Initial Stable Parameter Range

For m=-inf to m=1.0 F0 is an attractor with two real and two conjugated complex eigenvalues. The direction corresponding to the y axis and the stable equilibrium exhibits a slightly stronger attraction.

F1 has two conjugated complex eigenvalues with negative real components corresponding to a spiral attraction. One of the remaining two real eigenvalues is positive and one negative, making F1 a saddle.

The Phasespace is divided into two big areas: the bassin of the attractor F0 and the area attracted by (-inf/inf/-inf/-inf). the border between those two bassins is close to the stable relative equilibrium. States slightly displaced in one direction from the equilibrium evolve into F0, whereas states displaced in the opposite direction escape into infinity. The m value of this image is -7, the fourth dimension is encoded into the color of the streamlines.
This animation visualizes this scenario using extruded parallel coordinates. A short tube sweeps over the streamlines, the parallel coordinate view of the streamline vertices is shown in the inset window at the top right corner. In the parallel coordinate view the coordinates of a vertex are placed equally spaced along the y axis, the magnitude of the coordinate is used as the z component of the resulting vertex. Subsequent streamline vertices are displaced along the x axis.
F1 attracts nearby states locally, finally repelling them either into F0 or into infinity. The repelling direction of F1 forms a heteroclinic connection with F0. The image shows the Evolution of a few states around F0 and F1 for m=0.2
The evolution of streamlines around the heteroclinic connection can be seen in the following image:
The slow chhanges of the system´s behaviour without changing the global topology are depicted in this movie