fractals -6- dr christoph traxler.pdf

6. The Mandelbrot Set

6.1

Christoph Traxler 1

The Mandelbrot Set

! King of fractals

! Most beautiful mathematical monster

1

0

-1 -2 -1 0

Im

Re

The Mandelbrot Set

Christoph Traxler 2


6.2

Christoph Traxler 3

The Mandelbrot Set

Christoph Traxler 4

The Mandelbrot Set


6.3

Christoph Traxler 5

The Mandelbrot Set

Christoph Traxler 6

Ordering the Julia Sets Jc

! For each c of fc = z2+c there exists a particular Julia set Jc

! The set of all Julia sets is uncountable infinite ! How can Jc be classified with respect to c ?


6.4

Christoph Traxler 7


! Mandelbrot’s idea (1979): Create a map of the complex plane, which shows information about the Julia sets for each complex number c

! What is the classification criterion ?

Christoph Traxler 8

Jc is connected Jc is totally disconnected (Cantor dust)


! Dichotomy of Julia sets: Julia sets can be seperated into two categories:


6.5

Christoph Traxler 9


! Visualize the area of points for which the corresponding Julia set is connected

! Def.: The set M = {c ∈ C | Jc is a connected set }

is called Mandelbrot set ! Fact: Jc is connected if and only if the critical

orbit 0 → c → c2+c →... is bounded

Christoph Traxler 10

Critical Point of Jc

! Def.: If fc’(z) = 0, then z is called critical point ! fc’(z) = 2z ⇒ z = 0 is the critical point

for any Jc

! The orbit of the critical point z=0 is called critical orbit

! If and only if the critical orbit escapes to ∞ then Jc is Cantor dust


6.6


Calculating the Map

! Calculate the fate of the critical point z = 0 ! Iterate z → z2+c, z0 = 0 ! The orbit is not bounded and escapes to ∞ if |

z| > 2 ⇒ Jc is totally disconnected ⇒ c ∉ M ! Approximation of M with the pixel game,

accuracy depends on the number of iterations ! Visualization of the equipotentials of A(∞)


Calculating the Map

! The pixel game algorithm for M:

for(i=0; i<HEIGHT; i++) for(j=0; j<WIDTH; j++) { c = point4pixel(i,j); for(n=0,z=0; n<=NMAX; n++) { if(rad(z)> 2) break; z = z*z + c; } if(n > NMAX) setPixel(i,j,black); else setPixel(i,j,getCol(n)); }


6.7


Difference between Jc and M

! Julia set ! fc(z) = z2 + c ! c is a constant ! z is variable ! examine the orbit

z→fc(z)→fc2(z) →... for each point z

! Mandelbrot set ! f(z) = z2 + c ! c is variable ! z0 = 0 is constant ! examine the orbit

0→f(0)→f2 (0) →... for each c

First Mandelbrot Print



6.8


Zoom Sequence


Zoom Sequence


6.9

Zoom Sequence


Zoom Sequence


Seahorse Valley


6.10

Zoom Sequence

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

v=x6DD1k4BAUg&feature=related ! www.youtube.com/watch?

v=0jGaio87u3A&feature=related



Encirclement of Jc

! Approximation of Jc with deformed discs ! Start with a disc Sr that surrounds Jc and

calculate the sequence: Sk = { z | fc(z) ∈ Sk-1} ,with S0 = Sr

! This sequence of nested deformed discs converges to Jc

! The deformed discs correspond to equipotential curves of A(∞)


6.11


Encirclement of Jc

! One of the following conditions is true:

A) The whole sequence Sn , n → ∞, consits of Jordan curves (disc like shapes)

B) S0 ,...,Sk are Jordan curves but not the rest of the sequence Sk+1,...


Encirclement of Jc

! If Sk is a Jordan curve, then there are 3 possible cases for Sk+1 : 1) Sk+1 is also a Jordan curve and the

critical point 0 and point c are in the interior of Sk+1 , condition A is true

Sk+1

0


6.12


Encirclement of Jc

0

Sk+1

Sk+1 0

� 2) Sk+1 is made up of two Jordan curves, which touch exactly at the critical point, condition B is true

3) Sk+1 is made up of two disjoint Jordan curves and does not contain the critical point, condition B is true


Encirclement of Jc

! If condition A is true, then ! Kc contains the critical point and c (their orbits

don’t escape to ∞ ) ! Jc is connected

! If condition B is true, then ! Kc neither contains the critical point nor c

(their orbits escape to ∞ ) ! Jc is disconnected


6.13


Encirclement of Jc

Condition A is true Condition B is true


Equivalent Definitions of M

! The Mandelbrot set M can be defined in various ways: ! M = { c ∈ C | Jc is connected } = ! { c ∈ C | c ∈ Kc } = ! { c ∈ C | critical point 0 ∈ Kc } = ! { c ∈ C | critical point 0 ∉ A(∞) } = ! { c ∈ C | |fn(c)| ≤ 2 ∀ n = 0,1,2,... }


6.14


Level Sets

! The encirclement of Jc is a decomposition of A(∞) into subsets of points which converge to ∞ with the same speed

! These subsets are called level sets ! Each area of attraction of a periodic point can

be decomposed in this way ! Level sets visualize the behaviour of points

under iteration in an area of attraction


Level Sets

! Decomposition of an attraction area A(p) of an attractive periodic point p into a level set. ! Choose a target set T which contains p ! The level of each point z ∈ C is defined as:

lc(z) = k, if fci(z) ∉ T, i<k, and fck(z) ∈ T lc(z) = 0, else

! Level set Lk = { z | lc(z) = k }, k is the convergence speed of z


6.15


Level Sets

! Example: Julia set, Mandelbrot set, p = ∞, T = { z | |z| ≥1/ε }, ε very small


Level Sets

! Target set T arbitrary, for example: {c,z2}, ! A(0) is the unit circle, decomposition of A(0)

with T = Kc, c ∈ M


6.16


Level Sets

! Level sets as height fields


Level Sets



6.17

Level Sets




6.18


Level Sets



Binary Decomposition

! Extension of the level set method ! Subdivision of the target set into n parts

induces a subdivision of the level Lk into nk+1

parts ! Iteration of fc(z), z = r(cosϕ + isinϕ) doubles

the angle ϕ ⇒ a loop of a closed curve in Lk corresponds to nk+1 loops in T

! The borders of the subdivided parts are good approximations of field lines


6.19



! For each point z ∈ Lk : ! Draw a

black point if 0 ≤ angle(fck

(z)) ≤ π ! Draw a white

point else

T

L1




6.20




Properties of M

! M is connected ! M contains infinite many copies of itself ! These copies are connected with the main

body by strings ! The fractal dimension of M is 2 ! M is a “table of content” for all connected Julia

sets Jc


6.21


Properties of M


Components of M

! Cardioid - heart shaped region ! Buds - circle like shapes connected to the

cardioid or other buds ! Two components are connected by only one

point, called seed point ! Copies of M !   Buds correspond to Julia sets that bound the

area of attraction of a periodic point with a specific period


6.22

Components of M


Cardioid

Buds

Largest copy of M Seed points

1 2

3

3

4

5

5


P�

Classes of Jc with Respect to M

! If c is a point of the cardioid then Jc is a Jordan curve with a fixpoint (period = 1)


6.23



! If c is a seed point then P is an indifferent periodic point

P�

! If c lies on the border of M and is no seed point then P is an indifferent periodic point in the Siegel disc

! Points of Kc rotate on the Siegel disc arround P



P

Siegel disc


6.24



! If c lies on a connecting string then ∞ is the only fixpoint of Jc , Kc = ∅

! This class of Julia sets is called dendrites



! If c lies in a copy of M then Jc is a ombination of the Jc1 of the connecting string and the Jc2 of the main body where c2 has the same relative position in the main body


6.25



! If c lies outside of M then Jc is totally disconnected (Cantor dust)

! The density of the points decreases with increasing distance to M


Similarity Relation

! M has a similar structure in the environment of c as the corresponding Jc when magnified at the same position


6.26


Structural Stability of M

! Example: {C, rc} rc = ((z2+c-1) / (2z+c-2 ))2 ! rc describes magnetic phase transitions ! Attractive fixpoints: 1, ∞ ! Examine the orbit 0→ rc(c)→ rc

2(c)→ ... for all c and draw a map

What does M there ?



! Answer: rc has a local behaviour like fc =z2+c ! M is universal, - it can be discovered in

several dynamic systems ! Given {C, fp}, f arbitrary, p ∈ C → the map of

the parameter space contains a (maybe deformed) copy of M with high probability


6.27




On the Web

! Applet for Mandelbrot exploration: ! users.erols.com/ziring/mandel_applet.html

! Applet for Mandelbrot-Julia generator: ! math.bu.edu/DYSYS/applets/

JuliaIteration.html


6.28

Quest for 3D Mandelbrot Set

! Does not exist from a strict mathematical point of view ! There is no Gaussian volume ! Dichotomy of Quaternion Julia sets

determined by complex part alone ! z = z0 + z1i + z2j + z3k

! There is no subset of the Hamilton space that corresponds to connected Quaternion Julia sets


Quest for 3D Mandelbrot Set

! Generating height fields or some kind of rotational sweeps does not count

! Search for a non linear fractal that expands into 3D space and is somehow related to the Mandelbrot set. ! Fractal details along all axes

! Best approach: Mandelbulb (Daniel White) ! Using spherical coordinates (r, α, β) ! Higher order function: z8+c ! {x,y,z}n = rn{cos(θ)cos(φ),sin(θ)cos(φ),-sin(φ)}



6.29

Mandelbulb


Mandelbulb



6.30




6.31




6.32

Mandelbulb

! Website: ! www.skytopia.com/project/fractal/mandelbulb.html


fractals -6- dr christoph traxler.pdf

Documents