Chaos | Chapter 7 : Strange Attractors - The butterfly effect

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Chaos | Chapter 7 : Strange Attractors - The butterfly effect
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École Normale Supérieure de Lyon (ENS-Lyon)

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but we look at the movement of the atmosphere we quickly realized that is infinitely more complex than that of course
yes fears of fluid that had each altitude of each point of the Earth's surface Speight a city of pressure temperature and all of this various time is of course impossible to understand this practically in infinite an amount of it is almost as if we were in a space understand something about it we must make approximation they 1963 it would simple the simplified but following death he simplified such a degree but there's no guarantee that his equation has anything to do with it a His models of the atmosphere was reduced to just 3 numbers X Y and Z the evolution of the atmosphere was reduced to a simple equation each point x y z instead represents insisting yet the evolution of for example and is only sale first-quarter could represent a temperature the 2nd Boeing speed and the 3rd but met over here it is cold resumed Her here the opposite when we follow a trajectory of the field we're following the evolution of the weather did the weatherman just needs to solve differential equation this is what the Red Sox studied his this habit anything to do with the weather that is far from clear is what is often called the point used to try and understand the broad outlines of some complex behavior 5th In fact the
Renzulli had these sorts of graft look at because his computer in in 1963 was quite primitive he a brave but
subject to atmosphere represented
by the Centers these 2 pop there are extremely close together so close they are almost identical that's observe what happens to them 1st to evolution are almost indistinguishable you what's S a Split up significantly the 2 atmospheres become completely different this thing is chaos sensitive dependence on the initial conditions in 1972 the Reds was going to preserve his work at a prestigious conference the but he was late standing in the title of his lecture the organizer Philip Marley's was in a hurry the program the participants so he chose a little further particularly as the flat butterflies wings in Brazil for Natal in Texas the Butterfly Effect was born mathematical concepts are often not well understood by the general public the image of a small frail butterfly having influence on the world is very poetic the idea was very successful and has been modified there is always a butterfly but sometimes it comes from Africa and sometimes from China and is responsible for disasters in New York or in Chicago interest a year found its way into literature music and movies a list of films based upon this idea and it generates a lot of emotion involved the 2006 by Alejandro Gonzalez please using incident gunshot Morocco will change a lot of American couple a Mexican made a Japanese girl 8 an interplay of small causes large consequences how can determine the fate of human quick unfortunately only one-half of Lawrence's message made its way to the general public but King the statement that it is impossible to predict the future and practice how could scientists resigned themselves to such an admission of failure Lorenza's messages much richer I do here trajectories of Lawrence's 1 1 that do not start from very similar initial condition that's a racing the 1st 10 seconds of the movement and then let's observatory so be In what do we see the trajectories indeed very different they seem a a little crazy very unpredictable they accumulate on the scene butterfly-shaped object accumulation does not seem to depend on the initial position the the trajectories attracted to this butterfly this is what is referred to as the Lorenza track a stranger track here is a positive sign Vermont 1 famous as the butterfly instead of observer just trajectory but put in the little these balls each 1 of them representing a simplified after a while all accumulate same butterfly young came soon a very nice object that we can admire endlessly what do you do these are real problems for mathematicians and for scientists instead of described the future starting from a given initial position which we know impossible we will instead trial described track where does it look like How did the internal dynamics were in the 1970 Berman Guggenheim Williams proposed a simple model for trying to understand the Lorenz attractor Our an use a strip that folded it and
glued together a kid the toll of dead do we have a special object in states a mishandled is starting Our here we we draw In a little while later we are back at the start but we are added deferred here start position along the story but that means you the math 0 on the left one-half we're in the middle and while we are on the right if we start In return the . 2 small and we will return the . 2 x minus 1 X is larger than 1 In other words when the Carmen points where it meets a bubble at every turn except we need subtract 1 if the the result is Greater but it's a bit like the Horseshoe the dynamics in continuous time they replaced by dynamics in discrete mainly by the success of places where we cross the starting line am yet we start off From 1 one-third Our we arrived at two-thirds and then we arrive at 4th 30 but we must subtract 1 that is to say we arrive at once so we're back toward regional starting point after 2 but we have a periodic trajectories of period too here is a trajectory of period 18 I give after periodic trajectories after successfully the left and right here is following a certain does where it can be shown that all sequences are and of course not all trajectories of periodic whatever the event sequence of left and right there's a trajectory follows that yesterday what is the relationship between Lorenza tractors and the model data strips of paper the well it was until 2001 but the mathematician showed that the paper model actually describes the movement on the Lorenza attract brick each trajectory in the Lorenz attractor there's a trajectory that the behaves in exactly the same way Is it possible that the movement of the atmosphere can be reduced to continually doubling the number
x between 0 and 1 and subtracting 1 of the result of greater than 1 the Of course not all of this
is much too simplistic but it is an
illustration of the phenomena simple thing mathematicians why has Butterfly Effect become so popular perhaps because it gives us that were freed freedom a legacy of
cold sometimes leads to a kind of fatal Lorenzo butterfly claims that small as we are we can have an influence on the world good news for us