A Brief Introduction of Chaos 96

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????? A Brief Introduction of Chaos ????96?
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??????
.????,???? ?????????? -
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??????
.????(Butterfly Effect) -
????????????,???????????? - ???(Edward N.
Lorenz )??????
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??????
.?????????? - ??????????,?????????????????
-???????????????
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??????
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(Devaneys Definition of Chaos) fX?X i) f is
topologically transitive. transitive
for all non-empty open subsets U and V of X,
there exists a natural number k such that fk(U)nV
is nonempty. ii) The periodic points of f
are dense in X. The point x in X is a
periodic point of period n if fn(x)x, n
positive integer. iii) f has sensitive
dependence on initial conditions. if
there is a positive real number d (a sensitivity
constant) such that for every point x in X and
every neighborhood N of x there exists a point y
in N and a nonnegative integer n such that the
nth iterates fn(x) and fn(y) of x and y
respectively, are more than distanced apart.
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Devaneys Definition of Chaos
i) f is topologically transitive.
transitive for all non-empty open subsets U and
V of X, there exists a natural number k such that
fk(U)nV is nonempty.
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Devaneys Definition of Chaos
ii) The periodic points of f are dense in
X. The point x in X is a periodic point of
period n if fn(x)x. the least positive n for
which fn(x)x is called the prime period of
x. If for any two point a and b in X, a?b, altb,
and there exists a periodic point p that altpltb,
then we called the periodic points of f are dense
in X. (For example, Q dense in R where Q is the
set of rational number and R is the set of real
number.)
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Devaneys Definition of Chaos
iii) f has sensitive dependence on initial
conditions. if there is a positive real number
d (a sensitivity constant) such that for every
point x in X and every neighborhood N of x there
exists a point y in N and a nonnegative integer n
such that the nth iterates fn(x) and fn(y) of x
and y respectively, are more than distanced
apart. neighborhood N of x
for any egt0
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?????????
(Devaneys Definition of Chaos) fX?X i) f is
topologically transitive. transitive
for all non-empty open subsets U and V of X,
there exists a natural number k such that fk(U)nV
is nonempty. ii) The periodic points of f
are dense in X. The point x in X is a
periodic point of period n if fn(x)x, n
positive integer. iii) f has sensitive
dependence on initial conditions. if
there is a positive real number d (a sensitivity
constant) such that for every point x in X and
every neighborhood N of x there exists a point y
in N and a nonnegative integer n such that the
nth iterates fn(x) and fn(y) of x and y
respectively, are more than distanced apart.
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On Devaneys Definition of Chaos (1992) J. Banks,
J. Brooks, G. Cairns, G. Davis and P. Stacey
(i),(ii)gt(iii) If fX?X is transitive and has
dense periodic points then f has sensitive
dependence on initial conditions.
http//johnbanks.maths.latrobe.edu.au/chaos/
?J. Banks
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On Intervals, Transitivity Chaos (1994) Michel
Vellekoop and Raoul Berglund

• On intervals, (i)gt(ii),(iii)
• Let I be a, not necessarily finite, interval and
  f I?I a continuous and topologically transitive
  map. Then
• the periodic points of f are dense in I and
• (2) f has sensitive dependence on initial
  conditions.

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?????????(Iterative)
???????????????????? ?f(xn)xn1
???????????? ??????????
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Example f(xn1)4xn(1-xn)
brown x00.6 green x00.6001
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Example f(xn1)4xn(1-xn)
brown x00.37 green x00.3701
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Example f(xn1)4xn(1-xn)
i)????????????,??????????? ii)?????????????????,?
????????? (??? 0.6 ? 0.6001 ?????????),???????????
??????????
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?????????(Iterative)
?????????????? i) ????????? ii) ??????
???????????????????
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?????????(Iterative)
? ??????, ??????????,???????, ????????,
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?????????(Iterative)
???????????(James P. Crutchfield)? ??????????????
????,??????????????????????????
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??(fractal)-???????
???????????? ???????????? ????????????????????
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??(fractal)-???????
??????????????? ??????????
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??(fractal)-???????
?????????????????(Benoit Mandelbrot)?????? ??????
???? Answer???????
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??(fractal)-???????
??????
?????,??????,???? ????,?????????????,?????? ????
?,???????????????,??????, ?????????????? And
then ? ????????????????? ????????????????????
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???????(Space-filling Curve) By ??(Giuseppe Peano)
???????????????? ????????????Space-filling
Curve???? ?Space-filling Curve?????????????? ?Spac
e-filling Curve?????
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????(Koch Curve)
?????????, ??????????????????????????????, ???????
???????? Xn1 (3/4)Xn ????????? ????????????????
????? ???????????????????????????
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???????
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??????,????0 (???????????)
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??????? (????????)
Q??????????
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Special Thanks to..
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