Double separatrix loop

Fig. 13.3.2. Lamerey diagram corresponding to the double separatrix loop.

L2 corresponds to a double separatrix loop on the Mobius band. The saddle value (Tq is positive on this curve ... 

In the region Z i, there are no limit cycles. On the curve L4, upon moving from D towards D2j a stable limit cycle is born from a simple separatrix loop. An imstable double-loop limit cycle bifurcates from a double separatrix loop with (To > 0 on L2. Thus, in the region D3, there are two limit cycles one stable and the other is unstable. The unstable double limit cycle merges with the stable limit cycle on the curve Li. After that only one single-circuit unstable limit cycle remains in region D4. It adheres into the homoclinic loop on the curve L3. 

Fig. 13.7.22. The bifurcation diagram for the heteroclinic connection in Fig. 13.7.12 when A > 0, A2 < 0, i/i > 1, 1/2 < 1 and 1/11/2 > 1. The system has one simple periodic orbit in regions 1, 2, 3 and 5, two periodic orbits (one simple and one of double period) in region 4, and no periodic orbits elsewhere. The stable periodic orbit loses stability on the curve PD corresponding to a period-doubling (flip) bifurcation. The unstable limit cycle of double period becomes a double-circuit separatrix loop on L. The stable simple limit cycle terminates on Li.

Fig. C.7.11. Twisted (A < 0) and orientable A > 0) double homoclinic loops. The two-dimensional Poincare map has a distinctive hook-like shape after the separatrix value A becomes negative.

Another codimension-two homoclinic bifurcation in the Shimizu-Morioka model occurs at (a 0.605,6 0.549) on the curve H2 corresponding to the double homoclinic loops. At this point, the separatrix value A vanishes and the loops become twisted, i.e. we run into inclination-flip bifurcation [see Figs. 13.4.8 and C.7.11]. The geometry of the local two-dimensional Poincare map is shown in Fig. 13.4.5 and 13.4.6. To find out what our case corresponds to in terms of the classification in Sec. 13.6, we need also to determine the saddle index 1/ at this point. Again, as in the case of a homoclinic loop to the saddle-focus, it is very crucial to determine whether 1/ < 1/2 or 1/ > 1/2. Simple calculation shows that z/ > 1/2 for the given parameter values. Therefore,... 

The point NS. This point is of codimension two as <7 = 0 here. The behavior of trajectories near the homoclinic-8, as well as the structure of the bifurcation set near such a point depends on the separatrix value A (see formula (13.3.8)). Moreover, they do not depend only on whether A is positive (the loops are orientable) or negative (the loops are twisted), but it counts also whether A is smaller or larger than 1. If A < 1, the homoclinic-8 is stable , and unstable otherwise. To find out which case is ours, one can choose an initial point close sufficiently to the homoclinic-8 and follow numerically the trajectory that originates from it. If the figure-eight repels it (and this is the case in Chua s circuit), then A > 1. Observe that a curve of double cycles with multiplier 4-1 must originate from the point NS by virtue of Theorem 13.5. 

---