1 GEM2505M Frederick H. Willeboordse Taming Chaos

2 The Essence of Chaos Lecture 11

GEM2505M 3 Today’s Lecture Sensitive Dependence Stretch and Fold Homoclinic Points Chaos and Randomness Universality 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?

GEM2505M 4 Sensitive dependence on initial conditions means that initially tiny differences grow rapidly to the order of the system size. Sensitive Dependence 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?

GEM2505M 5 The answer to that question lies in stretch and fold. Stretch and Fold With layers Stretch Fold Back to the original shape

GEM2505M 6 If it funny once, it’s funny twice! Stretch and Fold Stretch Fold Back to the original shape

GEM2505M 7 The distance between points on opposite end of the bar. Stretch and Fold The distance grows exponentially!

GEM2505M 8 The distance between nearby points. Sensitive Dependence! Stretch and Fold The distance grows exponentially!

GEM2505M 9 Stretch and Fold On a line Stretch After Folding and Merging Start Here, after stretching and folding, the top and bottom layer are merged together (as is the case in a 1-D map).

GEM2505M 10 Stretch and Fold Mathematically The Tent Map if Bifurcation DiagramCobwebLyapunov Exponents

GEM2505M 11 Stretch and Fold In real life An excellent example of stretch and fold is the making dough!

GEM2505M 12 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. ? In 3 minutes, a croissant with how many layers can I make? 1.Between 10 and Between 100 and 1,000 3.Between 1,000 and 10,000 4.Between 10,000 and 100,000 Stretch and Fold Next I’ll use this dough to make a croissant.

GEM2505M 13 Stretch and Fold In the logistic map The same as stretch and fold with the stretch being nonlinear.

GEM2505M 14 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. Homoclinic Points Let us see what happens to a small area near the stable manifold After a few steps it will be near the fixed point.

GEM2505M 15 Homoclinic Points After arriving at the fixed point the rectangle will be stretched and pushed away along the unstable manifold. Eventually, it will be near the starting point again and overlap the original area. original square

GEM2505M 16 Homoclinic Points Hence we see stretching and folding at work. Note: in these simplified drawings other deformations due to the homoclinic points etc. have been ignored. original square Where does the luck go in this case

GEM2505M 17 Let us have a look at two time series: And analyze these with some standard methods Data: Dr. C. Ting Chaos and Randomness What is the relationship between chaos and randomness? Are they the same?

GEM2505M 18 No qualitative differences! Chaos and Randomness Power Spectra

GEM2505M 19 Chaos No qualitative differences! Chaos and Randomness Histograms

GEM2505M 20 Well these two look pretty much the same. Chaotic?? Random??? Chaotic?? Random??? Chaos and Randomness ? What do we have here? 1.Both are chaotic 2.Red is chaotic and blue is random 3.Red is random and blue is chaotic 4.Both are random

GEM2505M 21 Return map (plot x n+1 versus x n ) x n+1 = x 2 n y n y n+1 = x n White Noise Henon Map Deterministic Non-Deterministic Red is Chaotic and Blue is Random! Chaos and Randomness

GEM2505M 22 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. Universality 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.

GEM2505M 23 Universality Some Examples Experimental verifications of the Feigenbaum constant.

GEM2505M 24 Universality Some systems for which a sand pile is the standard model are composed of many parts. 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. this is why it’s called ‘self- organized criticality’

GEM2505M 25 Universality The minor events are much more common than the major events, but their underlying mechanism is the same. Self-organized criticality 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:

GEM2505M 26 Universality Drop one grain of sand slowly onto a circular surface and see what happens: Self-organized criticality 1.Grains stay close to where they land 2.Slowly form a pile with a gentle slope 3.Slope stops getting steeper Critical state is reached. Avalanches of all sizes occur.

GEM2505M 27 Universality If one plots the number of avalanches versus their size, one obtains a so-called power law. Self-organized criticality 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) Power law: Mass fluctuations Outcome of the sand pile experiment Avalanches are a big risk in alpine countries how steep is a pyramid?

GEM2505M 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. In a more general sense, nowadays, a power law describing the frequency of something versus its rank is often called a Zipf’s law. 10M 1M 100K M = Million, K = Thousand Frequency Population Frequency vs Rank

GEM2505M 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.

GEM2505M 30 Stretch and Fold Universality Key Points of the Day

GEM2505M 31 Is there a Zipf’s law for Innovation? Think about it! Stretch, Fold, Exercise, Fitness, Chaos is healthy!

GEM2505M 32 References