chaos and fractals

Chaos and Fractals

Frederick H. Willeboordsehttp://staff.science.nus.edu.sg/~

frederik

SP2171 Lecture Series: Symposium III, February 6, 2001

http://staff.science.nus.edu.sg/~frederik








Today's ProgramPart 1: Chaos and Fractals are ubiquitous

- a few examples

Part 2: Understanding Chaos and Fractals- Some Theory- Some Hands-on Applications

Part 3: Food for Thought

Part 1: A few examples ..The world is full of Chaos and Fractals!

The weather can be: ...chaoticThe ocean can be: ...chaoticOur lives can be: ...chaoticOur brains can be: ...chaoticOur coffee can be: ...chaotic

Well …

Chaos and Fractals in PhysicsThe motion of the planets is chaotic. In fact, even the sun, earth moon system cannot be solved analytically!

In fact, the roots of Chaos theory go back to Poincare who discovered ‘strange’ propertied when trying to solve the sun, earth moon system at the end of the 19th century

Chaos and Fractals in PhysicsTin Crystals

Molten tin solidifies in a pattern of tree-shaped crystals called dendrites as it cools under controlled circumstances.

From: Tipler, Physics for Scientists and Engineers, 4th Edition

Chaos and Fractals in PhysicsSnowflake

The hexagonal symmetry of a snowflake arises from a hexagonal symmetry in its lattice of hydrogen and oxygen atoms.

From: Tipler, Physics for Scientists and Engineers, 4th Edition

A nice example of how a simple underlying symmetry can lead to a complex structure

Chaos and Fractals in PhysicsThe red spot on Jupiter.

Can such a spot survive in a chaotic environment?

Chaos in and Fractals PhysicsAn experiment by Swinney et al

One of the great successes of experimental chaos studies.

A spot is reproduced.

Note: these are false colors.

Chaos and Fractals in ChemistryBeluzov-Zhabotinski reaction

Waves representing the concentration of a certain chemical(s).These can assume many patterns and can also be chaotic

Chaos and Fractals in GeologySatellite Image of a River Delta

Chaos and Fractals in BiologyDelicious!

Broccoli Romanesco is a cross between Broccoli and Cauliflower.

Chaos and Fractals in BiologyBroccoli Romanesco

Chaos and Fractals in BiologyWould we be alive without Chaos?

The venous and arterial system of a kidney

Chaos and Fractals in PaleontologyWould we be here without Chaos?

Evolutionary trees as cones of increasing diversity. From ‘Wonderful Life’ by Stephen Jay Gould who disagrees with this picture (that doesn’t matter as with regards to illustrating our point).

Chaos and Fractals in PaleontologyReplicate and Modify

Built from similar modified segments?

Chaos and Fractals in PaleontologyWould we be alive without Chaos?

Is there a relation to stretch and fold?

Simple? Complex?

Simple

Complex

The phenomena mentioned on the previous slide are very if not extremely complex. How can we ever understand them?

Chaos and Fractals can be generated with what appear to be almost trivial mathematical formulas…

Try to write anequation for this.

xn11 xn2

You could have done this in JCRight!??

Part 2:

Understanding Chaos and FractalsIn order to understand what’s going on, let us have a very brief look at what Chaos and Fractals are.

Chaos Fractal

Chaos

Are chaotic systems always chaotic?

What is Chaos?

Chaos is often a more ‘catchy’ name for non-linear dynamics.

No! Generally speaking, many researchers will call a system chaotic if it can be chaotic for certain parameters.

Dynamics = (roughly) the time evolutionof a system.

Non-linear = (roughly) the graphof the function is not a straight line.

Parameter = (roughly) a constant in an equation. E.g. the slope ofa line. This parameter can be adjusted.

Chaos

Chaos

Try it!

What is Chaos?

Quiz: Can I make a croissant with more than 15’000layers in 3 minutes?

Chaos

Chaos

Sensitive dependence on initial conditions

What is Chaos?

The key to understanding Chaos is the concept of stretch and fold. Or … Danish Pastry/Chinese Noodles

Two close by points always separate yet stay in the same volume. Inside a layer, two points will separate, but, due the folding, when cutting through layers, they will also stay close.

Quiz-answer: Can I make a croissant with more than 15’000layers in 3 minutes? – Yes: stretch and fold! Or perhaps I should say kneed and roll .

Chaos

The Butterfly EffectSensitive dependence on initial conditions is what gave the world the butterfly effect.

Chaos

The Butterfly EffectSensitive dependence on initial conditions is what gave the world the butterfly effect.

Chaos

The butterfly effect describes the notion that the flapping of the wings of a butterfly can ‘cause’ a typhoon at the other side of the world.

How? We saw with the stretch and fold Chinese Noodle/Danish Pastry example, where the distance between two points doubles each time, that a small distance/difference can grow extremely quickly.

Due to the sensitive dependence on initial conditions in non-linear systems (of which the weather is one), the small disturbance caused by the butterfly (where we consider the disturbance to be the difference with the ‘no-butterfly’ situation) in a similar way can grow to become a storm.

Logistic MapThe logistic map can be defined as:

xn11 xn2

Looks simple enough to me! What could be difficult about this?Let's see what happens when we increase the parameter alpha from 0 to 2.

Chaos

IterationIteration is just like our Danish Pastry/Chinese Noodles.

In math it means that you start with a certain value (given by you) calculate the result and then use this result as the starting value of a next calculation.

2n1n αx1x

223 αx1x

212 αx1x

201 αx1x

given

Chaos

Logistic MapThe so-called bifurcation diagram

xn11 xn2

Plot 200 successive values of xfor every value of As the nonlinearity

increases we sometimes encounter chaos

Chaos

Logistic MapWhat's so special about this?Let's have a closer look.

Let's enlarge this area

Chaos

Logistic MapHey! This looks almost the same!

Let's try this again...

Chaos

Logistic MapLet's enlarge a much smaller area!

Now let's enlarge this area

Hard to see, isn't it?

Chaos

Logistic MapThe same again!

Chaos

Logistic MapIndeed, the logmap repeats itself over and over again at ever smaller scalesWhat's more, this behaviour was found to be universal!

Yes, there's a fractal hidden in here.

Chaos

Chaos and RandomnessChaos is NOT randomness though it can look pretty random.

Chaos

Let us have a look at two time series:

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 50 100 150 200 250 300 350 400-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 50 100 150 200 250 300 350 400

And analyze these with some standard methods

Data: Dr. C. Ting

Chaos and RandomnessChaos

Power spectra

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 50 100 150 200 250 300 350 400

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 50 100 150 200 250 300 350 400

0 1000 2000 3000 4000 500015

10

5

0

5

frequency

PowerSpectrum

0 1000 2000 3000 4000 500020

15

10

5

0

5

frequency

PowerSpectrum

No qualitativedifferences!


Histograms

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 50 100 150 200 250 300 350 400

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 50 100 150 200 250 300 350 400



Autocorrelation Functions

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 50 100 150 200 250 300 350 400

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 50 100 150 200 250 300 350 400

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0 20 40 60 80 100 120 140

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0 20 40 60 80 100 120 140


Chaos and RandomnessWell these two look pretty much the same.

Chaos

Let us consider 4 options:

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 50 100 150 200 250 300 350 400

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 50 100 150 200 250 300 350 400

What do you think?

A: They are both ChaoticB: Red is Chaotic and Blue is RandomC: Blue is Random and Red is ChaoticD: They are both Random

Chaotic??

Random???

Chaotic?? Chaotic??

Chaotic??Chaotic??Random???

Random???

Random???


Return map (plot xn+1 versus xn )

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 50 100 150 200 250 300 350 400

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 50 100 150 200 250 300 350 400

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

xn+1 = 1.4 - x2n + 0.3 yn

yn+1 = xn

White Noise

Henon Map

Deterministic

Non-Deterministic

Red is Chaotic and Blue is Random! As we can see from the return map.

FractalsWhat are Fractals?

(roughly) a fractal is a self-similar geometrical object with a fractal dimension.

self-similar = when you look at a part, it just looks like the whole.

Fractal dimension = the dimension of the object is not an integer like1 or 2, but something like 0.63. (we’ll get back to what this means alittle later).

Fractals

FractalsThe Cantor Set

Take a line and remove the middle third, repeat this ad infinitum for the resulting lines.

This is the construction of theset!The set itself is the result ofthis construction.

Fractals

Remove middle third

Then remove middle third of what remains

And so on ad infinitum

FractalsFractal Dimension

Let us first look at a regular line and a regular square and see what happens when we copy the these and then paste them at 1/3 of their original size.

We see that our original objectcontains 3 of the reduced pieces.

We see that our original objectcontains 9 of the reduced pieces.

Apparently, we have:Ds

a 1

sa

D

# of pieces

Reduction factor

Dimension

Fractals

FractalsFractal Dimension

Now let us look at the Cantor set:

This time we see that our original objectcontains only 2 of the reduced pieces!

If we fill this into ourformula we obtain:

Dsa 1

OriginalReduced Copy

D3/112

6309.03log/2log D

A fractal dimension.. Strictly, this just one of several fractal dimensions, namely the self-similarity dimension.

Fractals

FractalsThe Mandelbrot Set

This set is defined as the collection of points c in the complex plane that does not escape to infinity for the equation:

czz nn 2

1

Note: The actual Mandelbrot set are just the black points in the middle!All the colored points escape (but after different numbers of iterations).

Fractals


Does this look like the logistic map? It should!!!

111 21 nn z

cz

cTake z to be real, divide both sides by c,

czz nn 2

1

then substitute nn zc

x 1 12

1 nn cxx

Defining: c

,

to obtain .

we find the logistic map from before2n1n αx1x

Fractals


The Madelbrot set is strictly speaking not self-similar in the same way as the Cantor set. It is quasi-self-similar (the copies of the whole are not exactly the same).

Here are some nice pictures from:http://www.geocities.com/CapeCanaveral/2854/

What I’d like to illustrate here is not so much that fractals can be used to generate beautiful pictures, but that a simple non-linear equation can be incredibly complex.

Fractals


Next, zoom into thisArea.

Fractals


Next, zoom into thisArea.


Fractals

Chaos and FractalsHow do they relate?

Fractals often occur in chaotic systems but the the two are not the same! Neither of they necessarily imply each other.

A fractal is a geometric object

Roughly:

Chaos is a dynamical attribute

Let us have a look at the logistic map again.

Chaos and FractalsHow do they relate? -> Not directly!

In the vertical direction we have the points on the orbit for a certain value of .

This orbit is chaotic, but if we look at the distribution, it is definitively not fractal. It approximately looks like this

-1 value of x +1

probability

Self Similar, adinfinitum. This can be used to generate a fractal.

Coupled Maps -Why on 'earth'Universality

Simplicity

The logistic map has shown us the power of universality. It is hoped that this universality is also relevant for Coupled Maps.Coupled Maps are the simplest spatially extended chaotic system with a continuous state (x-value)

A short detour into my research.Chaos

Coupled Maps -What they areThe coupled map discussed here is simply an array of logistic maps. The formula appears more complicated than it is.

Or in other words:

xn i 1 f xn i 2

f xn i1 f xn i1

f is the logistic map1

2

2

f( ) f( ) f( )Time n

Time n+1

Chaos

Coupled Maps -Phenomenology

Patterns with KinksFrozen Random PatternsPattern SelectionTravelling WavesSpatio-temporal Chaos

Even though coupled maps are conceptually very simple, they display a stunning variety of phenomena.

Chaos

Coupled Map have so-called Universality classes. It is hoped that these either represent essential real world phenomena or that they can lead us to a deeper understanding of real world phenomena.

Pattern with Kinks

No Chaos: lattice sites are attracted to the periodic orbits of thesingle logistic map.

Chaos

Frozen Random Pattern

Parts of the lattice are chaotic and parts of the lattice are periodic.The dynamics is dominated by the band structure of the logisticmap.

Chaos

Pattern Selection

Even though the nonlinearity has increased and the logistic map ischaotic for , the lattice is entirely periodic.

Chaos

Travelling Waves

The coupled map lattice is symmetric, yet here we see atravelling wave. This dynamical behaviour is highly non-trivial!

Chaos

Spatio-Temporal Chaos

Of course we have spatio-temporal chaos too. No order to be foundhere ... or ??? . No, despite the way it looks, this is far from random!

Chaos

My quasi-logoNow we can guess what it means

The logistic map, the building block of the coupled map lattice

The bifurcation diagram, the source of complexity

A coupled map lattice with travelling domain walls, chaos and orderly waves

The strength of the non-linearity

The strength of the coupling

Chaos

Simple? Complex?

Universality

How do these two seemingly contradictory aspects relate?

The study of Chaos shows that simplicity and complexity can be related by considering universal properties of simple iterative processes.

It was discovered that certain essential properties of chaotic systems are universal. This allows us to study a simple system and draw conclusion for a complex system

We can now come back to the question posed previously

IterationRepeat a (simple but non-linear) recipe over and over again

Understanding Chaos and Fractals

Some applications/hands-on demonstrations

Chinese Noodles

Double Pendulum

Video Feedback

Part 3:

Food for Thought

More Coffee!

(Scientists at work )

Chaos in BiologyEvolution?

Replicate and Modify – What does that mean?

Is the human body a fractal?

Could it be that fractals hold the key to how so much information can be stored in DNA?

Life?

Is the fact that Chaos ‘can’ look like randomness essential for life?Could it be that chaos is an ‘optimization algorithm’ for life in an unstable environment?

Chaos in Physics

But! Quantum Mechanics is a linear science!

Stunningly common at the macroscopic level

It was discovered that certain essential properties of chaotic systems are universal. This allows us to study a simple system and draw conclusion for a complex system

linear

non-linear

Chaos in MeteorologyWhat are the implications of the butterfly effect for the prediction of weather?

Sunny?

Rainy?

Perhaps it’s up in the clouds

Chaos and Philosophy

Good-bye?

DeterminismClassical (and in a sense also Quantum Physics) seems to imply that the world is deterministic. If we just had the super-equation, we could predict the future exactly. In essence, there is not freedom of the mind.

Can non-linear science contribute to the discussion on determinism versus free will?

Conclusion

Almost everything in our world is chaotic, yet order is also everywhere.Understanding this dichotomy is a fabulous challenge.The study of Chaos can help us on our way.

Chaos is fun!

chaos and fractals

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