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The Poincaré plane is taken perpendicular to the page, with the The above Poincaré sections using the steady data in the middle, but Most likely the temperature was changing slightly. Perhaps the parameters were drifting slightly from beginning to end. I then graphed the X (red), Y (green), and Z (blue)Īpparently the data is doing something strange at the beginning andĮnd this is not suprising as this comes from an experiment, and The data every 6.1 seconds (this being the period corresponding to a I then tried a different sort of Poincaré section I strobed I took a different Poincaré plane, one that cuts this Red points are coming out of the page, blue into the page. Going through the center of the attractor (this plane is perpendicular to theĪxis of the tube) this is shown in the second picture above.Ĭat 3d.026 | poincare -n 0.671 0.574 0.470 -d0 -VM3 -x 3.6 3.6 3.6 |Ĭat 3d.026 | poincare -n 0.671 0.574 0.470 -d1 -VM3 -x 3.6 3.6 3.6 | I took a Poincaré plane that cuts this tube parallel to the page, It in the picture below (left), we are looking down the length of the This attractor looks like a tube with a little bit of fuzz inside OfĬourse, I didn't expect that it would be. Values) but overall the map does not appear to be chaotic. (angles near theta=0 and theta=pi are mapped to several different Has a unique theta_n corresponding to it). Noisy, but clearly appears to be invertible (that is, each theta_n+2 Has rotated slightly (generally moving backwards). Thus, every second time the Poincaré plane is intersected, the orbit This results in a nice 1-D map which shows that this is Stretch them so that they roughly form a circle.Ĭalculate the angle theta each point makes (with respect Here's a plot of the points, in the Poincaré plane:Ĭonsider the blue points (points crossing in the direction into Program called poincare to handle all of this. The fundamental frequency which you could find using a power spectrum). Section, which can reveal structure of the attractor.Īnother sort of Poincaré section is when youĬonsider a natural period of the attractor (say, from Which occur in one direction (crossing from the "bottom" side Poincaré plane, although only plot the intersections Plot the intersections of the orbits and the The orbits which comprise the attractor cross the plane Perhaps fractal dimensions.įor an explanation of what these pages are all about, select topic 1 above.Īs I see it, the goal of a Poincaré section is to detectįor a Poincaré section, take an attractor and anĪrbitrary plane which cuts the attractor into two pieces. Mutual information to find delay coordinatesĩ.
POINTCARRÉ WIKIPEDIA SERIES
My Adventures in Chaotic Time Series Meet the time seriesģ.