Chaos | Chapter 6 : Chaos and the horseshoe - Smale in Copacabana

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Chaos | Chapter 6 : Chaos and the horseshoe - Smale in Copacabana
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École Normale Supérieure de Lyon (ENS-Lyon)

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Computer animation Chaos (cosmogony)
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Point (geometry) Trajectory Observational study Transformation (genetics) State of matter Multiplication sign Translation (relic) Kontraktion <Mathematik> Similarity (geometry) Heat transfer 2 (number) Square number Electric field Modulform Körper <Algebra> Rotation Scaling (geometry) Process (computing) Stress (mechanics) Algebraic structure Cartesian coordinate system Sequence Numerical analysis Computer animation Vector field Resultant Spacetime
Predictability Point (geometry) Axiom of choice Trajectory Free group Dynamical system Group action State of matter Multiplication sign Algebraic structure Chaos (cosmogony) Chaos (cosmogony) Mereology Sequence Mathematics Category of being Voting Exterior algebra Computer animation Structural stability Network topology Modulform Right angle Physical system
Mathematics Quantum state Computer animation Model theory Chaos (cosmogony)
nearly 90 60 young American Matt Stevens I was
working on the beach of coke when he made the
discovery he discovered portion
not the same In fact this
horseshoe abstract mathematical object it is
another simple idea attempts
to reduce the most elementary
expression we must 1st
explained old ideas being back by here a vector field on state there's but trajectory that starts on the green must travel all the way around before returning to the and then taking another trip and so each my taxes the we can observe the trajectories start that Wash and wait the dispute New he said a few seconds later this new point began transformed by we get there T of TO and so after each trip point is transformed by instead of studying the trajectory space study sequence of France this is much easier we replace the dynamics of the electric field the end of the transformation discrete 0 1 2 3 so this is often much easier stay here since recent use rotation by one-third the center of the Centel trajectory where sponsor fixed point the center of the other trajectories are also but it takes them 3 times longer starting this is because we are wrote here the transformation T is a similarity the trajectories are no longer or on the contrary they approach the central trajectory spiraling around we say essential trajectory here is an example of an unstable trajectory the transformation we get along with and stretches along the center of the fixed point so its trajectory hurry up you see the the trajectory of a horizontal approach for those on the vertical axis for now with the plot squares form of translation to comply the skill of traction and also when the square back from the original it is taken the Bush now watch the behavior square as we go back inverse transfer a scale contraction for this time all the other let's try to envision the future and the past of the square after 1 trip as we've seen it becomes a portion Of after 1 trip other way into the pack it also becomes a portion of that earned by one-quarter they choose small articles of intersection of where the are also pressed Field the behavior in the past the and this whole structures repeated an infinite number of that at a time of very complicated porch how understand but called script air industry up well the amazing thing is that we almost at the same results due debts until you're frequent final sequence of allowing reputation for example a theirs periodic this year we will point jobs until the president on at the start here
is on point of a always remains it is a fixed fight this it is my honor plant such from years a trajectory alternate path with voters this is not the in sequence of your choice a a a a a year so what so Wales vs . com exactly this group in its future we think that this is incredible every possible future it's true but 1 so I meant for at least 1 point prediction is impossible because everything is Free will rediscover start China anything is possible that's fine slogan for thank you Mister Smith Smeal realize realizes that is not always necessary know things exactly what understand a distorted picture is often not correct based he shows that or should state mind you this does not mean the trajectories of state on the contrary trajectories are very very sensitive to their initial what we here a structure 2 the store mister smells picture annual still recognized on the right we have the original on the left another 1st slightly modified it is being formed but after all we stretch the square erect and pulled it in the form of worship you can't do it any 1 of many different ways the it says the this is somewhat analogous situation but we study from the village where the plant at slightly different 2 almost identical 4 ships we can't do exactly the same as we did with with the original draw a priority he can be shown in the same way that check trees on his new martial arts still described sequences of a prime prime modified she is just as chaotic as the original chaos is really there indestructible a woman In fact still shows at the summit to force somehow identical the trajectories right and left follow the same choreography is at the same time the same EU Yukon we can match any trajectory the 2nd part to some trajectory on 1st fortune and vise versa it is in this sense of structure state we form or should they win not only does it remained it keeps the same dynamics individual trajectories are on state but the dynamics of the whole is an early in the 20th century my with characteristic explained how it is possible understated and system even if 1 does not know it if you ask me if it if I had to misfortune to know I only get there tried to cut and I would not be able to view it but as I am sure that question could North them us right to what more Rock is that right and so we correct coexistence of him and hence instability but individual checked with structural stability of global property absolutely I'm on stage with the world around me is stable very real