Visualizing Poincaré Maps together with the underlying flow

Project Duration: 1996, June - 1997, March.

This page assembles some results (figures, animation sequences, and more images) of work that is part of our research topic ``Visualization of Complex Dynamical Systems''. The figures are provided in JPEG format and the animation sequences are given as MPEG or Quicktime video clips.

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Figures in the paper (JPEG)

	Figure 1 (97k): An illustration of the Poincaré map definition. The figure shows how consecutive intersections of flow trajectories and the Poincaré section in the vicinity of a periodic orbit construct a Poincaré map P.
	Figure 2 (132k): Visualizing a non-linear saddle cycle. Light-grey arrows indicate one application of Poincaré map P for many points on the Poincaré section, whereas red spheres depict the repeated application of P to a few initial conditions (white spheres). Furthermore for one of these series the constructing trajectory is added for a better understanding of the correlation flow <-> Poincaré map. Refer to animation sequences 1, 2, and 3 for an animation of this image.
	Figure 3 (218k): Visualizing a non-hyperbolic saddle cycle. Again red spheres show the long-term development under repeated applications of Poincaré map P. Additionally spot noise is used to given an impression of the entirety of P (one application). See also Image 7 (no inset).
	Figure 4 (136k): Visualizing why Pq is sometimes more expressive than P. If cycles intersect Poincaré section S more than once before they close, it is usually better to visualize P(no. of intersections) than just P. Figure 6 illustrates such a situation.
	Figure 5a & Figure 5b (222k+180k): Visualizing {(xi, P(xi))} (left image) vs. {(xi, P3(xi))} (right image). These two images demonstrate what is illustrated in Figure 6. For both pictures spot noise was used for the visualization. The right image is much more expressive than the left one. Moreover the left image could be even misleading for the viewer!
	Figure 6 (189k): Evaluating the initial texture after two applications of W. This Figure illustrates the approximation of P by a warping function W.
	Figure 7a & Figure 7b (167k+153k): Visualizing supplementory information in 3D. Comparing these two images it can be seen that the planar Poincaré section visualization, i.e., spot noise, does not change, although the underlying flow is different. Embedding the visualization of Poincaré maps within the underlying flow helps to improve such a situation.
	Figure 8 (229k): Enhancing the 2D Poincaré map visualization by the use of 3D flow visualization icons. In this image the Poincaré map was visualized by the use of spot noise and red spheres for series of repeated applications of Poincaré map P. Additionally an orange streamsurface together with a green streamline was started at the Poincaré section to visualize the underlying flow.
	Figure 9a & Figure 9c (130k+127k): Figures resulting from one and eleven applications of warp function W, i.e., W(S) and W11(S). The warping function is used to approximate the Poincaré map P. This approach is useful since the calculation of W is significantly cheaper than the one of P.

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Animation Sequences (Quicktime)

		Sequence 1 (5.803k~=6M): Constructing the Poincaré map, part 1. See how a flow trajectory constructs the Poincaré map. To omit the problem of visual cluttering, we also allow to reduce the representation of the constructing trajectory to short parts nearby the Poincaré section. Skipping the trajectory completely allows to increase the number of applications to higher numbers.
		Sequence 2 (33.617k~=34M): How to lie with Poincaré maps. See patterns on the Poincaré map that does not exist in the formulas. These aliasing effects occur due to problematic relations between the frequency of rotation vs. number of windings. Refer to Image 2 (short description below) for a visualization(!) of this problem.
		Sequence A (900k): Visualizing the repeated applications of P by approximating it with a warping function. You can see the stable manifold of the saddle cycle in the beginning of the clip. Towards the end of this clip the instable manifold (curved) becomes visible. Note: of course, just the intersections of these manifolds and the Poincaré section are visualized.
		Sequence B (2.715k~=3M): This animation sequence shows that adding 3D flow cues to a Poincaré map visualization can enhance the presentation by aspects that cannot be seen from the 2D Poincaré section visualization. In this case the Poincaré map P stays constant, althought the underlying flow changes.
		Sequence C (3.954k~=4M): Animating a seedline where a streamsurface is started helps to understand the context of the Poincaré map visualization.
		Sequence D1 (3.190k~=3M): Animating the seedpoint xi of orbit {Pj(xi) | j>=0} helps to understand the overall behavior of map P.
		Sequence D2 (11.464k~=11M): Again an animation of seedpoints xi together with a visualization of resulting sequence {Pj(xi) | j>=0}.

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More Images (JPEG)

	Image 1 (352k): See Image 1 for a visualization of the 3D autocatalator. It is a dynamical system which exhibits mixed-mode oscillations, i.e., oscillations of alternating small and large amplitute heights. This phenomenon is often encountered in `real world' system that are found, e.g., in chemistry. See pages ``Visualization of Mixed Mode Oscillations'' and ``Streamarrows -- Results'' for further visualization of this system.
	Image 2 (428k): This image is related to animation sequences named ``How to lie with Poincaré maps'' (33M, be aware!). It shows how a visualization of one single sequence {Pj(xi) | j>=0} consisting just of the red spheres is misleading. Placing arrows that visualize P1 onto the Poincaré section helps to understand the `real' behavior of the system.
	Image 3 (517k): This image is a visualization of a cycle(!) embedded within the torus-like invariant manifold. It crosses the Poincaré map lots of times before it closes! For the sake of clarity there is no additional visualization on the Poincaré section. Note: there are an infinite number of cycles of the same structure within this torus-like manifold.
	Image 4 (266k): See Image 4 for another parameter setting of our system RTorus. The streamsurface is started on a Poincaré map, which is itself generated on the basis of a rather complicated cycle (white tube).
	Image 5 (38k): An example of a 2D Poincaré map visualization. See URL http://www.physics.cornell.edu/sethna/teaching/sss/jupiter/Web/L5Poinc.htm for the source of this image.
	Image 6a & Image 6a (2*61k): Two hand-drawn examples of embedding the visualization of a Poincaré map within the underlying 3D flow. These figures were taken from the book ``Dynamics - The Geometry of Behavior'' by R. H. Abraham and C. D. Shaw, second edition, Addison-Wesley, 1992.
	Image 7 (218k): Visualizing a non-hyperbolic saddle cycle. Again red spheres show the long-term development under repeated applications of Poincaré map P. Additionally spot noise is used to given an impression of the entirety of P (one application).

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Helwig Löffelmann, last update on September 24, 1997.