Chaos | Chapter 4 : Oscillations - The swing

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Chaos | Chapter 4 : Oscillations - The swing
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École Normale Supérieure de Lyon (ENS-Lyon)

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Rotation Point (geometry) Trajectory Trail Greatest element Correspondence (mathematics) Direction (geometry) Spiral Chaos (cosmogony) Oscillation Thermodynamic equilibrium Measurement Theory Numerical analysis Vector potential Frequency Vector field Friction Velocity Cylinder (geometry) Vertex (graph theory) Gravitation Circle Position operator
Point (geometry) Addition Trajectory Group action Dot product Multiplication sign Moment (mathematics) Model theory Water vapor Chaos (cosmogony) Limit (category theory) Measurement Numerical analysis Frequency Computer animation Vector field Circle Cycle (graph theory)
Point (geometry) Trajectory Dynamical system Multiplication sign Rule of inference Theory Grothendieck topology Dimensional analysis Plane (geometry) Modulform Boundary value problem Theorem Circle Position operator Physical system Area Curve Line (geometry) Limit (category theory) Thermodynamic equilibrium Measurement Computer animation Vector field Vector space Order (biology) Cycle (graph theory) Game theory
Mathematics Quantum state Computer animation Model theory Chaos (cosmogony)
Mathematics Computer animation Chaos (cosmogony)
simple things 1st end In the 16th century galileo observed swing lamp and measures oscillation period a taking his own pots room I can't really describe this weekend all the tip of the pendulum moves on the Blue Circle if said a pack a day all my this place is Circle on Saunders likes every point of the circle represents a position of potential but the red points for example is the position at the bottom and have do I am it's speed given by the Red sector is described by a number the numbers positive if it rotates in 1 direction a negative rotates in the other we can represent his number in the Saunders vertical position a here so a single player on the cylinder describe as both the position and speed door the 1st scored on a circle houses with the Swedes and the 2nd vertical quarter for the past so now that's for gravity work and wash this week I knew more for now we will assume there's no friction off the swinging will never had I know a lot if the initial position is low mail an initial velocity is not too large my than the pendulum moose periodically we all know back GA where can if we wanted pasta perhaps too fast it will go also whereas it is still a periodic movement but not the same way look at the trajectory from the cylinder but the a were how a hay I it follows the vector field I get am L observe the more realistic situation where the friction Gropep be oscillations slowly decreases and then finally the pendulum slows to a stop Emma remember aerosols theory am I the swing is now back in its natural position at the bottom Ed book observed the corresponding vector field on the side trajectories eventually spiral into the resting it is equilibrium position is called a stable track or did a swim with friction is not very much fun because it eventually
stops so we have to give this with the plush To make you go really hot this is also the point of view of the gentleman represented by the painter frown on us he seems very interested in this way Roo let's imagine that the pendulum in addition to grab infection Norway is also subjected 2 pushing for coming from a small mosque found this is a very modern pendulum it comes with a jet engine Group for example suppose nozzle actuates only when the speed in the height of the pendulum are lower than a certain value where at if we started slowly we are pushed it we accelerate and accelerate until pushing stops because the pendulum is going too fast a patrician takes over it slows down and slows down until Bush comes the rescue again and so on the motion finally stabilizers on a periodic regime point calls this a limit set my this is the opposite of chaos in a way the periodic pushing everything is in sync at all head I'm more pain did here is what is called the these portrait of the silent the limit cycle is red we see is spiraling trajectory that gradually becomes the periodic trajectory the daily met head here is a simple example that illustrates all of this in college the lot vault terror model dating from the 1930 s right so to population shared for example rabbits and foxes we will instead a measure calm with stocks and water b for you do see the Ducks leading the water when there are only a few ducks than a only a few lives the water lily population therefore grows rapidly e when there are a few ducks in any water that is the well fed and their number increases off on I I am a book are Moog Road go on for so long and got an inhabited by a lot of taste large water shed it I however my hair a move back a by by when there are fewer and fewer Waterloo the Ducks have less and less to eat and their number decreases so we have come full circle and can start again Inc we can require situation at any given moment in time using a point on a play the 1st quarter represents a number of water and the 2nd represents the number of dots
In fact we get objectors shields and the Over time the dioxin follow trajectory of the vector field and Richard limit cycle the populations of ducks in Waterloo is eventually Ostlie periodic the belief that any motion perhaps after a short transition period eventually
stabilizes thereby stopping costly periodically his company's sites for a very long time where the 1st theorems in the theory of dynamical systems i . 3 in the late 19th century seem to justify this is about vector fields in the plane a measure that too few looks like In fact we don't know that to everywhere but we know how behaves close to this circle mail take a trajectory that enters the deaths share a maiden name where do they go where the point grave in Nixon said there are 2 possible cases a the trajectory either get very close to an equilibrium position as we see here this is what we saw for the day and pendulum 4 you must approach a limit cycle the say J. a if you say aII a head-to-head had I 0 copper the brewers How do we approve such as am here is a main ID a How do calm Tom take a point on the circle Air In observe the trajectory to the point entering the stocks that stopped at a point he look at all the trajectories in the vicinity of Tom Is it possible that the trajectory continues returns the synergy of look the can the supposed saying he returns to nearby park Chiu the Ark of the trajectory between games you but it's lined UP forms a closed curve NO How it Chan this is the
boundary of a certain area John in blue yeah I am can get we can see that a trajectory that starts from Q enters the blue area and they can get us anymore it's trap In order to exit each across our GQ however trajectories cannot cross have a rule L why not well it to trajectories passed the same point that this would Koshy Lifshits there is only a single trajectory to any given point really the trajectory starting to cannot escape the line of duty either there's vector coming in mock going out Sharon you can see the trajectory from may well return very close to the case but then his condemned to never escape we say in the situation that there is no car this is the main idea of the Quecreek antics and this year marks the beginning of what is now called qualitative theory of dynamical systems Todd came even if we have only an imperfect knowledge of a vector we can often understand the behavior of his trajectory In the case of the plane everything goes well the trajectories eventually become periodic for the approach a point of equilibrium but 0 . soon discovered that his name is valid only for vector Hilton to dimension that is to say for very small systems 4 3 dimensions of higher we will see the visit which can be much much much more complicated In very beautiful known when simple limit cycles welcome to the world