Bifurcation of the homoclinic

In Sec. 13.5 we consider the bifurcation of the homoclinic loop of a saddle without any restrictions on the dimensions of its stable and imstable manifolds. We prove a theorem which gives the conditions for the birth of a single periodic orbit from the loop [134], and also formulate (without proof) a theorem on complex dynamics in a neighborhood of a homoclinic loop to a saddle-focus. Here, we show how the non-local center manifold theorem (Chap. 6 of Part I) can be used for simple saddles to reduce our analysis to known results (Theorem 13.6). 

Fig. 13.7.2. The bifurcations of the homoclinic-8 in the symmetric case. An outward breakdown of both homoclinic loops gives birth to a large symmetric periodic orbit. When the loops split inwards, a periodic orbit bifurcates from each of the loops.

Figure 14. Characterization of the homoclinic bifurcation near pL for x = 0.57, T p) = 0[ n pL — p)], where T is the period of the instantaneous angular velocity 0(r) = d o/dr. The solid line is the best fit to the theoretically calculated values ( ). Inset time evolution of n(r) at p = 1.02025. A perfect agreement with the calculated values (fiUed circles) is obtained with the parameterizations a + b n pL — p) for a 8.232 and b —2.406 (solid line).

The two kinds of expansions of U around F and S are suitable to describe respectively the Hopf bifurcation and the bifurcation of the limit cycle from the homoclinic trajectory. The results are qualitatively in agreement with those obtained by BAESENS and NICOLIS but our approach does not lead to the exact condition of existence of, the homo-clinic trajectory. 

A straightforward generalization of two-dimensional bifurcations was developed soon after. So were some natural modifications such as, for instance, the bifurcation of a two-dimensional invariant torus from a periodic orbit. Also it became evident that the bifurcation of a homoclinic loop in high-dimensional space does not always lead to the birth of only a periodic orbit. A question which remained open for a long time was could there be other codimension-one bifurcations of periodic orbits Only one new bifurcation has so far been discovered recently in connection with the so-called blue-sky catastrophe as found in [152]. All these high-dimensional bifurcations are presented in detail in Part II of this book. 

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). 

In Chap. 13 we will consider the bifurcations of a homoclinic loop to a saddle equilibrium state. We start with the two-dimensional case. First of all, we investigate the question of the stability of the separatrix loop in the generic case (non-zero saddle value), as well as in the case of a zero saddle value. Next, we elaborate on the cases of arbitrarily finite codimensions where the so-called Dulac sequence is constructed, which allows one to determine the stability of the loop via the sign of the first non-zero term in this sequence. 

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.

We study some homoclinic bifurcations of codimension two in Secs. 13.6. In Sec. 13.7, we review the results on the bifurcations of a homoclinic-8, and on the simplest heteroclinic cycles. 

If rf and FF coincide, then F = F = F is called a separatrix loop or a homoclinic loop). The closure T of the separatrix loop is an invariant closed set r = OUF. Our goal of this section is to describe the behavior of trajectories in a sufficiently small neighborhood of F (the bifurcations of the separatrix loop will be analyzed in the following section). 

Fig. 13.3.3, Bifurcation diagram of the homoclinic loop to a saddle with zero saddle value in the non-orient able case (—1 < A < 0). The governing parameters are the same as in Fig. 13.3.1.

If (2/1 j 2/2) is a solution of this system, then (2/2,2/1) is a solution as well. There is also the solution 2/1 = 2/2 = 2/o where yo is the imique fixed point of the map (13.3.8), which always exists for p > 0. Therefore, to prove that there are no saddle-node orbits of period two, it suffices to check that system (13.3.8) has no more than three solutions, including multiplicity. This verification will be performed in Sec. 13.6 for a more general system (see (13.6.26)), corresponding to the bifurcation of a homoclinic loop of a multi-dimensional saddle with... 

This result gives us the last known principal (codimension one) stability boundary for periodic orbits. We will see below (Theorems 13.9 and 13.10) that the other cases of bifurcations of a homoclinic loop lead either to complex dynamics (infinitely many periodic orbits), or to the birth of a single saddle periodic orbit. 

Fig. 13.6.1. The inclination-flip bifurcation A = 0) is due to a violation of the transversality of the intersection of and at the points of the homoclinic loop F.

We will analyze the following three cases of codimension-two bifurcations of such homoclinic loops. 

In general, the bifurcation of a homoclinic butterfly is of codimension two. However, the Lorenz equation is symmetric with respect to the transformation (x y z) <-)> (—X, —y z). In such systems the existence of one homoclinic loop automatically implies the existence of another loop which is a symmetrical image of the other one. Therefore, the homoclinic butterfly is a codimension-one phenomenon for the systems with symmetry. 

In order to resolve this problem, it was proposed in [138] to study the homoclinic butterflies in the Cases A, B and C above. Namely, it was established that the bifurcation of a homoclinic butterfly results in the immediate appearance of a Lorenz attractor when... 

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. 

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