Chaos | Chapter 9 : Chaotic or not - Research today

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Chaos | Chapter 9 : Chaotic or not - Research today
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École Normale Supérieure de Lyon (ENS-Lyon)

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Point (geometry) Trajectory Trail Dynamical system Statistics Group action Multiplication sign 3 (number) Chaos (cosmogony) Parameter (computer programming) Mereology Attractor 19 (number) Frequency Sign (mathematics) Plane (geometry) Permanent Natural number Bifurcation theory Series (mathematics) Initial value problem Rule of inference Chaos (cosmogony) Lace Measurement Numerical analysis Computer animation Vector field Vector space Order (biology) Game theory Near-ring Maß <Mathematik>
Point (geometry) Trajectory Trail Statistics Variance Water vapor Measurement Numerical analysis Computer animation Order (biology) Modulform Initial value problem Set theory Spacetime Condition number
Trajectory Statistics Meeting/Interview Multiplication sign Water vapor Mathematician Evolute Initial value problem Stability theory
Modulform Spacetime
Arithmetic mean Computer animation Object (grammar)
Mathematics Quantum state Computer animation Model theory Chaos (cosmogony)
Mathematics Computer animation Chaos (cosmogony)
there are many kinds of dynamical systems some are complicated others are not watch this vector field and that yield depends on parameter let's start by choosing a equal to 0 . 3 and watch what happens for a large number of initial condition after a while all attractive a single periodic trajectory no chaos here scenes there certainly is an SRP as all initial conditions wind-up moving along the skirt but slowly changing parameters do players a burial you have heard we see that is equal to 0 . 3 3 5 chaotic trajectories splits apart back still no chaos but now the chaotic trajectories is about twice as long and then critique for the series we believe the attracting trajectory doubles a 2nd time then things Peter the problem of an Obama for a equal 0 . 4 0 5 detractors complicated chaos but surprise if we continue to increase the parameters some time without warning the chaotic dynamics simplified returning to just a single periodic work how can we understand these bifurcation but is the most common behavior in nature chaotic non chaotic it is not clear enough for centuries we had no idea that chaos could exist today we see everywhere for each value of a pecan record-high the attractor meets a place we draw the intersection in red as the premise a varies you picture that looks like a piece of lace this is called a bifurcation dies they a game came to a head very pretty but not so easy to understand they game a a game at a time when they can hear some others as he seeks to establish resort their universal but they often start a studying exam they then hope but what they see the simple example will hold Rockies but Chen I recall the idea of low-rent and side will always SRP measure proportion of time their trajectory spend all converges told that does not depend on the initial condition is it reasonable to expect that this always happens they that would need to know it's unfortunate it is not true as we will see a small example discovered watch this vector field in the plane there are 3 points of equally a unit in Yuma it was the 1st time permanent but there are a lot of time but but the pair the trajectory there is no law but after while it approaches appointed but and then moves on the other hand when it approaches slows down and remains near that's part a while but then it speeds up what's the other but trimming there long the ACGIH those the 1st good where there's an even longer and sovereign consider the little green let's keep track of the transit time spent In this the trajectory stays such a long period but portion of the time is close 1200 then we leave the disk for an even longer period of time so the proportion of time Green also nearly 2 0 the climbs back on nearly a hundred per cent and then falls back to to 0 and so on as you see there's no there's no sign well on measures but what should we do Shall we abandon this idea said the rent I was mistaken well known the example we've just seen but it's very special it would change the vector just a little like that there then the problem goes away the new trajectories travel eventually Trojans chaotic Jack statistically Everything now his peers a man from the new vector field has of cyanide well Oh unless you so the question is whether these letters show us not for all that and is from and again we must temper part watch this it can and more pain and Obama D a man the visit rejected all is well human lives a butterfly has v Maine who became as initial conditions slowly along the Blue access everything is should In the trajectory always accumulates on the order track and even if it takes a little while we can easily said that statistics are not much but all of a sudden without
water was surprised the Gentry Cuellar elsewhere it has not been forged space has led the to regions give what VARs the point in the 1st region trajectory human wanted track if we start in the 2nd week and we end up in other words there is no S were being measures since the long-term behavior depends on the initial condition in fact there are 2 research initial conditions
statistics of the 1st measure other condition Rick no I don't that it was later that takes the form of a full set of problems that get resolved will allow for a global vision of 1 of these the situation we're just In order to have only on a number of initial conditions should be dropped 1 of these attract an 2 traffickers should have a son who will fold described how some college statistics of the typical trajectory that fall into that trap a whole group of
mathematicians harder work of salt water Creek by fight global picture seems is this picture too optimistic time will tell today we no longer think of turned the evolution of the individual trajectory but rather as he Of the collection sensitivity trajectories between initial conditions compensated for
by a kind of statistics stability Of the collection of Tracor e e e e e e e
e did before being 1783 not foresee this when he wrote this magnificent center about a complex and chaotic world that 1 has to try I understand you could In what way because we call within minutes that we were all spaces and the continued a succession of movements or matter Christian any form giving up 4 fingers printed so Everything is me all going
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