Poincaré maps and visualization

Dynamical systems often exhibit cyclic or quasi-cyclic behavior, e.g., food chains, oscillating chemical reactions, weather models based on the period of one year, etc. The cyclic property of such systems usually dominates the character of the behavior. Since the periodic or quasi-periodic behavior usually is known in advance, the local changes turn after turn are much more interesting.

In cases like these Poincaré maps, a technique used by mathematicians, becomes useful. A planar cross-section, called the Poincaré section, is placed orthogonal to the periodic flow. Consecutive intersections of flow trajectories are related via the Poincaré map, i.e., a discrete dynamical system of one dimension less than the cyclic flow. This Poincaré map inherits many important properties from the periodic or quasi-periodic flow. See Fig. 5.1 for an illustration of this relation between the 3D flow and its 2D Poincaré map.

The visualization of periodic or quasi-periodic dynamical systems can be done on the basis of Poincaré maps. Several possibilities are given:

Visualizing $\left\{\mathbf{p}(\mathbf{x}_i)\ \vert\ 0\!\le\!{}i\!\le\!{}m\right\}$ -

A direct visualization of Poincaré map p is to visually correlate x and p(x). This can be done by placing small arrows on the Poincaré section with the tail aligned with x and the head coinciding with p(x). See Fig. 5.4 for a visualization of a non-linear saddle cycle where this technique was used.

Visualizing $\mathbf{p}(\mathcal{S})$ -

Instead of plotting a rather small number of arrows for a discrete set of pairs (x_i,p(x_i)), a continuous representation of $\mathbf{p}(\mathcal{S})$ by the use of adapted spot noise can be used. Elliptic spots are placed on the spot noise texture such that the focal points of the ellipses coincide with x and p(x). Using a high number of spots a direct representation of continuous $\mathbf{p}(\mathcal{S})$ is achieved (see Fig. 5.6).

Visualizing $\left\{\mathbf{p}^j(\mathbf{x}_0)\ \vert\ j\!\ge\!0\right\}$ -

Often the long-term behavior of a dynamical system is of special interest. Thus the visualization of the repeated application of Poincaré map p is also very important. Instead of showing many arrows representing one application of p to many points of the Poincaré section, $\left\{\mathbf{p}^j(\mathbf{x}_0)\ \vert\ j\!\ge\!0\right\}$, i.e., the repeated application of p to one specific initial condition is represented. The discrete orbit can be shown, for instance, as set of small spheres. See again Fig. 5.4, where a few orbits are included to visualize the long-term evolution induced by the dynamical system.

Visualizing p^q instead of p¹ -

Sometimes, the investigation of p^q is more useful than visualizing p itself. Usually this is the case, if a q-loop cycle, i.e., a cycle that closes after q revolutions, dominates the behavior of the dynamical system. Visualization on the basis of Poincaré maps should take this into account (see Fig. 5.8).

Visualizing $\left\{\mathbf{p}^j(\mathcal{S})\ \vert\ j\!\ge\!0\right\}$ -

Using animation even the repeated application of Poincaré map p to the continuous sub-sets of Poincaré section $\mathcal{S}$, i.e., $\{\mathbf{p}^j(\mathcal{S})\ \vert\ j\!\ge\!0\}$, can be visualized. For efficiency reasons, a texture on sub-set $\mathcal{S}$ is transformed using a warp approximation of p in discrete time steps for each application of p. The vectors of the warp approximation are derived from the Poincaré map p.

Adding Flow Visualization -

It is also helpful to combine Poincaré section visualization with flow visualization. The relation between Poincaré map and continuous flow is depicted. This helps to keep the periodic or quasi-periodic structure in mind when investigating the Poincaré map (see the figure on page ).

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