Fractals -5- Dr Christoph Traxler.pdf

7/27/2019 Fractals -5- Dr Christoph Traxler.pdf

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5. Julia Sets

5.1

Christoph Traxler 1

Iteration of Nonlinear Functions

! Gaston Julia (1893 - 1978)

! Pierre Fatou (1878 - 1929)

! Examination of dynamic

systems {C,f}, where f is

a polynomial or

rational function

! Examples: {C,z2} , {C,(z-2)2/z}

Christoph Traxler 2

Complex Numbers

! z = a + bi = r (cosϕ + i sinϕ)

! z1 + z2 = (a1+a2) + (b1+b2)i! z1 * z2 = (a1a2 + b1b2) + (a1b2+a2b1)i

! zn = r n(cos nϕ + i sin nϕ)

! z1/n = r 1/n (cos(ϕ+360k)/n + i sin(ϕ+360k)/n),

k = 0,1,...,n-1

ϕ

r

Im

Re

za

b



5. Julia Sets

5.2

Christoph Traxler 3

The Dynamic System {C,z2}

! Partition of the complex plane into 3 sets:

! |z|<1: orbit converges to the attractive point 0

! |z|=1: all points stay on the unit circle

! |z|>1: orbit converges to the attractive point ∞

Christoph Traxler 4

Areas of Attraction

! Area of attraction: Set of all points, which

converge to an attractive fixpoint

! Decomposition of C into

areas of attraction

! The border betweenareas of attraction is

called Julia set

(frequently a fractal set)



5. Julia Sets

5.3

Christoph Traxler 5

! Def.: {X,f} is a dynamic system, x∈X is a

fixpoint of f, then

A(x) = { y ∈ X | lim f n(y) = x )

is called area of attraction of x

! A(x) contains all points y, which have an orbit

{y,f(y),f 2(y),...} that converges to x

Areas of Attraction

Christoph Traxler 6

Areas of Attraction

! Def.: {X,f} is a dynamic system, x∈X is an

attractive periodic point of f with period n, i.e.

c = { x,f(x),...,f n(x) = x }, then the area of

attraction of the cycle c is defined as:

! A’(f i(x)) is the area of attraction of the fixpoint

f i(x) in the system {X,f n}

n

i

i x f Ac A

0

))(()(=

ʹ=



5. Julia Sets

5.4

Christoph Traxler 7

x,f(x),f 2(x) attracting

fixpoints of {X,f 3}

c = {x, f(x), f 2(x) } attracting

cycle of {X,f}

f 2(x)

f(x)xx=f 3(x) f(x)

f 2(x)

Areas of Attraction

! Example: x periodic point with period 3

Christoph Traxler 8

Areas of Attraction

! Example: x periodic point with period 3

! Area of attraction of the cycle

c = ∪ areas of attraction of x,f(x),f 2(x)

Julia set = bd A(c)

A’(x)

A’(f(x))

A’(f 2(x))



5. Julia Sets

5.5

Christoph Traxler 9

Julia set of a fixpoint Julia set of a periodic point

Julia Sets for {C, z2+c}

! The Julia set for {C, z2} is the unit circle

! The Julia sets for the dynamic systems

{C,z2+c}, c ∈ C are frequently fractal sets

Christoph Traxler 10


c = 0.001 c = -0.57+0.39i

c = -1 c = -0.79+0.14i



5. Julia Sets

5.6



c = 0.35+0.11i

c = -0.25+0.64i

c = -i

c = -0.66i


Classification of Fixpoints

! A fixpoint x of a dynamic system {X,f}

(X = R or X = C) is called:

superattractive ⇔ | f’(x) | = 0

attractive ⇔ | f’(x) | < 1

indifferent ⇔ | f’(x) | = 1repelling ⇔ | f’(x) | > 1

! |f’(x)| determines how fast a point converges

or escapes

! The same classification applies to periodic

points



5. Julia Sets

5.7


Rational Julia Sets

! Examination of the dynamic system

{C,r},where r is a rational function, i.e.

r(x) = p(x)/q(x), p and q are polynomials

! The Julia set for {C,r} is the set of all repelling

periodic points of r together with their

accumulation points

! Thus the Julia set is the closure of the set of all repelling periodic points


Rational Julia Sets

! Properties of the Julia sets Jr :! The set of all repelling periodic points P is

dense in Jr (Jr is the closure of P)

! Jr ≠ ∅ and contains uncountably many points

! The Julia sets for {C,r} and {C,r k

} (k=1,2,...)are identical

! r(Jr ) = Jr = r -1(Jr ), i.e. Jr is invariant under r



5. Julia Sets

5.8


Rational Julia Sets

! Properties of the Julia sets Jr :! For any x∈Jr the inverse orbit is dense in Jr

! The inverse mapping r -1 is multivalued, ⇒ the

backward (inverse) orbits can be conceived

as tree of points

xr -1 r -1

r -1

r -1r -1

r -1r -1

r -1

r -1

r -1r -1

r -1


Rational Julia Sets

! Properties of the Julia sets Jr :! If c is an attractive cycle of r, then

A(c) = Fr = C\Jr and bd A(c) = Jr

Fr is called Fatou set,

bd A(c) denotes the boundary of A(c)

! Jr is a chaotic fractal attractor

! Jr is a closed set ⇒ there are no isolated

points



5. Julia Sets

5.9


Rational Julia Sets

! If r has more attractive fixpoints a,b,c,... then

bdA(a) = Jr = bdA(b) = Jr = bdA(c) = ..., i.e the

boundaries of all areas of attraction coincide


Rational Julia Sets

! Dynamic behaviour of {C, r}:

! In areas of attraction: Orbits converge to the

attractive periodic point

! In the Julia set: Orbits are chaotic and stay in

the Julia set! Self similarity of Julia sets: Jr consist of

distorted (non linear) copies of the whole set



5. Julia Sets

5.10


Rational Julia Sets

! Newton’s iteration:

! Find the zero points of a differentiable

function f(z)

! z0 ... starting point

! zn+1 = zn - f(zn) / f’(zn) = N(zn)

! The zero points ni of f(z) are attracting

fixpoints of N(zn)


Rational Julia Sets


! z0∈ C: Does the sequence {Nn(z0)} converge

to one of the fixpoints of N(zn) ?

! If it does, which fixpoint is the limit ?

! How the areas of attraction look like ?! Example: f(z) = z3-1

2

3

2

3

3

12

3

1)(

n

n

n

n

nn

z

z

z

z z z N

+

=

−

−=



5. Julia Sets

5.11


Rational Julia Sets


! Example: f(z) = z3-1

Fixpoints: 1, (1,2π/3), (1,4π/3)

! JN = bd A(1) = bd A(1, 2π/3) = bd A(1, 4π/3) is

the set of points which do not converge to any

fixpoint of N(z)


Rational Julia Sets



5. Julia Sets

5.12


Rational Julia Sets



! Special case of {C, r}, r(z)=p(z)/q(z),

q(z) = 1, p(z) = z2+c, c ∈ C

! All properties of Julia sets Jr of {C, r} can be

transfered to the Julia sets Jc

of {C, z2+c}

! The Julia sets Jc are strongly related to the

Mandelbrot set



5. Julia Sets

5.13


Julia sets for {C, z2+c}

! Properties:

! Point ∞ is always a fixpoint of f c = z2+c

! Ac(∞) = {z ∈ C | lim f cn(z) = ∞ }

! Jc = bd Ac(∞), Kc = C \ (Ac(∞) ∪ Jc)

! Jc subdivides C into Ac(∞) and Kc ! If ∞ is the only fixpoint, then Kc = ∅

! f c

has at most one attractive periodic point


Visualization of Jc

! The pixel game:

! Operates on a M×N pixel raster that covers

an area of the complex plane, i.e. each pixel

(i,j) is associated with a complex number z ij

! Check for each pixel if zij belongs to Ac(∞)! If |f cn(z)| > max(|c|,2), 0 ≤ n ≤ nmax, then z

belongs to A*c(∞) and A*c(∞) ⊆ Ac(∞)

! The value n can be used to produce color

images



5. Julia Sets

5.14


Visualization of Jc

! The pixel game:

for(i=0; i<HEIGHT; i++)

for(j=0; j<WIDTH; j++)

{

z = point4pixel(i,j);

for(n=0, n<=NMAX; n++)

{

if(rad(z)> max(abs(c),2)) break;

z = z*z + c;}

if(n > NMAX) setPixel(i,j,black);

else setPixel(i,j,colortab[n%MAXCOL]);

}


Visualization of Jc

! The pixel game:

! Good approximation of Ac(∞) if n is large

! It is easy to generate color pictures

! Computation time depends on image

resolution and nmax! Cannot be used for rational Julia sets

! Color map can be interpreted as height field to

produce a 3D rendering



5. Julia Sets

5.15


Visualization of Jc

C = 0.12 - 0.6i


Visualization of Jc

C = 0.3+0.49i

C = -0.039 + 0.695i



5. Julia Sets

5.16


Visualization of Jc


C = -0.6 + 0.6i



5. Julia Sets

5.17


Visualization of Jc

3D-rendering: Ac(∞) as height fieldJulia set as height field


Visualization of Jc

! The stochastic method:

! Julia sets are attractors of the general IFS:

! Approximation of Jc by a random sequence of

points {z0, f r1(z0), f r2(z0),...,f r max(z0)},

r i ∈ {1, 2}

c z f c z f f f C −−=−= 2121 },,;{



5. Julia Sets

5.18


Visualization of Jc

! The stochastic method :

! f 1, f 2 non linear ⇒ uneven convergence

! Choice of the starting point ?

! The IFS has 2 fixpoints (z2-z+c=0), one is

attractive, the other repelling and lies in Jc,

⇒ take this one


Visualization of Jc

! Progress of the stochastic method

c = 0.12+0.74i



5. Julia Sets

5.19


Visualization of Jc

! Adaptive cut:

! Given:

! Using the Hutchinson operator W = f 1 ∪ f 2:

! Uneven convergence, high computation time

! Consider the local convergence to cut off the

transformation tree

c z f c z f f f C −−=−= 2121 },,;{

arbitrary I I W J n

nc )(lim

∞→

=


Visualization of Jc

! Progress of

adaptive cut

for c = -1:



5. Julia Sets

5.20


Visualization of Jc

! The stochastic method is fast and can be

used to obtain a preview

! The pixel game is the most flexible method

and gives the best results

! The stochastic method & the adaptive cut

method can be used for rational Julia sets as

well


Quaternion Julia Sets

! Ouaternions are extensions of complexnumbers:

! z = z0 + z1i + z2 j + z3k! z0 is the real part and

! z0 + z1i is the complex part of z! Almost all rules for R and C hold in thequaternion space H

! The only exception is the multiplication,which is not commutative



5. Julia Sets

5.21



! The Julia sets JHc for {H, z2+c } are 4D-objects

! The subset H0 = {z ∈ H | z3=0} is invariant

under z2+c

! Operating with H0 results in a 3D-Julia set,

which is a subset of the 4D-set



! Rendering is not trivial (sophisticated

visualization techniques required)

! Volume rendering to find the border of

attraction areas in 4D space

! The cross section with the complex plane is

the conventional Julia set Jc for {C, z2+c }



5. Julia Sets

5.22







5. Julia Sets

5.23





3D-cross section

of a 4D-Julia set

with cut open

2D-cross section

showing thecorresponding

Julia set of the

complex plane



5. Julia Sets



Quaternion Julia Sets - Links

! Keenan Crane:

! http://www.cs.caltech.edu/~keenan/

project_qjulia.html

! YouTube:! www.youtube.com/watch?v=gruJ0S3TTtI

! www.youtube.com/watch?v=VkmqT6MQoDE


Documents

Fractals -5- Dr Christoph Traxler.pdf