fractals and chaos theoryies

8/4/2019 Fractals and Chaos Theoryies

1/40

Fractals and Chaos

Theory

Ruslan Kazantsev

Rovaniemi Polytechnic, Finland


2/40

Chaos Theory about disorder

NOT denying of determinism

NOT denying of ordered systems

NOT announcement about useless ofcomplicated systems

Chaos is main point of order


3/40

What is the chaos theory?

Learning about complicated nonlineardynamic systems

Nonlinear recursion and algorithms

Dynamic variable and noncyclic


4/40

Wrong interpretations

Society drew attention to the chaostheory because of such movies asJurassic Park. And because of such

things people are increasing the fearof chaos theory.

Because of it appeared a lot of wrong

interpretations of chaos theory


5/40

Chaos Theory about disorder

Truth that small changes could givehuge consequences.

Concept: impossible to find exactprediction of condition, but it givesgeneral condition of system

Task is in modeling the system basedon behavior of similar systems.


6/40

Usage of Chaos Theory

Useful to have a look to thingshappening in the world differentfrom traditional view

Instead of X-Y graph -> phase-spatialdiagrams

Instead of exact position of point ->

general condition of system


7/40

Usage of Chaos Theory

Simulation of biological systems (most chaoticsystems in the world)

Systems of dynamic equations were used forsimulating everything from population growth

and epidemics to arrhythmic heart beating Every system could be simulated: stock

exchange, even drops falling from the pipe

Fractal archivation claims in future coefficient

of compression 600:1 Movie industry couldnt have realistic

landscapes (clouds, rocks, shadows) withouttechnology of fractal graphics


8/40

Brownian motion and its

adaptation Brownian motion for example accidental

and chaotic motion of dust particles,weighted in water.

Output: frequency diagram

Could be transformed in music

Could be used for landscapecreating


9/40

Motion of billiard ball

The slightestmistake in angleof first kick willfollow to huge

disposition afterfew collisions.

Impossible to

predict after 6-7hits

Only way is toshow angle and

length to each hit


10/40

Motion of billiard ball

Every single loop ordispersion areapresents ball behavior

Area of picture, where

are results of oneexperiment is calledattraction area.

This self-similarity will

last forever, if enlargepicture for long, wellstill have same forms.=> this will be

FRACTAL


11/40

Fusion of determined fractals

Fractals are predictable.

Fractals are made with aim to predictsystems in nature (for example migrationof birds)


12/40

Tree simulation using Brownian motionand fractal called Pythagor Tree

Order of leaves andbranches is

complicated andrandom, BUT can beemulated by shortprogram of 12 rows.

Firstly, we need togenerate PythagorTree.


13/40


On this stageBrownian motion isnot used.

Now, every sectionis the centre ofsymmetry

Instead of lines are

rectangles. But it still looks like

artificial


14/40


Now Brownianmotion is usedto makerandomization

Numbers arerounded-up to 2

rank instead of39


15/40


Rounded-up to 7

rank

Now it looks like

logarithmic spiral.


16/40


To avoid spiral weuse Brownian

motion twice to theleft and only once tothe right

Now numbers are

rounded-up to 24rank


17/40

Fractals and world around

Branching, leaves on trees, veins in hand,curving river, stock exchange all thesethings are fractals.

Programmers and IT specialists go crazywith fractals. Because, in spite of itsbeauty and complexity, they can begenerated with easy formulas.

Discovery of fractals was discovery of newart aesthetics, science and math, and alsorevolution in humans world perception.


18/40

What are fractals in reality?

Fractal geometric figure definitepart of which is repeating changingits size => principle of self-similarity.

There are a lot of types of fractalsNot just complicated figures

generated by computers.

Almost everything which seems to becasual could be fractal, even cloud orlittle molecule of oxygen.


19/40

How chaos is chaotic?

Fractals part of chaos theory.

Chaotic behaviour, so they seemdisorderly and casual.

A lot of aspects of self-similarity insidefractal.

Aim of studying fractals and chaos topredict regularity in systems, whichmight be absolutely chaotic.

All world around is fractal-like


20/40


21/40

Practical usage of fractals

Computer systems (Fractal archivation, picturecompressing without pixelization)

Liquid mechanics Modulating of turbulent stream

Modulating of tongues of flame

Porous material has fractal structure Telecommunications (antennas have fractal form)

Surface physics (for description of surface curvature)

Medicine

Biosensor interaction Heart beating

Biology (description of population model)


22/40

Fractal dimension: hidden dimensions

Mandelbrot called not intactdimensions fractal dimensions (forexample 2.76)

Euclid geometry claims that space isstraight and flat.

Object which has 3 dimensionscorrectly is impossible

Examples: Great Britain coastline,human body


23/40

Deterministic fractals

First opened fractals.

Self-similarity because of method ofgeneration

Classic fractals, geometric fractals, linearfractals

Creation starts from initiator basicpicture

Process of iteration adding basic pictureto every result


24/40

Sierpinskij lattice

Triangles made ofinterconnection of middlepoints of large trianglecut from main triangle,generating triangle with

large amount of holes.

Initiator large triangle.

Generator process ofcutting triangles similar

to given triangle.

Fractal dimension is1.584962501


25/40

Sierpinskij sponge

Plane fractal cellwithout square, butwith unlimited ties

Would be used as

buildingconstructions


26/40

Sierpinskij fractal

Dont mix up thisfractal withSierpinskij lattice.

Initiator andgenerator are thesame.



27/40

Koch Curve

One of the most typical fractals.

Invented by german mathematic Helge fonKoch

Initiator straight line. Generator equilateral triangle.

Mandelbrot was making experiments with

Koch Curve and had as a result KochIslands, Koch Crosses, Koch Crystals, andalso Koch Curve in 3D

Fractal dimension is 1.261859507


28/40

Mandelbrot fractal

Variant of Koch Curve

Initiator andgenerator are

different from Kochs,but idea is still thesame.

Fractal takes half ofplane.



29/40

Snow Crystal and Star

This objects are classical fractals. Initiator and generator is one figure


30/40

Minkovskij sausage

Inventor is GermanMinkovskij.

Initiator and generator arequite sophisticated, aremade of row of straight

corners and segments withdifferent length.

Initiator has 8 parts.

Fractal dimension is 1.5


31/40

Labyrinth

Sometimes called H-tree.

Initiator and generator hasshape of letter H

To see it easier the H form isnot painted in the picture.

Because of changing

thickness, dimension on thetip is 2.0, but elementsbetween tips it is changingfrom 1.333 to 1.6667


32/40

Darer pentagon

Pentagon asinitiator

Isosceles triangle

as generator Hexagon is a

variant of thisfractal (David Star)

Fractal dimensionis 1.86171


33/40

Dragon curve

Invented by Italianmathematic GiuseppePiano.

Looks like Minkovskijsausage, because hasthe same generatorand easier initiator.

Mandelbrot called itRiver of DoubleDragon.

Fractal dimension is

1.5236


34/40

Hilbert curve

Looks like labyrinth,but letter U is usedand width is not

changing. Fractal dimension is

2.0

Endless iterationcould take all plane.


35/40

Box

Very simple fractal

Made by addingsquares to the top

of other squares. Initiator and

generator andsquares.

Fractal dimensionis 1.892789261


36/40

Sophisticated fractals

Most fractals which you can meet ina real life are not deterministic.

Not linear and not compiled fromperiodic geometrical forms.

Practically even enlarged part ofsophisticated fractal is different from

initial fractal. They looks the samebut not almost identical.


37/40

Sophisticated fractals

Are generated by nonlinear algebraicequations.

Zn+1=Zn + C

Solution involvescomplex and supposednumbers

Self-similarity ondifferent scale levels

Stable results black,for different speeddifferent color


38/40

Mandelbrot multitude

Most widespreadsophisticated fractal

Zn+1=Zna+C

Z and C complexnumbers

a any positive

number.


39/40

Mandelbrot multitude

Z=Z*tg(Z+C).

Because of Tangentfunction it looks like

Apple. If we switch Cosine it

will look like AirBubbles.

So there are differentproperties forMandelbrot multitude.


40/40

Zhulia multitude Has the same

formula asMandelbrotmultitude.

If building fractal

with different initialpoints, we will havedifferent pictures.

Every dot inMandelbrot

multitudecorresponds toZhulia multitude

fractals and chaos theoryies

Documents