GEM2505M

1. Taming Chaos GEM2505M Frederick H. Willeboordse frederik@chaos.nus.edu.sg

2. The Essence of Chaos Lecture 11

3. Today’s Lecture The Story Now that we have obtained some understanding of the phenomena encountered in chaotic and complex systems, let us bring the essential points together. What are the key features of such systems? • Sensitive Dependence • Stretch and Fold • Homoclinic Points • Chaos and Randomness • Universality

4. Sensitive Dependence Sensitive dependence on initial conditions means that initially tiny differences grow rapidly to the order of the system size. As a consequence, in real life, systems that display sensitive dependence on initial conditions cannot be predicted long term due to the inevitable presence of noise. But there is a problem here. Real systems are not infinitely big! How can errors keep on growing in a finite system?

5. Stretch and Fold The answer to that question lies in stretch and fold. With layers Stretch Fold Back to the original shape

6. Stretch and Fold If it funny once, it’s funny twice! Stretch Fold Back to the original shape

7. Stretch and Fold The distance between points on opposite end of the bar. The distance grows exponentially!

8. Stretch and Fold The distance between nearby points. Sensitive Dependence! The distance grows exponentially!

9. Stretch and Fold Here, after stretching and folding, the top and bottom layer are merged together (as is the case in a 1-D map). On a line Stretch Start After Folding and Merging

10. Stretch and Fold Mathematically The Tent Map if if Bifurcation Diagram Cobweb Lyapunov Exponents

11. Stretch and Fold In real life An excellent example of stretch and fold is the making dough!

12. Stretch and Fold Give me some flower, water and a tiny bit of oil. After some mixing and kneading, I’ll have a hopefully nice piece of dough. Next I’ll use this dough to make a croissant. In 3 minutes, a croissant with how many layers can I make? ? • Between 10 and 100 • Between 100 and 1,000 • Between 1,000 and 10,000 • Between 10,000 and 100,000

13. 1 1 1 0 0 -1 -1 -1 Stretch and Fold In the logistic map The same as stretch and fold with the stretch being nonlinear.

14. Homoclinic Points Now that we have seen stretch and fold at work, we can get a bit a better understanding of why the homoclinic points lead to chaotic orbits. Let us see what happens to a small area near the stable manifold After a few steps it will be near the fixed point.

15. Homoclinic Points After arriving at the fixed point the rectangle will be stretched and pushed away along the unstable manifold. original square Eventually, it will be near the starting point again and overlap the original area.

16. Homoclinic Points Hence we see stretching and folding at work. Where does the luck go in this case  original square Note: in these simplified drawings other deformations due to the homoclinic points etc. have been ignored.

17. Chaos and Randomness What is the relationship between chaos and randomness? Are they the same? Let us have a look at two time series: Data: Dr. C. Ting And analyze these with some standard methods

18. Chaos and Randomness Power Spectra No qualitativedifferences!

19. Chaos Chaos and Randomness Histograms No qualitativedifferences!

20. Chaos and Randomness Random??? Chaotic?? Chaotic?? Chaotic?? Well these two look pretty much the same. Random??? Random??? Random??? Chaotic?? Chaotic?? ? What do we have here? • Both are chaotic • Red is chaotic and blue is random • Red is random and blue is chaotic • Both are random

21. Chaos and Randomness Return map(plot xn+1versus xn ) Red is Chaotic and Blue is Random! Henon Map Deterministic xn+1 = 1.4 - x2n + 0.3 yn yn+1 = xn White Noise Non-Deterministic

22. Universality A key motivation for the study of chaos is the notion of universality. In this context it means that a certain feature or a certain constant is applicable to a whole range of systems which are said to be a class of systems. It is important to note that universality in this sense does not mean everywhere in all conceivable cases. The most well known universal constant in chaos theory is the Feigenbaum constant. It applies to all single hump functions.

23. Universality Some Examples Experimental verifications of the Feigenbaum constant.

24. Universality Self-organized criticality Some systems for which a sand pile is the standard model are composed of many parts. this is why it’s called ‘self-organized criticality’ A sand pile turns out to naturally evolve to a critical state in which a small event can trigger a chain reaction of tumbling sand grains. This chain reaction can stop rapidly but can also become a so-called catastrophe where a large number of sand grains forms an avalanche.

25. Universality Self-organized criticality The minor events are much more common than the major events, but their underlying mechanism is the same. An essential aspect of self-organized critical systems is that their global features do not depend on the details of the components dynamics. Let us look a bit more closely at a sand pile:

26. Universality Self-organized criticality Drop one grain of sand slowly onto a circular surface and see what happens: • Grains stay close to where they land • Slowly form a pile with a gentle slope • Slope stops getting steeper Critical state is reached. Avalanches of all sizes occur.

27. Universality Self-organized criticality Mass fluctuations If one plots the number of avalanches versus their size, one obtains a so-called power law. Outcome of the sand pile experiment how steep is a pyramid? Power law: When a quantity’s parameter dependence is a straight line in a log-log plot. L(d) = c d^a (with c and a constants) Avalanches are a big risk in alpine countries

28. Universality Zipf’s law An interesting power law, known as Zipf’s law is the relative ranking of cities in the world around 1920 versus their population. 10M Frequency vs Rank Population 1M In a more general sense, nowadays, a power law describing the frequency of something versus its rank is often called a Zipf’s law. 100K 1 10 100 Frequency M = Million, K = Thousand

29. Universality Zipf’s law For example, Zipf also discovered a power law for the occurrences of words in the English language. The most common word is ‘the’ with a frequency of about 9%. The tenth most common word ‘I’ has a frequency of 1%. This independent of the text as long as the text is long enough. E.g. it holds as well for Ulysses as for news papers.

30. Key Points of the Day • Stretch and Fold • Universality

31. Think about it! • Is there a Zipf’s law for Innovation? Stretch, Fold, Exercise, Fitness, Chaos is healthy!

32. References http://www.cmp.caltech.edu/~mcc/Chaos_Course/Lesson4/Demo1.html http://www.expm.t.u-tokyo.ac.jp/~kanamaru/Chaos/e/