The nonsmooth saddle-node bifurcation

We can distinguish two types of nonsmooth saddle-node bifurcations: one-sided and two-sided. This distinction is best observed in the lift. The one-side nonsmooth saddle-node bifurcation is shown in Figure 9: An attracting and a repelling invariant curve seem to collide in a dense orbit, leading to unidirectional diffusive motion.

Figure 9:

One-sided nonsmooth saddle-node bifurcation with

= 0.001 and

= 2.58: (left) before the collision (K = 0.865): attractors in black, blue and green, repellors in red, purple and orange; (right) after the collision (K = 0.868): attractor with the same initial condition as the black attracting invariant curve (left); repellor with the same initial condition as the purple repelling invariant curve. The attractor moves upwards in forward time and the repellor moves downwards in reverse time. The rotation number is nonzero.

In the two-sided nonsmooth saddle-node bifurcation the attractor collides both with the repellor immediately above and below it. This leads to bidirectional diffusive motion as shown in Figure 10.

Figure 10:

Two-sided nonsmooth saddle-node bifurcation with

= 0 and

= 2.58 (cf. the transition from the white to the blue region in Figure 11: (left) before the collision (K = 0.927): attractors in black, blue and green, repellors in red, purple and orange; (right) after the collision (K = 0.928): SNA with the same initial condition as the black attracting invariant curve (left). The strange repellor is not shown.

We find the nonsmooth saddle-node bifurcations on the boundary of the phase-locked region. The two-sided nonsmooth saddle-node bifurcation is the transition from the white region to the blue region in Figure 11 (left).


Figure 11:	Section = 0 of the third (left) and fourth (right) overlap region. In the region of large phase sensitivity the attractor is very wrinkled or even an SNA.

Both one-sided and two-sided nonsmooth saddle-node bifurcations take place in the red region in (, K, )-space; see Figure 1 and the enlargement of the third region of overlap in Figure 12. The one-sided bifurcation happens when the tongue boundary has non-negligable height.


Figure 12:	A small part of the boundary of the phase-locked region with zero rotation number; in [2.52, 2.62] runs from right to left, K in [0, 1] also from right to left, and in [0, 0.004] from bottom to top; compare Figure 11 (left). A grid of 80 X 160 points in the (, K)-plane is taken. Red regions correspond to transitions to SNAs (the yellow spots are due to the finithe threshold).

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Created by Hinke Osinga
Last modified: Wed May 17 16:11:04 2000