On-edge homoclinic loop

We remark that Theorem 12.1 remains valid also in the case where the separatrix enters an edge of the node region, i.e. F C. However, a complete bifurcation analysis in this case requires an additional governing parameter. It is introduced in the following way. Let us build a cross-section Sq to the on-edge homoclinic loop F, i.e. we define So = y = d, a < d/2, as depicted in Fig. 12.1.4. At the bifurcation point, the separatrix F intersects Sq at some... 

As in Theorem 12.2, it is easy to construct a small neighborhood V of the on-edge homoclinic loop F U O such that for all small p and e, the forward trajectory of any point in V stays there forever. 

Fig. 12.1.6. Bifurcation diagram for the on-edge homoclinic loop to a saddle-node equilibrium.

Fig. 12.1.7. The boundary of the stability region of the periodic orbit born at the bifurcation of the on>edge homoclinic loop to a degenerate saddle-node corresponds to the homoclinic loop of the border saddle equilibrium state Oi.

Remark. This statement remains valid (with obvious modifications) also in the case of the on-edge homoclinic loop to a degenerate saddle-node. In this case, /i is a vector of parameters (of dimension equal to the number of zero Lyapunov values plus one), and an additional bifurcation parameter e is introduced as before. A stable periodic orbit exists when the saddle-node disappears (the region /j> Dq m our notations), or when e > hhomi/j) fjL Do. Here, the surface e = hhomifJ ) corresponds to the homoclinic loop of the border saddle equilibrium Oi, as illustrated in Fig. 12.1.7. 

The non-smooth case appears, for example, when Wq touches the strong-stable manifold Wq, as shown in Fig. 12.2.2. The latter, in turn, may be detected via a small time-periodic perturbation of a system with an on-edge homoclinic loop to a saddle-node (see the previous section). Generically, the non-transversality of with respect to is also preserved under small smooth perturbations (say, if the tangency between and the corresponding leaf of is quadratic). 

In Chap. 12 we will study the global bifurcations of the disappearance of saddle-node equilibrium states and periodic orbits. First, we present a multidimensional analogue of a theorem by Andronov and Leontovich on the birth of a stable limit cycle from the separatrix loop of a saddle-node on the plane. Compared with the original proof in [130], our proof is drastically simplified due to the use of the invariant foliation technique. We also consider the case when a homoclinic loop to the saddle-node equilibrium enters the edge of the node region (non-transverse case). 

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