fractals -4- dr christoph traxler
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4. Chaotic Behavior of Fractal Attractors
4.1
Christoph Traxler 1
Chaotic Behavior of Fractal Attractors
! Study the dynamic behavior of fractals! Understand the stochastic method! Assignment of an address to the points of
an attractor A ∞ ! Examination of the address space with
dynamic system theory!
Chaotic properties of the address spacecan be transfered to the attractor
Christoph Traxler 2
Addresses of Fractal Attractors
! Given: IFS {X; f 1,f 2,...,f n} with the attractor A∞ = f 1(A∞) ∪ ... ∪ f n(A∞), decomposed in nsubsets
! Definition of the address of a point x ∈A∞:
x ∈ f a(A∞) ⇒ address of x: a … x ∈ f af b(A∞) ⇒ address of x: ab ...
! Infinite many indices are necessary toidentify a point x ∈A∞ exactly
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4. Chaotic Behavior of Fractal Attractors
4.2
Christoph Traxler 3
Addresses of Fractal Attractors
Hierarchy tree
Address tree∑2
1 ... 2 ...
11 ... 12 ... 21 ... 22 ...
A∞
f 1(A∞) f 2(A∞)∪
f 1f 1(A∞) f 1f 2(A∞)
∪
f 2f 1(A∞) f 2f 2(A∞)
∪
Christoph Traxler 4
Addresses of Fractal Attractors
! Subtriangles of the Sierpinski gasket and thecorresponding addresses of ∑3
1 2
3
1211
13
21 22
23
31
33
32f 1f 3
f 1 ∪ f 2 ∪ f 3
f 3
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4. Chaotic Behavior of Fractal Attractors
4.3
Christoph Traxler 5
Addresses of Fractal Attractors
! Example: Addresses of subtriangles & points
33333
33231
312333313222
23133323311113133
11213
3111113333
Christoph Traxler 6
Address Space
! Def.: The metric address space (∑n,d S ) for anIFS {X; f 1,f 2,...,f n} is defined by
∑n = {σ1 σ2 ... σ i ... | σ i ∈ {1 ... n} }and the Symbol Metric
α = α 1 ... α m ...β = β1 ... βm ...
∑∞
=+
−=
1 )1(),(
ii
ii
sn
d β α
β α
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4. Chaotic Behavior of Fractal Attractors
4.4
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Address Space
! Def.: The mapping φ:∑n→ A∞ is defined asφ(σ) = f σ1f σ2 ... f σm ... (x) = a ∈ A∞,σ ∈ ∑n, x arbitrary
! φ(σ) is independent of the starting point x
! φ is a continuous mapping, similar addresses
correspond to nearby points
Christoph Traxler 8
Address Space
! The metric spaces ( ∑n, d S) and (A ∞,d) havethe same properties
! Thus the dynamic behaviour of A ∞ can beanalyzed by examining the dynamicbehaviour of ∑n
! φ-1(a) = { σ ∈ ∑n: φ(σ) = a}, a ∈ A∞, set of alladdresses for a point a
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4. Chaotic Behavior of Fractal Attractors
4.5
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Address Space
! If each point of A ∞ has a unique address,then A ∞ is called totally disconnected
! The Sierpinski gasket consists of infinitemany branching (touching) points, thus onlycorner points have a unique address
Totallydisconnected
Just touching Overlapping
Christoph Traxler 10
Dynamic Systems
! Def.: Let (X,d) be a metric space andf: X→X a function, then {X, f} is calleddynamic system
! Def.: The sequence {f n(x)} = {x,f(x),f 2(x),...}
is called orbit of x ∈X
! Example: ( ∑n, Τ) is a dynamic system,Τ: ∑n→∑n is called shift function anddefined as Τ(σ1 σ2 σ3 σ4 ...) = σ2 σ3 σ4 ...
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4. Chaotic Behavior of Fractal Attractors
4.6
Christoph Traxler 11
Dynamic Systems
! An IFS {X; f 1,f 2,...,f n} defines a specialdynamic system {H(X), W }, where W is theHutchinson operator
! The most interesting dynamic systemsoperate with non linear functions (Julia sets,Mandelbrot set, strange attractors)
Christoph Traxler 12
Dynamic Systems
! Orbit of {R,f}:the fixpointsare theintersection
points of themedian withthe graph
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4. Chaotic Behavior of Fractal Attractors
4.7
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Fixpoints
! Def.: {X,f} is a dynamic system, a point p ∈ Xis called periodic point if there exists a number n > 0, so that f n(p) = p{p, f(p),...,f n(p)} is called cycle of p andn is called cycle length or period of p
! p periodic in {X,f} ⇔ p is fixpoint of {X,f n}
! Minimal period of p = min{n | f n(p) = p }
Christoph Traxler 14
Fixpoints
! Def.: A fixpoint x=f(x) of {X,f} is calledattractive , if there exists an ε > 0, so that f is acontraction mapping in B(x, ε)repelling , if there exists an ε > 0, s>1, so thatd(x,f(y)) ≥ s • d(x,y), ∀ y∈B(x, ε)
attractive point repelling point
x
f
f f x
f
f
f
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4. Chaotic Behavior of Fractal Attractors
4.8
Christoph Traxler 15
Fixpoints
! The dynamic system {H(X), W } of an IFShas exactly one fixpoint, no periodic points, allorbits converge to A ∞
! Usually dynamic systems have severalfixpoints
! A periodic point p with period n is calledattractive (repelling), if p is an attractive(repelling) fixpoint of {X,f n}
Christoph Traxler 16
Fixpoints
! Example: {R,f}, f(x) = λx(1-x), λ < 3
attractive point
repelling point
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4. Chaotic Behavior of Fractal Attractors
4.9
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Fixpoints
! Example: {R,f}, f(x) = λx(1-x), λ > 3
repelling point
www.geogebra.org/de/upload/files/dynamische_arbeitsblaetter/lwolf/chaos/start.html repelling point
Christoph Traxler 18
IFS Attractor as Dynamic Systems
! Def.: {X; f 1,f 2,...,f n} is an IFS, the shifttransformation S: A ∞ → A∞ is defined asS(a) = f i
-1(a), ∀ a∈ f i(A∞){A∞, S} is a dynamic system
! f m(A∞) ∩ f n(A∞) ≠ ∅ ⇒ S(a) is ambiguous∀ a ∈ f m(A∞) ∩ f n(A∞)
! Orbits of points from A ∞ can be examinedin {A∞ ,S} (backward orbits)
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4. Chaotic Behavior of Fractal Attractors
4.10
Christoph Traxler 19
IFS Attractor as Dynamic Systems
! Relation between { ∑n,Τ} and {A ∞ ,S} ?! Function Τ has the same effect in ∑n as
S in A ∞ ! The relation is established by φ:∑n→ A∞
φ(Τ(σ)) = S( φ(σ))
∑n
∑n
A∞
A∞
Τ S
φ
φ
Christoph Traxler 20
IFS Attractor as Dynamic Systems
! The knowledge of the dynamic behavior of {∑n,Τ} can be applied to {A ∞,S} as well
! It is much easier to examine the propertiesof {∑n,Τ} than of {A ∞,S}
! {∑n,Τ} contains no geometric but onlytopological information about {A ∞,S}
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4. Chaotic Behavior of Fractal Attractors
4.11
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1231322231322
31322
1322
322
2222
{∑n,Τ} {A∞ ,S}- 1
a 1 = f 1-1(a 0)
a 2 = f 2-1(a 1)
a 3 = f 3-1(a 2)
a 4 = f 1-1(a 3)
a 5 = f 3-1(a 4)a 6 = f 2-1(a 5) = a 5
a 0
Orbits of IFS Attractors
! Example:
a 0
a 1
a 2
a 3
a 4
a 5 = a 6
Christoph Traxler 22
Orbits of IFS Attractors
! Examples:
Cantor set, orbit of thepoint s = 123111213212
Orbit in Barnsely‘s fern
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4. Chaotic Behavior of Fractal Attractors
4.12
Christoph Traxler 23
Chaos in Dynamic Systems
! Def.: A dynamic system {X,f} is called chaotic if:
(1) {X,f} is transitive
(2) {X,f} is sensitive with respect to thestarting conditions
(3) The set of periodic orbits of f isdense in X
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Chaos in Dynamic Systems
! Def.: {X,f} is transitive, if there exists a number n for the open sets U, V ⊆X so thatU ∩ f n (V) ≠ ∅
! Orbits of points taken from an arbitrary smallsubset reach every part of X
U
V
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4. Chaotic Behavior of Fractal Attractors
4.13
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Chaos in Dynamic Systems
! {∑n,Τ} is transitive:! V open set ⇔ ∀ ω ∈ V exists a
B(σ ,ε) = {ω: d( σ ,ω) < ε}, B(σ ,ε) ⊆V! Example n=2:
σ = σ1σ2 ... σm ...ω1 = σ1σ2 ... σm 11 ...ω2 = σ1σ2 ... σm 12 ...
ω3 = σ1σ2 ... σm 21 ...ω4 = σ1σ2 ... σm 22 ...
! The ωi can be continued with any possiblecombination of symbol
Christoph Traxler 26
Chaos in Dynamic Systems
! {∑n,Τ} is transitive:
! ω ∈B(σ ,ε) ⇒ ω = σ1σ2 ... σmω1ω2 ...,ωi ∈ {1,…,n}
⇒Τm
(ω) = ω1 ω2 ...⇒Τ m (B(σ ,ε)) = ∑
⇒Τ m (V) ∩ U ≠ ∅, ∀ U,V ⊆X
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4. Chaotic Behavior of Fractal Attractors
4.14
Christoph Traxler 27
Chaos in Dynamic Systems
! Def.: {X,f} is sensitive towards startingconditions if there exists a number d > 0, sothat ∀ x ∈X and ∀ B(x, ε), ε > 0, ∃ y ∈B(x, ε)and n > 0, so that d(f n(x), f n(y)) > d
! Nearby orbits move away from each other
x
y f n
(y)
f n(x)
B(x, ε)
Christoph Traxler 28
Chaos in Dynamic Systems
! {∑n,Τ} is sensitive:! µ = σ1σ2 ... σm1... Τm (µ) = 1...! ν = σ1σ2 ... σm2 ... Τm (ν) = 2 ...! dS(µ,ν) ≈ 1/(n+1) m ! dS(Τm(µ),Τm(ν)) ≈ 1/(n+1)! µ and ν are the starting conditions for Τ
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4. Chaotic Behavior of Fractal Attractors
4.15
Christoph Traxler 29
Chaos in Dynamic Systems
! Def.: (X,d) is a metric space, a set S ⊆X iscalled dense in X, if for each point x ∈X thereexists a sequence {s n} in S, with lim s n=x(X is called closure of S)
! Example: The set of rational numbers Q isdense in R
Christoph Traxler 30
Chaos in Dynamic Systems
! The set of periodic points P of { ∑n,Τ} isdense in ∑n:
! Arbitrary address σ & sequence of periodicaddresses
σ1σ2 ... σm ...s 1 = σ1
s 2 = σ1σ2 Μ
s m = σ1σ2 ... σm
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4. Chaotic Behavior of Fractal Attractors
4.16
Christoph Traxler 31
Chaos in Dynamic Systems
! Fixpoints of ∑n are repelling points:! Example n = 2:
! µ = 1 fixpoint! ν = 1112 ... Arbitrary! dS(µ,ν) ≈ 1/34 d S (Τ3(µ),Τ3(ν)) ≈ 1/3! Periodic points of ∑n are repelling points
! Periodic points are dense in ∑n ⇒ ∑n isdensely covered by repelling points
Christoph Traxler 32
Chaos in Dynamic Systems
! {∑n,Τ} is a chaotic dynamic system, because:
(1) It is transitive
(2) It is sensitive towards starting conditions(3) Set of periodic orbits of Τ is dense in ∑n
{∑n,Τ} chaotic {A∞,S} chaoticφ
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4. Chaotic Behavior of Fractal Attractors
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Back to the Stochastic Method
! The attractor A ∞ of an IFS {X; f 1,f 2,...,f n} isapproximated by a random sequence of pointsxn = f i(xn-1 ), i ∈ {1,...,n}
! Example: n=2, Random sequence 112 ...1:Point x 0 x1=f 1(x0) x 2=f 1(x1) x 3=f 2(x2) … xmax
Address σ 1σ 11 σ 211 σ … σmax
! The sequence {x max ,...,x 3,x2,x1,x0} is an orbit of the chaotic system {A ∞,S}
Christoph Traxler 34
Back to the Stochastic Method
! Dynamic system { ∑n,Τ}, starting point σ : ! σ periodic, quasiperiodic:
! Orbit {T n(σ)} converges to periodic repellingpoint
! σ not periodic:! Orbit {T n(σ)} comes closer to a periodic
repelling point from which it moves away! New points of A ∞ are always generated
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4. Chaotic Behavior of Fractal Attractors
4.18
Christoph Traxler 35
Back to the Stochastic Method
! How frequent are starting points σ with a tooshort period ?
! U(p) - number of periodic orbits with minimalperiod p
∑ ⋅−=
pdividesk
p pk U k n pU /))(()(
Christoph Traxler 36
Back to the Stochastic Method
! Example { ∑2,Τ} :! U(1) = 2 fixpoints 1, 2! U(2) = 1 fixpoint 12! U(3) = 2! U(10) = 99! U(15) = 2182! U(20) = 52377! U(p) is a fast increasing function
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4. Chaotic Behavior of Fractal Attractors
4.19
Christoph Traxler 37
Back to the Stochastic Method
! If the starting point σ is periodic, then itsperiod is very long with high probability
! Orbits of chaotic dynamic systems aredistributed among the whole attractor
! {x, f 1(x), f 2f 1(x), f 2f 1f 1(x), ...} x ∈A∞, covers A ∞ ! {p, f 1(p), f 2f 1(p), f 2f 1f 1(p), ...} p ∉A∞, converges
to {x, f 1(x), f 2f 1(x), f 2f 1f 1(x), ...}
Christoph Traxler 38
Conclusion
! Examination of the chaotic behavior of fractal attractors of an IFS {X; f 1,f 2,...,f n}
! Introduction of the address space ∑n and thedynamic system { ∑n,Τ}
! Relation between { ∑n,Τ} and {A ∞,S} → chaotic properties of { ∑n,Τ} can betransfered to {A ∞,S}
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4. Chaotic Behavior of Fractal Attractors
Christoph Traxler 39
Conclusion
! The stochastic method can be analyzed with(backward) orbits in {A ∞,S}
! {A∞,S} is chaotic ⇒ orbit is distributedamong the whole attractor with very highprobability
! The random orbit generates a good
approximation of A ∞