a4mula 1 Posted July 5 (edited) I am currently reading a paper about complexity. The author starts with a brief introduction to chaos. In the paper it's stated that chaos is defined by stretching and folding. The paper goes on to equate a baker crafting a croissant. The visualization of how two starting initial conditions can quickly diverge is clear in the example. I'm curious if this analogy goes further than just baking however. Can we equate the stretching of the dough to time and the folding of the dough to space? I appreciate any thoughts, and I apologize if direct links and quotes such as this are any type of misstep. Quoted Source: https://necsi.edu/chaos-complexity-and-entropy Quote And how does nonlinearity manufacture fractals and chaos? There is one and only one answer: stretching and folding. All flows and all maps that manufacture fractals do it by stretching and folding. Let’s look at a simple example. Think of a pastry chef making a croissant. She puts down the dough and stretches it with a rolling-pin. Then she puts a layer of butter on it and folds it. She rolls and stretches it again, puts another layer of butter, and folds it again. And so on ad infinitum, or almost. What you get is an object, a delicious croissant, which is a fractal in the direction perpendicular to the table, with a very large (quasi-infinite) number of layers. This is the way all dynamical chaos works! It is easy to see how the sensitivity to initialconditions comes about. Consider two points close to each other in the initial dough. When the chef rolls the dough the first time, they get farther apart, unless they happen to be on the same vertical, which is very unlikely. Next time she rolls the dough, they get farther apart again. Eventually it is bound to happen that, in one of the folding operations, our two points end up in different layers. After that, they have completely different histories, and if they come close again it will be pure accident. All this is pretty simple and pretty obvious! But for physicists in the 1970’s it was new. And it turns out to be very, very useful. And where is the nonlinearity in this? It is in the folding. Linear equations of motion have solutions whose behavior does not change with amplitude. If they start stretching, they stretch forever; they never fold. It is the nonlinearity that folds. Edited July 5 by a4mula 1 Share this post Link to post Share on other sites

studiot 2060 Posted July 5 9 hours ago, a4mula said: I am currently reading a paper about complexity. The author starts with a brief introduction to chaos. In the paper it's stated that chaos is defined by stretching and folding. The paper goes on to equate a baker crafting a croissant. The visualization of how two starting initial conditions can quickly diverge is clear in the example. I'm curious if this analogy goes further than just baking however. Can we equate the stretching of the dough to time and the folding of the dough to space? I appreciate any thoughts, and I apologize if direct links and quotes such as this are any type of misstep. Quoted Source: https://necsi.edu/chaos-complexity-and-entropy Thank you for introducing this interesting subject and welcome to ScienceForums. +1 I see the subtitle of this paper is "A Physics talk for Non Physicists". It was published in 2000 As such the author (who is an aged theoretical physicist) bends the rules to provide an analogy. Further he admits that the definition of mathematical complexity is not yet fixed. Although he does observe that Chaotic phenomena can arise in simple ones as well complex ones, he does link Chaos and Complexity inextricably. We do not do that these days. They are regarded as separate phenomena. Indeed the mathematical term 'complex' , as opposed to 'simplex' has been was originally introduced in one school of algebraic geometry and algebraic topology has developed in its own right. I don't agree that the specific folding example you highlight underlies all Chaos, for instance in the phase space of a system there may be chaotic regions and non chaotic regions. This will not happen with dough rolling as it (tries to) treat all regions the same. We have a curent discussion thread on aspects of Chaos here you might like to look at. Finally what is your particular interest in this subject, Maths , Physics or General? 0 Share this post Link to post Share on other sites

Strange 4267 Posted July 5 It doesn't strike me as a very useful analogy as you end with a croissant every time. (Half joking; half serious) 0 Share this post Link to post Share on other sites