Visualizing Poincaré map  p

  

Figure 5.4: Visualizing a non-linear saddle cycle.

Since Poincaré map  p maps x onto p(x), both lying in $\mathcal{S}$, a visualization based on a directed stroke connecting x and p(x) has been implemented in AVS (see Chapt. 8). A module named FLOW generates a set of arrows on the Poincaré section, which start at a point  x_i and end in a correlated point  p(x_i). See Fig. 5.4 for a visualization of a non-linear saddle cycle where this technique was used. $\mathcal{S}$ is represented as a semi-transparent disk. A set of light-grey arrows is placed on $\mathcal{S}$ to visualize  p¹. Sequences of consecutive applications of p are visualized by the use of small red spheres, representing $\left\{\mathbf{p}^j(\mathbf{x})\ \vert\ j\!\ge\!0\right\}$ , whereby the sphere depicting x= p⁰(x) is colored differently from the others. The visualization of the sequence $\left\{\mathbf{p}^j(\mathbf{x}_0)\ \vert\ j\!\ge\!0\right\}$ with x₀ close to the origin of phase space is combined with a visualization of the constructing flow trajectory.

  

Figure 5.5: Visualizing an cycle attractor using spot noise.

We also adapted spot noise [87] to Poincaré maps. We place elliptic spots onto $\mathcal{S}$ such that the focal points of the ellipses coincide with x_i and p(x_i), respectively. See Fig. 5.5 for an example. This choice is due to the fact that no directional information should be encoded, when  p(x_i)= x_i. In this case both focal points coincide and the elliptic spot degenerates to a circular spot. Images rendered with this method are well suited to visualize the entirety of $\mathbf{p}(\mathcal{S})$ within one still image. See Fig. 5.6 for a visualization of a non-hyperbolic saddle cycle (3 stable and 3 unstable manifolds) where spot noise was used for visualization. Similar as in Fig. 5.4, six sequences  $\left\{\mathbf{p}^j(\mathbf{x}_i)\ \vert\ j\!\ge\!0\right\}_{i\in\left\{1,2,\ldots ,6\right\}}$ are visualized by the use of white and red spheres.

  

Figure 5.6: Visualizing a non-hyperbolic saddle cycle.

The results of the previous techniques are now embedded into a 3D visualization of the underlying flow. We therefore represent Poincaré section  $\mathcal{S}$ as a semi-transparent disk placed within the flow and realize the arrows and spot noise as a texture of this disk (see Figs. 5.4 and 5.6). Semi-transparency was used for the map to allow the viewer to see through. This improves the understanding of the context of map  p.

mailto:helwig@cg.tuwien.ac.at.