Separatrix value

In the same section we give the bifurcation diagrams for the codimension two case with a first zero saddle value and a non-zero first separatrix value (the second term of the Dulac sequence) at the bifurcation point. Leontovich s method is based on the construction of a Poincare map, which allows one to consider homoclinic loops on non-orientable two-dimensional surfaces as well, where a small-neighborhood of the separatrix loop may be a Mobius band. Here, we discuss the bifurcation diagrams for both cases. 

Let us now consider the case of codimension two in more detail. Recall that this case is distinguished by two conditions the first is the existence of a separatrix loop, and the second condition is the vanishing of the first saddle value t7o while the first separatrix value s is non-zero. The latter is equivalent to i4 1. We will assume that A <1 because the case A> 1 follows directly by a reversion of time. 

We will define below the quantity A in terms of the Poincare map. It is an analogue of the separatrix value A introduced in Secs. 13.1 and 13.2 for the two-dimensional case. Recall that A is always non-zero in dimension two. However, in the multi-dimensional case the non-vanishing of. 4 is an essential assumption. 

The bifurcation unfoldings for Cases B and C are identical and shown in Fig. 13.6.4. Here, p is the splitting parameter of the homoclinic loop, and A is the separatrix value. Since in Sec. 13.4 the separatrix value A was defined only when the loop does not belong to we must specify its meaning for Case C. 

Recall that the non-leading manifold is (n — l)-dimensional. It partitions into two components. If the loop F lies in then a small perturbation may make it miss so that it enters the saddle from either component of We will show (Subsec. 13.6.2) that when the loop is moved from one component to the other, it is accompanied by a change in the sign of the separatrix value A. 

In the case of a saddle (the leading characteristic exponent Ai is real), the bifurcation diagram depends on the signs of the separatrix values Ai and A2, as well as on the way the homoclinic loops F1 and F2 enter the saddle at t = -f 00. Let us consider first the case where F1 and F2 enter the saddle tangentially to each other, i.e. bifurcations of the stable homoclinic butterfly. 

The bifurcation diagrams for the case where both Ox and O2 are saddles are shown in Figs. 13.7.12-13.7.15. Here, if both the separatrix values are positive, the only possible heteroclinic connections are the original ones which exist at... 

The separatrix values Ai 2 on the heteroclinic orbits are defined in the same way as in the case of homoclinic loops. Note that both cases Ai > 0, A2 > 0 ... 

All these intersection points correspond to a non-orientable homoclinic loop of O2 (the separatrix value on the loop is negative), and they accumulate at the point where the separatrix value vanishes on the loop. The segment of L2 between this point and the point P (the point Q at A2 < 0) corresponds to... 

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. 

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,... 

A fragment of its (r, cr) bifurcation diagram is shown in Fig. C.7.14. Detect the points where the path cr = 10 intersect the curve HB of the homoclinic butterfly and the curve LA on which the one-dimensional separatrices of the saddle tend to the saddle periodic orbits. Find the point on the curve LA above which the Lorenz attractor does not arise upon crossing LA towards larger values of r. The dashed line passing through the T-point in Fig. C.7.14 corresponds to the moment of the creation of the hooks in the two-dimensional Poincare map when the separatrix value varishes A — 0 (see discussion on the Shimizu-Morioka model). ... 

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