西沢 清子
KIYOKO NISHIZAWA
Graduate School of Science, Josai University
E-mail : kiyoko@matb.josai.ac.jp

Keywords and phrases: complex dynamical systems –chaos –topological conjugate –topologically transitive –sensitive dependence –blending-1

Introduction

Chaotic dynamical systems theory is an attractive and important branch of mathematics of interest to scientists in many disciplines. Although there has been no universallyaccepted mathematical definition of chaos, the most widely utilized definition of chaos isdue to R. Devaney([Dev89]). He isolates three properties as beeing the essential featuresof chaos.
Definition of Choas (R. Devaney, 1989)
Let $Xl$) americ space. A continuous function from $X$ to itself is said to he chaotic
on $X$ if lias the following three properties:
$f$

$e$

$f$

1.

is $topologicall$ } transitive: that is, for all non-empty open sets and of X. there
exists a natural number such thatis non-empty.
2. periodic points are dense in the space.
3. lias sensitive dependence on initial conditions; that is, there is a positive constant(sensitive constant)such that for ever$ry$ point of $X$ and eve neighborhood $N$of there exists a point in $N$ and a nonnegative integer such that -th iterates$f^{n}(x)$ and $f^{n}(y)$ are more than distanceapart.
$f$

$V$

$U$

$\backslash \cdot$$k$$f^{k}(U)\mathrm{r}\urcorner$$\mathrm{L}^{f}$$f$$\delta$$x$$x$$r\tau/$

$n$

$y$

$n$

$\delta$

In acertain sense, Condition 1. is an irreducibilty condition, Condition 2. an elem entof regularity, and Condition 3. an element of unpredictability.
Hereafter we call Condition 1. transitivity, Condition 2. density, and Condition 3.
sensitivity.
It is proper for Chaos to be topological property and invarinat under the action of
topological conjugate

67Transitivity(i.e. 1.) and density (i.e.2.) are topological conditions, but unfortunatelysensitivity(Le.3.) is depend to amertic.
On the otherhand, the sensitivity condition which is famous as “butterfly cffect”
is considered as being the central idea in chaos. This condition captures the idea thatin chaotic systems small errors in experimental readings eventually lead to large scale
divergence. And it is easily understood.
Further sensitivity or expansiveness of adynamical system are recognaized as formalizations of the notion of unpredictability in the presence of chaos. S. MacEachern andL. Berliner in [MB93], sharpen these two concepts under the case of acompact set of thereal line.
The requirement of density is less intuitive than sensitivity, but it appeals to thoselooking for patterns or somewhat regularity within aseeming random system: density
implies that there is order in chaos.
The irony of Devaney’s definition is that the more understood each condition is, themore redundant it is in relationship to the other two.
Banks et al. pointed out in [BBCDS92] aredunduncy :despite its popular appeal,sensitivity is mathematically redundant: transitivity and density imply sensitivity.
Their elegant result makes clear that chaos is aproperty relying only on the topological,
and not the metric, properties of aspace.
Vellekoop and Berglund showed in [VB94] amore redunduncy: namely on intervals,
transitivity is equal to chaos.
Crannell points out that this result of Banks et al. yields asomewhat less intuitive”,definition of chaos, and asks in [Cran95] “why transitivity? -why not something
and suggested aslightly more natural concept, blending, as an alternative to transitivity.
Recently Touhey introduced in [TOu97] another definition of chaos, equivalent to Devney’s one.
The subject of this paper is to survey these papers on aredundancy in the definitionof chaos by Devaney in his text [Dev89], and ones on other definitions of chaos.
$(^{11\mathrm{s}\mathrm{e}?},‘$2Redundancey 1n the definitions of Chaos
Banks et al. point out in [BBCDS92] aredunduncy in the definition of chaos by Devaney:
Theorem Let $X$ be americ space and acontinuous function on X. If is topologieasily ansitive and has dense periodic points, then has sensitive dependence on initialconditions.
Namely, transitivity and density of Periodic points imply sensitivity.
And in the same volume, Assaf IV and Gadbois [AG92] show that this is only redundancy for ageneral map:
$f$

$f$

$tr$

$f$

68(A) transitivity and sensitivity $clo$ not imply density, and (B) density andsensitivitl. do not imply transitivity.

Theorem

For the case of (A), there is afollowing example ([AG92]):
Example : Let $X=S^{1}\backslash \{\exp(i2\pi p/q) : p, q\in \mathbb{Z}\}$ and $f(\exp(i\theta)=\exp(i2\theta)$ .
The set of periodic points of is empty.
$\mathrm{A}$$f$

For the case of (B), there are following examples Bl, and B2 on an interval:
Let $X$ be acylinder $S^{1}\cross[01]$ with the induced ‘ taxicab”Example Bl ([AG92]):
metric and $g(\exp(i\theta, t)=$ ( $\exp$ (i2?), ).
can not intersect with ,underIf we takeanditerations of .
$i$

$t$

$U=S^{1} \cross[0, \frac{1}{2})$$\iota^{r}$$V=S^{1} \cross[\frac{1}{2},1]\backslash \downarrow,\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{n}l^{\gamma}$$g$

Example B2 ([VB94]):

Let X

and f be$=\mathbb{R}^{1}$,if$0 \leq x<\frac{1}{3}$:$-3x+2$if$\frac{1}{3}\leq x<\frac{2}{3}$:$3x-2$

if$\frac{2}{3}\leq x<1$:

$f(x-1)$

if $1\leq x\leq 2$ .

$3x$$f(x)=\{$. If we take $U=[0,1)$ andhas sensitivity, because of its expansivenss$V=$ $(1, 2]$ ,thencan not intersect with und iterations of .
$f$$|_{dx}^{\lrcorner d}|=3$

$V$

$U$

2.1

On an interval, transitivity$‘$)$\mathrm{r}$$f$$=\mathrm{C}\mathrm{h}\mathrm{a}\mathrm{o}\mathrm{s}$If we restrict our attension to maps on an interval, astronger result can be obtainedby M. Vellekoop and R. Berglund [VB94]:
Proposition
Let I be anot necessarily finite, interval and acontinuous function from I to itself. If
has (topological) transitivity, then has (1) density and (2) sensitivity.
Namely, for maps on an interval, both sensitivity and density are redundant conditionsin the definition of chaos.
And further they note that there are no other trivialities in Devaney’s definition whenrestricted to intervals: on intervals, neither density nor sensitivity is enough to ensure anyof the other conditions of chaos.
We have an example B2 on an interval showing that density and sensitivity do not
imply transitivity.
And in [VB94], they give an example on an interval satisfying that sensitivity does not
imply density:
$f$

$f$

$f$

69
Example C ([VB94]):

Let I$=[0, \frac{3}{4}]$$\frac{3}{2}x$$f(x)=\{$

and f be,$\frac{3}{2}(1-x)$

if$0 \leq x<\frac{1}{2}$:if$\frac{1}{2}\leq x<\frac{3}{4}$.

And in [VB94], they note the identity map on an interval as an example satisfying thatdensity does not imply sensitivity. However we are thirsty for anon-linear example on aninterval of this case.

3Another definitions of Chaos
For one dimensional dynamical systems, the result by M. Vellekoop and R. Berglundleaves us only the study of the tansitivity.
Crannell points out that these redundant results yields asomewhat less intuitive definition of chaos, and asks in [Cran95] “why transitivity? –why not some thing else? ”.
Of the three conditions (transitivity, density, and sensitivity), sensitivity is clearly themost easily understood. In order to restore this vost sense of intuitiveness, she suggests aslightly more natural concept, blending, as an alternative to transitivity.

Transitivity or Blending

3.1Definition([Cran95]):
Let $M$ be asubset of and acontinuous function on M. is weakly blending if, for any
pair of non-empty open sets and in $M$ , there is some $k>0$ so that.
And is strongly blending if, for any pair of non-empty open sets and in , there
is some $k>0$ so that $f^{k}(U)\cap f^{k}(V)$ contains non-empty open subset.
It is clear that blending functions are not necessarily transitive, and transitive functions
are not necessarily blending. She gives examples as follows:
Example, given by $f(\theta)=\theta+k$ , where:
Let $X=S^{1}$ and :is irrational.
In this case, is transitive but not strongly or weakly blending.
$\mathbb{R}^{n}$

$f$

$f$

$V$

$U$

$f^{k}(U)\cap f^{k}(V)\neq\emptyset$
$V$

$U$

$f$

$\Lambda f$$\partial$$f$$\mathrm{D}([\mathrm{C}\mathrm{r}\mathrm{a}\mathrm{n}95])$$\mathrm{k}\mathrm{b}^{\prime 1}-arrow S^{1}$$\frac{k}{\pi}$$f$:
Let $X=[-11]$ an(l.f : $Xarrow X$ satisfying:$|f’(x)|>2$ , on except at the vertices of ;and
each vertex of the graph of likes alternately on the line
andThis function is strongly blending but not transitive.

Example$\bullet$$\bullet$$\mathrm{E}([\mathrm{C}\mathrm{r}\mathrm{a}\mathrm{n}95])$

$f$

$f$

$y= \frac{x}{2}$$y=- \frac{x}{2}$.

And she gives an example on an interval satisfying that weakly blending does not imply
transitivity, and nor strongly blending :70$\fbox\chi\backslash 1$:Graph of afunction of EX.

Example F ([Cran95]):$\mathrm{F}$which is strongly blending but not transitive

Let I $=[-1,$ 1] and f. be$-2x-2$ ,$f(x)=\{$ $2x$ ,$\mathrm{i}\mathrm{f}-1\leq_{-}x\leq-\frac{1}{9,\sim}$,$\mathrm{i}\mathrm{f}-\frac{1}{2}\leq x\leq\frac{1}{2}$$2-2x$ ,if$.\sim\underline{1},$$\leq x\leq 1$::.

She remarks that these two examples(i.e., , E) have not density. Therefore, adding
density condition to blending conditions, she obtains the following theorem:
Theorem 1[Cran95]:
and acontinuous function on $M$ with dense periodic points.
Let $M$ be asubset ofThen if is strongly blending, is also transitive.
Namely, density and strongly bending condition imply chaos.
$\mathrm{D}$$\mathbb{R}^{n}$

$f$

$f$

And if$\Lambda f$

is resticted to acompact set inTheorem 2[Cran95]:
Let I be acompact subset ofareppeling fixed point. Then

3.2

$f$

. she gives another theorem:

and acontinuous function on I with transitivity aJld
is weakly blending.

$\mathbb{R}^{1}$$f$$\mathbb{R}^{1}$$f$

Yet another definition of chaos

P. Touhey proposes anew and natural definition of chaos in [TOu97], equivalent toDevaney’s. He reformulates the two topological conditions of transitivity and densityof chaos.
of periodic points as asingle condition that yields asimple, concice
He use his definition to give acharcterization of chaos that restores the lost sense of$\mathrm{d}\mathrm{e},\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}_{1}\mathrm{i}\mathrm{o}\mathrm{n}$71intuitiveness: amap : $Xarrow X$ is chaotic on $X$ if and only if it mixes together, via
periodic cycles, any finite number of non-empty open subsets in infinitely many ways.
Definition of Choas [TOu97]
Let $X$ be americ space. A continuous function from $X$ to itself said to be “CHAOTIC”
on $X$ if given and , non-empty open sets in $X$ , there exists aperiodic point $p\in U$
and anon-negative integer such that $f^{k}(p)\in V$ .
Namely, every pair of non-empty open subsets of $X$ shares aperiodic point.
Theorem [TOu97] :Let $X$ be americ space, acontinuous function on X. The following are equiv alent:is “CHAOTIC”,
is chaotic,
any finite collection of non-empty open subsets of $X$ shares aperiodic point,
any finite collection of non-empty open subsets of $X$ shares infinitely many periodicorbits.
$f$

$f$

$U$

$V$

$k$

$f$$\bullet$$f$$\bullet$$f$$\bullet$$\bullet$

参考文献[Dev89]R. Devaney. An Introduction to Chaotic Dynamical Systems.
Addison -Wesley, 1989.

[BBCDS92]

J. Banks, J. Brooks, C. Crains, G. Davis, P. Stacy.
On Devaney’s Definition of Chaos. Amer. Math. Monthly. 99 :332-334,
1992.

[AG92]

D. Assaf, S. Gadbois. Definition of Chaos. Amer. Math. Monthly, 99 :865,
1992.

[MB93]

S. MacEachern, L. Berliner. Aperiodic Chaotic Orbits.
Monthly, 100 :237 –241, 1993.

[VB94]

M. Vellekoop, R. Berglund. On Intervals, Transitivity
Math. Monthly, 101 :353-355, 1994.

[Cran95]

A. Crannell. The Role of Transitivity in Devaney’s definition of Chaos.
Amer. Math. $Monthl_{t}v$ , 102 : 788-793, 1995.

[TOu97]

P. Touhey Yet Another Definition of Chaos. Amer. Math. Monthly, 104 :
411 -414, 1997.

Amer. Math.

$=\mathrm{C}\mathrm{h}\mathrm{a}\mathrm{o}\mathrm{s}$. Amer.