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This Concept Map, created with IHMC CmapTools, has information related to: b5. Poincare Sections, A Poincare section is like a paper screen set up across the paths of the trajectory. Every time the trajectory pieces the paper screen, it leaves a hole in the paper, or a dot. The Poincare section therefore shows a pattern of dots. ???? The number of dots corresponds to the period of the attractor, i.e. an attractor of period two shows two dots., The Poincare MAP is a rule, function, graph or model that tells where the trajectory will next cross the Poincare section, given the location of the previous crossing. ???? The quantity of data to be managed is significantly less., A Poincare section is a geometric picture of a trajectory’s activity at a cross section of the attractor. ???? The section is a way of sampling or probing the system to get details of the revisits., Poincare sections through chaotic attractors can look like a heterogeneous cloud, with no apparent order. ???? Scattered here and there are large zones with no points, or conditions, which the trajectory avoids. New points tend to pop up in haphazard locations all over the attractor. The structure becomes better defined as the system evolves., A Poincare section is a cross section of an attractor, taken transverse to the flow. It consists of a bunch of dots. Each dot represents the trajectory’s intersection with the cross-section. ???? The cross section is a slice or section of the attractor transverse to the flow of the trajectory., 2. Toroidal attractors: ???? The torus is an imaginary surface shaped like a three-dimensional doughnut., Recurring visits to a cross-section taken through the doughnut produce a dotted ring, or circle, or deformed loop. ???? Information not otherwise available emerges about the attractor’s internal structure. The attractor’s cross-sectional dynamics emerge much more clearly., 3. Chaotic attractors: ???? Poincare sections through chaotic, or strange attractors, show a relatively complex geometry. They are made up of an infinite number of points, in contrast to the finite number of points through most aperiodic attractors., A Poincare section is a cross section of an attractor, taken transverse to the flow. It consists of a bunch of dots. Each dot represents the trajectory’s intersection with the cross-section. ???? Poincare sections can only apply to attractors with three or fewer dimensions., Usually only dots representing one side of the (circular) attractor are included. ???? The number of dots corresponds to the period of the attractor, i.e. an attractor of period two shows two dots., Patterns ???? The patterns of the Poincare section fall into three categories. The pattern of the Poincare section indicates the type of attractor., There is a distinctive fine structure or orderly and fractal-like texture, which typically has layers, like a flakey pastry crust. ???? Poincare sections through chaotic attractors can look like a heterogeneous cloud, with no apparent order., Maps ???? The Poincare MAP is a rule, function, graph or model that tells where the trajectory will next cross the Poincare section, given the location of the previous crossing., The Poincare section defines a certain range of conditions, or values of phase space variables, that the system revisits from time to time. ???? A Poincare section is only proper for a continuous system, even if it is sampled at discrete times., A Poincare section is a cross section of an attractor, taken transverse to the flow. It consists of a bunch of dots. Each dot represents the trajectory’s intersection with the cross-section. ???? Each dot has an address, the phase space coordinate values of the intersection of the trajectory with the slicing plane., The patterns of the Poincare section fall into three categories. The pattern of the Poincare section indicates the type of attractor. ???? 1. Periodic attractors:, Maps ???? The return map is a rule, function, or model that tells where the trajectory will next cross a particular Poincare section, given the phase space location of the previous crossing at that section, b5. Poincare Sections ???? A Poincare section is like a paper screen set up across the paths of the trajectory. Every time the trajectory pieces the paper screen, it leaves a hole in the paper, or a dot. The Poincare section therefore shows a pattern of dots., The patterns of the Poincare section fall into three categories. The pattern of the Poincare section indicates the type of attractor. ???? 2. Toroidal attractors:, The Poincare section is a different way of looking at an attractor. It can indicate the type of attractor; however, it cannot tell us any thing about the rest of the attractor, such as its shape or location in phase space, etc. ???? A Poincare section is a geometric picture of a trajectory’s activity at a cross section of the attractor.