Fractals -4- Dr Christoph Traxler

7/27/2019 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




4.2

Christoph Traxler 3


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


! 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




4.3

Christoph Traxler 5


! 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




4.4

Christoph Traxler 7

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




4.5

Christoph Traxler 9

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 ...




4.6


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)


Dynamic Systems

! Orbit of {R,f}:the fixpointsare theintersection

points of themedian withthe graph




4.7


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 }


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




4.8


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}


Fixpoints

! Example: {R,f}, f(x) = λx(1-x), λ < 3

attractive point

repelling point




4.9


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


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)




4.10



! 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



! 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}




4.11


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


Orbits of IFS Attractors

! Examples:

Cantor set, orbit of thepoint s = 123111213212

Orbit in Barnsely‘s fern




4.12


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



! 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




4.13



! {∑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



! {∑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




4.14



! 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)



! {∑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




4.15



! 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



! 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




4.16



! 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



! {∑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




4.17


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}



! 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




4.18



! 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 /))(()(



! 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




4.19



! 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), ...}


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}





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 ∞

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Fractals -4- Dr Christoph Traxler