1. Chaos and Fractals Frederick H. Willeboordse http://staff.science.nus.edu.sg/~frederik SP2171 Lecture Series: Symposium III, February 6, 2001

2. Today's Program Part 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

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

4. Chaos and Fractals in Physics The 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 19 th century

5. Chaos and 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

6. Chaos and 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

7. Chaos and Fractals in Physics The red spot on Jupiter. Can such a spot survive in a chaotic environment?

8. 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.

9. Chaos and Fractals in Chemistry Beluzov-Zhabotinski reaction Waves representing the concentration of a certain chemical(s). These can assume many patterns and can also be chaotic

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

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

12. Chaos and Fractals in Biology Broccoli Romanesco

13. Chaos and Fractals in Biology Would we be alive without Chaos? The venous and arterial system of a kidney

14. Chaos and Fractals in Paleontology Would 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).

15. Chaos and Fractals in Paleontology Replicate and Modify Built from similar modified segments?

16. Chaos and Fractals in Paleontology Would we be alive without Chaos? Is there a relation to stretch and fold?

17. 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 an equation for this. You could have done this in JC Right!??

18. Part 2: Understanding Chaos and Fractals In order to understand what’s going on, let us have a very brief look at what Chaos and Fractals are. Chaos Fractal

19. 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

20. Try it! What is Chaos? Quiz: Can I make a croissant with more than 15’000 layers in 3 minutes? Chaos

21. 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 15’000 layers in 3 minutes? – Yes: stretch and fold! Or perhaps I should say kneed and roll. Chaos

22. The Butterfly Effect Sensitive dependence on initial conditions is what gave the world the butterfly effect. Chaos

23. The Butterfly Effect Sensitive 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.

24. Logistic Map The logistic map can be defined as: 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

25. Iteration Iteration 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. given Chaos

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

27. Logistic Map What's so special about this? Let's have a closer look. Let's enlarge this area Chaos

28. Logistic Map Hey! This looks almost the same! Let's try this again... Chaos

29. Logistic Map Let's enlarge a much smaller area! Now let's enlarge this area Hard to see, isn't it? Chaos

30. Logistic Map The same again! Chaos

31. Logistic Map Indeed, the logmap repeats itself over and over again at ever smaller scales What's more, this behaviour was found to be universal! Yes, there's a fractal hidden in here. Chaos

32. Chaos and Randomness Chaos is NOT randomness though it can look pretty random. Chaos Let us have a look at two time series: And analyze these with some standard methods Data: Dr. C. Ting

33. Chaos and Randomness Chaos Power spectra No qualitative differences!

34. Chaos and Randomness Chaos Histograms No qualitative differences!

35. Chaos and Randomness Chaos Autocorrelation Functions No qualitative differences!

36. Chaos and Randomness Well these two look pretty much the same. Chaos Let us consider 4 options: What do you think? A: They are both Chaotic B: Red is Chaotic and Blue is Random C: Blue is Random and Red is Chaotic D: They are both Random Chaotic?? Random??? Chaotic?? Random???

37. Chaos and Randomness Chaos Return map (plot x n+1 versus x n ) x n+1 = 1.4 - x 2 n + 0.3 y n y n+1 = x n White Noise Henon Map Deterministic Non-Deterministic Red is Chaotic and Blue is Random! As we can see from the return map.

38. 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). Fractals

39. 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. Fractals Remove middle third Then remove middle third of what remains And so on ad infinitum

40. 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 Fractals

41. 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. Fractals

42. 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). Fractals

43. The Mandelbrot Set Does this look like the logistic map? It should!!! Take z to be real, divide both sides by c, then substitute Defining:, to obtain. we find the logistic map from before Fractals

44. 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 incredibly complex. Fractals

45. The Mandelbrot Set Next, zoom into this Area. Fractals

46. The Mandelbrot Set Next, zoom into this Area.

47. Fractals The Mandelbrot Set Next, zoom into this Area.

48. Fractals The Mandelbrot Set Fractals

49. Chaos and Fractals 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.

50. Chaos and Fractals 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.

51. 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

52. 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 Chaos

53. 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. 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.

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

55. 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. Chaos

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

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

58. 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! Chaos

59. My quasi-logo Now 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

60. 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 Iteration Repeat a (simple but non-linear) recipe over and over again

61. Understanding Chaos and Fractals Some applications/hands-on demonstrations Chinese Noodles Double Pendulum Video Feedback

62. Part 3: Food for Thought More Coffee! (Scientists at work )

63. Chaos in Biology Evolution? 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?

64. 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

65. Chaos in Meteorology What are the implications of the butterfly effect for the prediction of weather? Sunny? Rainy? Perhaps it’s up in the clouds

66. 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 not freedom of the mind. Can non-linear science contribute to the discussion on determinism versus free will?

67. 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!