Chaos | Chapter 8 : Statistics - Lorenz' mill

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Chaos | Chapter 8 : Statistics - Lorenz' mill
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École Normale Supérieure de Lyon (ENS-Lyon)

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Point (geometry) Area Twin prime State of matter Model theory 3 (number) Chaos (cosmogony) Chaos (cosmogony) Event horizon Sequence Area Frequency Computer animation Oval Lorenz curve Condition number
Trajectory Computer animation Multiplication sign
Trajectory Computer animation Average Multiplication sign Calculation Moment (mathematics) Measurement
Trajectory Computer animation Average Multiplication sign
Trajectory Computer animation Limit (category theory) Initial value problem
Trajectory Computer animation Order (biology)
Trail Trajectory Statistics Presentation of a group Dynamical system State of matter Multiplication sign Water vapor Chaos (cosmogony) Parameter (computer programming) Drop (liquid) Event horizon 2 (number) Hypothesis Product (business) Frequency Different (Kate Ryan album) Average Term (mathematics) Physical law Position operator Initial value problem Set theory Condition number Predictability Distribution (mathematics) Process (computing) Total S.A. Evolute Limit (category theory) Measurement Sequence Numerical analysis Degree (graph theory) Computer animation Physicist Network topology Gravitation Right angle Identical particles Resultant
Mathematics Quantum state Computer animation Model theory Chaos (cosmogony)
Mathematics Computer animation Chaos (cosmogony)
listen to end Iran's telling us about his concept of chaos and 3 steps if a single flap over butterflies wings can be instrumental in causing a tornado so also in all of the previous and subsequent collapse of the twin as can the flaps of the wings of millions of other butterflies not to mention the activities of memorable much more powerful creatures including our own speech we've already seen the public understands this 1st point very well above flap over butterflies where can be instrumental because when became equally well instrumental in preventing it yes but the 3rd point is the most important in this 1 provides more precise were I am proposing Over the years minuscule disturbances either increase nor decrease the frequency of occurrences of various weather events such as 28 of the most do is to modify the sequence in which these events occur let's return to Lorenza's model remember each point in states represents state the patents In some areas the weather's fine and there's a supposedly area cost following hurricane is in this little ball let's take an initial climate conditions and let's observe the
trajectory that resort occasionally the hurricane occurs
that is when the trajectory enters the ball that's measure the portion of the time between 0 and T when hurricane it seems difficult to
guess which specific moments the trajectory enters the average time spent inside the converges took over as interval of time measure becomes larger and here the average is 5 . 1 per cent he
now we take different initial conditions and do the same calculation well again the
trajectory spends a certain amount of its time inside the average tends to How about that average is the same as 4 or 5 . 1 per cent this
trial with yet another trajectory it works we
obtain the same limit and yet the trajectories of very different they are sensitive to initial conditions the flat the
whims of the butterfly yet another ball he and a 3rd perhaps
course he weighs or smell as a
trajectory it crosses the balls in a certain order To
see what happens that's what preach checkers each starting from a different conditional condition look on
every trajectory situations hurricane snow he weighs alternate incomprehensible In the different ways for each trajectory impossible under state but the proportion the yellow green and came from drop into the same limits humor The at 5 . 1 14 . 0 3 7 . 4 per cent it is we will abandon involved yellow preparing from a bag filled those portion listen Iran's peaceful disturbances increase or decrease the frequency of events such as tornadoes half mostly to modify the sequence in which these events this is a scientific stick with the purpose of the forecast is changed we now try to predict averages statistics and probability the idea is that these statistics are perhaps insensitive to initial condition they are unaffected by the flapping of the wings of Brazilian butterflies this hypothesis as yet unproven Of the possible coexistence of ideological chaos where the future movements sensitive to initial conditions With a statistical stability in sensitive to initial conditions took a long time before related mathematically the Reds was probably a little embarrassed by simplistic To say the least theoretical side of his 3 parameter atmosphere With the help of 2 physicists Howard in markets he developed a real physical that it is real even if is still simplistic and far from true whether or not here is a mail it consists of a wheel with buckets suspended evenly around each but it has a hole in the Nevada and the water will run out of it runs more quickly when the water level we talk about at a time and watch them sir sometimes turns the right and sometimes the less we seem to be totally incapable of predicted which way it will be heard in just a few sect movement seems totally erratic but predicts chaos he there is a relationship between them and the Lorenza track let's choose 3 members to describe for instance angular velocity conviction quartet of its center of gravity evolution of these numbers trees a yellow cur a is not credible Farmville moves like a butterfly he that's changed initial position in percept the forget all the buckets are initially the left we to degrees what further all the the right wheel is only 1 . 9 9 9 6 degrees we wouldn't be good microscope yes there is clearly a sensitive dependence on condition after a while to mills have completely different here SEF Lorenza statistical statement is true for the right to Mills off from almost the same position we Let's measure their speed for example 25 times per 2nd for 5 thousand seconds so a 125 thousand observations in total for each observation we know the rotational speed of the wheel 1 simple idea is to do it statisticians do we use a bar chart illustrate the distribution we divide the possible speeds and 35 equal intervals we count the number of times that a measured speed falls in each interval here is the result of some intervals appeared to be visited more than others are 2 Mills behave in very different ways they have a sensitive dependence on initial conditions but we see that 2 sets of data although different are statistically identical or the bar charts tend to become identical they work Iran's seems to be right when the statistics trajectory are it is sensitive to initial conditions we say that dynamics haven't signed well aware 4 SRB measure the goal of the forecaster is term these statistics for example let's observed the evolution of the temperature over time the temperature being 1 of the quarters we job departure as we did with them each temperature we milked proportion of observations that fall within this even if we consider trajectory there are quite different from what after a while the charts tend to become identical Lorenza tractor hasn't SRB measure it is as if the temperature was changing it rent but with very specific problem but there is a lot left to do mathematically and physically we must Monday's probably ends distribution or to deal the say we're along with hair and let us hope that we had done justice to Lurette could do not merely saying that if future depends heavily on the present for him said that he also and 1 for contributed the following nite I'm focusing on statistical issues time can still make production