Chaos in addition to Fractals Frederick H. Willeboordse http://staff.science.nus.edu.sg/~fr

Chaos in addition to Fractals Frederick H. Willeboordse http://staff.science.nus.edu.sg/~fr www.phwiki.com

Chaos in addition to Fractals Frederick H. Willeboordse http://staff.science.nus.edu.sg/~fr

Donovan, Mark, Morning Show Host has reference to this Academic Journal, PHwiki organized this Journal Chaos in addition to Fractals Frederick H. Willeboordse http://staff.science.nus.edu.sg/~frederik SP2171 Lecture Series: Symposium III, February 6, 2001 Today’s Program Part 1: Chaos in addition to Fractals are ubiquitous – a few examples Part 2: Underst in addition to ing Chaos in addition to Fractals – Some Theory – Some H in addition to s-on Applications Part 3: Food as long as Thought Part 1: A few examples The world is full of Chaos in addition to 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

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Chaos in addition to 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 19th century Chaos in addition to 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 as long as Scientists in addition to Engineers, 4th Edition Chaos in addition to Fractals in Physics Snowflake The hexagonal symmetry of a snowflake arises from a hexagonal symmetry in its lattice of hydrogen in addition to oxygen atoms. From: Tipler, Physics as long as Scientists in addition to Engineers, 4th Edition A nice example of how a simple underlying symmetry can lead to a complex structure

Chaos in addition to Fractals in Physics The red spot on Jupiter. Can such a spot survive in a chaotic environment Chaos in in addition to 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. Chaos in addition to Fractals in Chemistry Beluzov-Zhabotinski reaction Waves representing the concentration of a certain chemical(s). These can assume many patterns in addition to can also be chaotic

Chaos in addition to Fractals in Geology Satellite Image of a River Delta Chaos in addition to Fractals in Biology Delicious! Broccoli Romanesco is a cross between Broccoli in addition to Cauliflower. Chaos in addition to Fractals in Biology Broccoli Romanesco

Chaos in addition to Fractals in Biology Would we be alive without Chaos The venous in addition to arterial system of a kidney Chaos in addition to 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). Chaos in addition to Fractals in Paleontology Replicate in addition to Modify Built from similar modified segments

Chaos in addition to Fractals in Paleontology Would we be alive without Chaos Is there a relation to stretch in addition to fold Simple Complex Simple Complex The phenomena mentioned on the previous slide are very if not extremely complex. How can we ever underst in addition to them Chaos in addition to Fractals can be generated with what appear to be almost trivial mathematical as long as mulas Try to write an equation as long as this. You could have done this in JC Right! Part 2: Underst in addition to ing Chaos in addition to Fractals In order to underst in addition to what’s going on, let us have a very brief look at what Chaos in addition to Fractals are. Chaos Fractal

Chaos Are chaotic systems always chaotic What is Chaos Chaos is often a more ‘catchy’ name as long as non-linear dynamics. No! Generally speaking, many researchers will call a system chaotic if it can be chaotic as long as 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 Chaos Try it! What is Chaos Quiz: Can I make a croissant with more than 15’000 layers in 3 minutes Chaos Chaos Sensitive dependence on initial conditions What is Chaos The key to underst in addition to ing Chaos is the concept of stretch in addition to 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 in addition to fold! Or perhaps I should say kneed in addition to roll . Chaos

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

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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 in addition to then use this result as the starting value of a next calculation. given Chaos Logistic Map The so-called bifurcation diagram Plot 200 successive values of x as long as every value of a As the nonlinearity increases we sometimes encounter chaos Chaos Logistic Map What’s so special about this Let’s have a closer look. Let’s enlarge this area Chaos

Logistic Map Hey! This looks almost the same! Let’s try this again Chaos Logistic Map Let’s enlarge a much smaller area! Now let’s enlarge this area Hard to see, isn’t it Chaos Logistic Map The same again! Chaos

Conclusion Almost everything in our world is chaotic, yet order is also everywhere. Underst in addition to ing this dichotomy is a fabulous challenge. The study of Chaos can help us on our way. Chaos is fun!

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