Study of Bifurcations

Bifurcations are events within parameter space of the dynamical system, where the behavior of the system changes. For example, if a parameter of the system is continuously changed from value $a$ to $b$, the topology of the basins of attraction may change at some point (figure 7.6). Bifurcations are often caused by contacts between structures as some parameter of the dynamical system is changed. Frequently, the process can not be investigated analytically, thus numerical simulation is required. In 3D the detection of where and how such contacts happen is extremely difficult, if only 2D sections are available as a visualization method.

For the analysis of bifurcations, sequences of up to hundreds of volumes are computed for different values of the bifurcation parameter. For investigation, the data produced by the simulation is loaded into the viewer (the disk-cache implemented by RTVR is extremely useful for large sequences of volumes). To ease the detection of contacts between objects, distance information can be mapped to voxel opacity or color, as shown in the sequence depicted in figure 7.4, which illustrates a contact bifurcation between an attractor and the boundary of it's basin.

Figure 7.4: Visualization of a contact bifurcation: as the bifurcation parameter is changed, the attractor approaches the boundary of it's basin (green). After the contact, which happens at the shared boundary to the blue basin, the green basin merges with the blue basin (all the states previously converging to the chaotic attractor now converge to the point attractor of the blue basin). The chaotic attractor disappears, only the point attractor of the blue basin remains. The distance between attractor and boundary is color-coded on the boundary. Red parts indicate proximity of the attractor, and thus regions which probably are responsible for the contact. The blue spots on the outer boundary of the green basin are aliasing artefacts due to the discretization of phase-space. The blue basin is extremely close (below the resolution of the sampled volume) to the outer boundary of the green basin.

Figures 7.5, 7.6, and 7.7 show the visualization of another type of contact bifurcation - the contact of a basin with a specific part of a critical surface ($\widehat{CS}$), and the resulting change in topology, namely the creation of disjoint basin parts (figure 7.6). Figure 7.5, top left shows the basins of four attractors (three inner basins and one outer basin) together with three sheets of critical surface structures $\widehat{CS}_{-1}$. If $\widehat{CS}_{-1}$ is iterated once, by the application of the difference equations which define the map, $\widehat{CS}$ is obtained, which is the folded structure depicted in 7.5 top right. The sheets of $\widehat{CS}$ subdivide phase space into regions $Z_n$ with different properties (a different number of pre-images for all the inner points). The crossing of a basin boundary into one of this regions (a region with more pre-images) may cause the appearance of disjoint parts of the basin.

Identifying the contact from 2D slices only is an extremely difficult task, a pure 3D visualization is also not well-suited for this purpose, as the critical surfaces are folded in a complex way in the area of interest (figure 7.5, sections). The capabilities of RTVR allow to efficiently create a meaningful visualization which depicts the location of the contact (figure 7.7), by depicting only objects which are involved in the contact: one basin, one sheet of the $\widehat{CS}$, and the boundaries of one of the $Z$ regions. Additionally, the distance of voxels of the basin boundary to the $\widehat{CS}$ is mapped to color, and a clipping plane is applied to reveal insight into the region of crossing.

Figure: $\widehat{CS}_{-1}$ (top left image, three sheets $\widehat{CS}_{-1}^{(a)}$, $\widehat{CS}_{-1}^{(b)}$, and $\widehat{CS}_{-1}^{(c)}$, depicted together with basins of attraction) and $\widehat{CS}$ (top right image, three sheets $\widehat{CS}^{(a)}$, $CS^{(b)}$, and $\widehat{CS}^{(c)}$, depicted together with separated zones $Z_0$, $Z_2$, $Z_4$, $Z_6$, and $Z_8$) visualized in 3D - the images below the top row show planar intersections at different (increasing) depth values for both 3D illustrations.

Figure 7.6: Basins of attraction (four attractors except $\infty$) visualized before and after the contact bifurcation - the creation of disjoint parts of one basin (depicted in cyan) is clearly visible.

Figure: The basin of attraction which causes the contact bifurcation is visualized in 3D together with responsible parts of the $\widehat{CS}$.

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