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

2. Today's Talk Introduction: Chaos and Fractals are ubiquitous - a few examples Relevance:What can we learn from the study of Chaos What are ChaosThe Butterfly effect and Fractals:Self-similar geometric shapes My Research:Coupled Maps, Recent Results Conclusion:Relevance revisited, Chaos is fun!

3. Introduction A few examples.. The world is full of Chaos! The weather can be:...chaotic The ocean can be:...chaotic Our lives can be:...chaotic Our brains can be:...chaotic Our coffee can be:...chaotic Well … Introduction

4. 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

5. Chaos in Physics The 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 19 th century. Introduction

6. 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, 4 th Edition Introduction

7. 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, 4 th Edition A nice example of how a simple underlying symmetry can lead to a complex structure Introduction

8. 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

9. Chaos in and Fractals Physics An experiment by Swinney et al One of the great successes of experimental chaos studies. A spot is reproduced. Note: these are false colors. Introduction

10. Chaos and Fractals in Geology Satellite Image of a River Delta Introduction

11. Fractals in Biology Delicious! Broccoli Romanesco is a cross between Broccoli and Cauliflower. Introduction

12. Fractals in Biology The venous and arterial system of a kidney Introduction

13. Relevance What 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

14. 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

15. 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

16. Chaos and Fractals in Paleontology Is evolution a matter of Replicate and Modify? Built from similar modified segments? Relevance

17. What are Chaos and Fractals Chaos Fractal Chaos and Fractals

18. 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 evolution of a system. Non-linear = (roughly) the graph of the function is not a straight line. Parameter = (roughly) a constant in an equation. E.g. the slope of a line. This parameter can be adjusted. Chaos and Fractals

19. Chaos What is Chaos? A croissant with how many layers can I make in 3 minutes? Chaos and Fractals

20. 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

21. The Butterfly Effect Sensitive 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

22. Logistic Map The logistic map can be defined as: A deceptively simple equation. Successive points can be found through iteration. Chaos and Fractals

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

24. Logistic Map The so-called bifurcation diagram Plot 200 successive values of x for every value of a As the nonlinearity increases we sometimes encounter chaos Chaos and Fractals

25. Logistic Map The bifurcation diagram displays unexpected properties. Let's enlarge this area Chaos and Fractals

26. Logistic Map An almost identical diagram! Let's try this again... Chaos and Fractals

27. Logistic Map This time, let us enlarge a much smaller area. Now let's enlarge this area Hard to see, isn't it? Chaos and Fractals

28. Logistic Map A virtually identical diagram again! Chaos and Fractals

29. Logistic Map Indeed, 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

30. Fractals What 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 like 1 or 2, but something like 0.63. (we’ll get back to what this means a little later). Chaos and Fractals

31. Fractals The Cantor Set Take a line and remove the middle third, repeat this ad infinitum for the resulting lines. This is the construction of the set! The set itself is the result of this construction. Remove middle third Then remove middle third of what remains And so on ad infinitum Chaos and Fractals

32. Fractals Fractal 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 object contains 3 of the reduced pieces. We see that our original object contains 9 of the reduced pieces. Apparently, we have: # of pieces Reduction factor Dimension Chaos and Fractals

33. Fractals Fractal Dimension Now let us look at the Cantor set: This time we see that our original object contains only 2 of the reduced pieces! If we fill this into our formula we obtain: Original Reduced Copy A fractal dimension.. Strictly, this just one of several fractal dimensions, namely the self-similarity dimension. Chaos and Fractals

34. Fractals The 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: 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

35. Fractals The Mandelbrot Set 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

36. Fractals The Mandelbrot Set Next, zoom into this Area. Chaos and Fractals

37. Fractals The Mandelbrot Set Next, zoom into this Area. Chaos and Fractals

38. Fractals The Mandelbrot Set Next, zoom into this Area. Chaos and Fractals

39. Fractals The Mandelbrot Set Chaos and Fractals

40. How 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

41. How 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, ad infinitum. This can be used to generate a fractal. Chaos and Fractals

42. 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.

43. 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 map f( ) Time n Time n+1 My Research

44. Coupled Maps -Phenomenology Patterns with Kinks Frozen Random Patterns Pattern Selection Travelling Waves Spatio-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

45. Pattern with Kinks No Chaos: lattice sites are attracted to the periodic orbits of the single logistic map. My Research

46. 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 logistic map. My Research

47. Pattern Selection Even though the nonlinearity has increased and the logistic map is chaotic for , the lattice is entirely periodic. My Research

48. Travelling Waves The coupled map lattice is symmetric, yet here we see a travelling wave. This dynamical behaviour is highly non- trivial! My Research

49. Spatio-Temporal Chaos Of course we have spatio-temporal chaos too. No order to be found here... or ???. No, despite the way it looks, this is far from random! My Research

50. 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

51. 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

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

53. My Research 0 ~0 Maximum Growth Fast Growth Slow Contractio n ~1 Maximum Growth Growth Contractio n ~2 Maximum Growth Slow Growth Fast Contractio n self neighbors = 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. Function, if is set to a certain value = critical size

54. 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.”

55. Fractals and Financial Markets Possible Extension of my Research 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.

56. Fractals and Financial Markets Possible Extension of my Research The ‘volatility’ can be adjusted by modifying the shape of the generator.

57. Fractals and Financial Markets Possible Extension of my Research 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.

58. Fractals and Financial Markets Possible Extension of my Research MPT – based on random walk IBM MPT – other random processes MPT – fractal Brownian motion US$ - DM rate Fractal MPT = Modern Portfolio Theory Real Fake

59. 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

60. 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. 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? Growth Conclusion

61. 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

62. Many 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

63. Thank you! More Coffee! (Scientists at work )