Chaos, Fractals and their Relevance to Understanding Nature Frederick H. Willeboordse.

Chaos, Fractals and their Relevance to Understanding

Nature

Frederick H. Willeboordse

Today's TalkIntroduction: Chaos and Fractals are ubiquitous

- a few examples

Relevance: What can we learn from the studyof Chaos

What are Chaos The Butterfly effectand Fractals: Self-similar geometric shapes

My Research: Coupled Maps, Recent Results

Conclusion: Relevance revisited, Chaos is fun!

IntroductionA few examples ..

The world is full of Chaos!

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

Well …

Introduction

However ..Are mountains chaotic?

Is a tree chaotic?

This wave is probably more obviously chaotic.

In general, Chaos, Fractals and Complexity are not the same. However, they not only often appear together, the study of one can also be very valuable for understanding the other.

Chaos? Fractal?

Introduction

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

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

Introduction

Fractals in Physics

Tin 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

Introduction

Fractals in Physics

Snowflake

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

Introduction

Chaos and Fractals in Physics

The red spot on Jupiter.

Can such a spot survive in a chaotic environment?

Can such a spot exist without Chaos?

Introduction

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.

Introduction

Chaos and Fractals in GeologySatellite Image of a River Delta

Introduction

Fractals in BiologyDelicious!

Broccoli Romanesco is a cross between Broccoli and Cauliflower.

Introduction

Fractals in Biology

The venous and arterial system of a kidney

Introduction

RelevanceWhat can we learn from the study of Chaos

When taking a very broad view, I believe that there are two main issues.

1. How can complex structures arise from simple recipes?

2. How can we understand that systems with simple underlying laws can be very complex?

One answer: Iteration

One answer: Butterfly effect

How can DNA make this?

Does a butterfly really cause a storm?

Relevance

How can the body make this?

It was recently found that humans only have about 30,000 genes. That is a stunningly small number. Imagine what a computer program with 30,000 lines can do (not much).

Perhaps, DNA is a recipe book.

If so, the understanding of the dynamics of simple recipes is of fundamental importance to the life sciences.

One could pose the question: Would we be alive without Chaos?

Relevance

Does a butterfly really cause a storm?

The dynamics of complex systems even when they are based on rather simple equations is … well .. complex.

How is this possible?

Gaining insight into the mechanism might be helpful for understanding how natural systems can efficiently find ‘unlikely’ solutions.

One could pose the question: Would we be here without Chaos?

Relevance

Chaos and Fractals in PaleontologyIs evolution a matter of Replicate and Modify?

Built from similar modified segments?

Relevance

What are Chaos and Fractals

Chaos Fractal

Chaos and Fractals

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 and Fractals

ChaosWhat is Chaos?

A croissant with how many layers can I make in 3 minutes?

Chaos and Fractals

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.

Chaos and Fractals

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

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 is this possible? As was shown 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.

Chaos and Fractals

Logistic Map

The logistic map can be defined as:

A deceptively simple equation.

Successive points can be found through iteration.

2n1n αx1x

Chaos and Fractals

Iteration

In math it means that one starts with a certain value, calculates the result and then use this result as the starting value of a next calculation.

2n1n αx1x

223 αx1x

212 αx1x

201 αx1x

given

Iteration is just like our Danish Pastry/Chinese Noodles example. A recipe is repeated over and over again.

Chaos and Fractals

Logistic MapThe so-called bifurcation diagram

Plot 200 successive values of xfor every value of a

As the nonlinearity increases we sometimes encounter chaos

2n1n αx1x

Chaos and Fractals

Logistic MapThe bifurcation diagram displays

unexpected properties.

Let's enlarge this area

Chaos and Fractals

Logistic MapAn almost identical diagram!

Let's try this again...

Chaos and Fractals

Logistic MapThis time, let us enlarge a much smaller area.

Now let's enlarge this area

Hard to see, isn't it?

Chaos and Fractals

Logistic MapA virtually identical diagram again!

Chaos and Fractals

Logistic MapIndeed, the logistic map repeats itself over and over again at ever smaller scales

What's more, this behaviour was found to be universal!

That is to say, a very large number of systems will behave in the same way as the logistic map. Therefore studying the ‘paradigm’ of the logistic map will provide insight into the dynamics of large class of possibly much more complicated systems.

Chaos and Fractals

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).

Chaos and 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.

Remove middle third

Then remove middle third of what remains

And so on ad infinitum

Chaos and Fractals

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

a1

s

a

D

# of pieces

Reduction factor

Dimension

Chaos and 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

D3/1

12

6309.03log/2log D

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

Chaos and 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).

Chaos and 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 surprisingly complex.

Chaos and Fractals


Next, zoom into thisArea.

Chaos and Fractals


Chaos and 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 Fractals

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.

Chaos and Fractals

My Research: Coupled Maps

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)

My Research

Note: This section is somewhat more technical. What I would like to illustrate is that a simple combination of simple elements can lead to rather surprising non-trivial dynamics.

Coupled Maps -What they are

The coupled map discussed here is simply an array of logistic maps. The formula appears more complicated than it is.

Or in other words:

f is the logistic mapf( ) f( ) f( )Time n

Time n+1

)()(2

)()1( 111

i

nin

in

in xfxfxfx

2

2

1

My Research

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.

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.

My Research

Pattern with Kinks

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

My Research

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.

My Research

Pattern Selection

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

My Research

Travelling Waves

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

My Research

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!

My Research

My quasi-logo

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

My Research

Recent Findings

My Research

I have recently found that many different types of coupled maps display identical pattern sequences. I have also found that the patterns scale (often linearly) when increasing the coupling range.

Long wave length solutions

Identical patterns for many models

My Research

This allowed me to introduce the following tent coupled map which is much simpler than the coupled map shown before.

in

in

in

in xxxx 11

1 1

It is hoped that this map will be useful in further analyzing the universality of the pattern sequences.

The tent coupled map could possibly also have an interpretation relevant to business:

My Research

in

in

in

in xxxx 11

1 1

0

~0 Maximum Growth

Fast Growth

Slow Contraction

~1 Maximum Growth

Growth Contraction

~2 Maximum Growth

Slow Growth

Fast Contraction

self

neighbors

11 in

in xx

inx

= other businesses

= own business

Here, -1 is the smallest possible size, 0 maximum fitness, and 1 the maximum size.

It would be interesting to see whether some aspects of the dynamics of this map can be found in the actual business world.

*xxi *xxi 11

*

21

1

in

in xx

x

Function, if 11 in

in xx

is set to a certain value

= critical size

Fractals and Financial Markets

Possible Extension of my Research

Early 1999, Mandelbrot (the father of fractals) wrote a somewhat provocative article in Scientific American in which he is not particularly kind to Modern Portfolio theory.

“The risk-reducing formulas behind portfolio theory rely on a number of demanding and ultimately unfounded premises.”

“Volatility – far from being a static entity to be ignored or easily compensated for – is at the very heart of what goes on in the financial markets.”



What I would like to illustrate is not whether he is justified in his statements but that the ideas emerging from the study of chaos and fractals can be of great value when trying to understand the dynamics of financial markets.

Mandelbrot uses the fractal at the right as an example of his approach.



The ‘volatility’ can be adjusted by modifying the shape of the generator.



Pick the Fake!

Depicted are 8 series. Three supposedly correspond to the Modern Portfolio Theory, and the rest is either extracted from real data or generated by a fractal model.



MPT – based on random walk

IBM

MPT – other random processes

MPT – fractal Brownian motion

US$ - DM rate

Fractal

Fractal

Fractal

MPT = Modern Portfolio Theory

Rea

lF

ake

Fak

e

Relevance Revisited and Conclusion

Simplicity and complexity, how do these two seemingly contradictory notions relate?

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

Chaos in Physics

Chaos provides conceptual insights.

In Physics, the relevance of Chaos is perhaps the most obvious. The butterfly effect has changed the way many perceive nature.

Conclusion

Chaos in the Life Sciences

Evolution

Evolution could be a matter of replication and modification, ergo an iterative process where the recipe to be repeated can change.

DNA could be a recipe book that provides guidance as to how cells should replicate and modify during the growth of an organism.

LifeIs 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?

Growth

Conclusion

Chaos and Philosophy

Good-bye?

Determinism

Classical (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 no freedom of the mind.

However, when combining Heisenberg’s uncertainty principle with the butterfly effect, physics may not imply determinism after all.

Conclusion

ConclusionMany things in our world are chaotic, yet order is also everywhere.

Understanding this dichotomy is not only a fabulous challenge but also has profound implications for how we perceive the world.

The study of Chaos has implications for probably all academic fields.

Conclusion

Thank you!

More Coffee!

(Scientists at work )

Chaos, Fractals and their Relevance to Understanding Nature Frederick H. Willeboordse.

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