Next: The bifurcation structure in the (, K)-plane

The internal structure of the main tongue for K = 0.8

If we cut a section along K = 0.8 in Figure 1 we get an idea of the complex bifurcation structure inside the phase-locked region with zero rotation number. The tongue seems to consist of a set of bubbles that overlap slightly; see Figure 2

Figure 2:	The bifurcation structure for K = 0.8 of the phase-locked region with zero rotation number (left) with detail of the first overlap (right).

Along the line = 0 we see that a second attractor gets created in a pitchfork bifurcation as increases. Another pitchfork bifurcation destroys the attractor again. However, in the second region of overlap, the situation is more complicated, as can be seen in Figure 3 (right), Figure 4 and Figure 5.

Figure 3:	Sketch of the bifurcations along the line = 0 and K = 0.8 in the first (left) and the second (right) overlap. Shown are invariant circles (represented by one point) versus . Closed curves represent stable and dashed curves represent unstable circles. The colors correspond with the colors of the bifurcation curves in Figure 2, Figure 4, and Figure 5.

Figure 4:	The bifurcation structure for K = 0.8 in the second overlap (left) seems to be the same as in Figure 2 (right). However, an enlargement (right) shows that the structure is much more complicated.

Further enlargements of the second overlap in Figure 5 show the full complexity in the main tongue.

Figure 5:	Details of the bifurcation structure for K = 0.8 in the second overlap. The last pitchfork bifurcation on the line = 0 is subcritical (right) as opposed to supercritical in Figure 2 (right).

Next: The bifurcation structure in the (, K)-plane

Created by Hinke Osinga
Last modified: Wed May 17 16:11:12 2000