Double homoclinic loop

Since A < 0, the Poincare map is decreasing. The new feature in this case is that such maps may have orbits of period two, which correspond to the so-called double limit cycles. They may appear via a period-doubling bifurcation (a fixed point with a multiplier equal to —1) or via a bifurcation of a double homoclinic loop. The latter corresponds to the period-two point of the Poincare map at 2/ = 0 (see Fig. 13.3.2). 

Thus, the saddle-node bifurcation leads to the appearance of a pair of fixed points, stable and unstable, whereas the period-doubling leads to the appearance of an unstable orbit of period two. An unstable fixed point vanishes at y = 0 (corresponding to a single-circuit homoclinic loop) when 0 = 0 and 0 = 7T. An unstable orbit of period two approaches y = 0 (a double homoclinic loop) when... 

A simple homoclinic loop corresponds to /x = 0. A double homoclinic loop... 

It is not hard to conclude from numerical experiments, which reveal the manner in which the separatrices converge to the homoclinic butterfly that A must be within the range (0,1). In this case, when a < 0, everything is simple the homoclinic butterfly splits into either two stable periodic orbits (Fig. C.7.8(g)), or just one stable symmetric periodic orbit (Fig. C.7.8(i)). It follows from Sec. 13.6 that when <7 > 0, two bifurcation curves originate from this codimensiomtwo point. They correspond to the saddle-node bifurcation (Fig. C.7.8(d)) and to the double homoclinic loop (Fig. C.7.8(f)). The... 

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,... 

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. 

Let us describe the essential bifurcations in this system on the path 6 = 2 as fjL increases. On the left of the curve AH, the equilibrium state 0 is stable. It undergoes the super-critical Andronov-Hopf bifurcation on the curve AH. The stable periodic orbit becomes a saddle through the period-doubling bifurcation that occurs on the curve PD. Figure C.6.7 shows the unstable manifold of the saddle periodic orbit homeomorphic to a Mobius band. As a increases further, the saddle periodic orbit becomes the homoclinic loop to the saddle point 0(0,0,0,) at a 5.545. What can one say about the multipliers of the periodic orbit as it gets closer do the loop Can the saddle periodic orbit shown in this figure get pulled apart from the double stable orbit after the fiip bifurcation In other words, in what ways are such paired orbits linked in in R ... 

The point NSF a = 0 corresponds to a neutral saddle-focus. At this codimension-two point the dynamics of the trajectories near the homoclinic loops to the saddle-focus becomes chaotic. This bifurcation indeed proceeds the origin of the chaotic double scroll attractor in Chua s circuit. In the general case, this bifurcation was first considered in [29]. The complete unfolding of... 

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. 

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