Simple homoclinic loop

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

In the other remaining cases, quasiminimal attractors do not appear. In the case > 0 and A2 (see Fig. 13.7.7), there exist cycles only with the codes 1, 2 and 21, A = 1,2,... (here 1 " denotes the word consisting of k ones ), and the parameter plane is partitioned into a countable number of regions by the curves Ci, C2, C12, C 2ifc and 0121 (A = 1,2,...). Note that these curves accumulate onto the negative /i2 semi-axis where the separatrix Fi forms a simple homoclinic loop and the separatrix F2 tends to the loop as t —> +00. 

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). 

We should, however, stress that such a reduction to the two-dimensional case is not always possible. In particular, it cannot be performed when the equilibrium state is a saddle-focus. Moreover, under certain conditions, we run into an important new phenomenon when infinitely many saddle periodic orbits coexist in a neighborhood of a homoclinic loop to a saddle-focus. Hence, the problem of finding the stability boundaries of periodic orbits in multidimensional systems requires a complete and incisive analysis of all cases of homoclinic loops of codimension one, both with simple and complex dynamics. This problem was solved by L. Shilnikov in the sixties. 

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. 

Geometrically, this means that the linearized flow near the saddle contracts two-dimensional areas. This implies, in turn, that the local map between any two cross-sections is a contraction (see the proof of Lemma 13.3). By this reason, the dynamics of the system near such homoclinic loop remains simple. 

The situation which we consider here is a particular case of Theorem 13.9 of the next section. It follows from this theorem (applied to the system in the reversed time) that a single saddle periodic orbit L is born from a homoclinic loop it has an m-dimensional stable manifold and a two-dimensional unstable manifold. This result is similar to Theorem 13.6. Note, however, that in the case of a negative saddle value the main result (the birth of a unique stable limit cycle) holds without any additional non-degeneracy requirements (the leading stable eigenvalue Ai is nowhere required to be simple or real). On the contrary, when the saddle value is positive, a violation of the non-degeneracy assumptions (1) and (2) leads to more bifurcations. We will study this problem in Sec. 13.6. 

This theorem is a part of a more general assertion [including also the case of a multi-dimensional unstable manifold as well as saddle-foci of types (2,1) and (2, 2)] on complex dynamics near the homoclinic loop of a saddle-focus [136]. Condition p < 1 also known as the Shilnikov condition is very important here, because the structure of the phase space near the homoclinic loop is drastically changed in comparison to the case p > 1 covered by Theorem 13.6. The main bifurcations in the boundary case p = 1, when a small perturbation trigging the system into a homoclinic explosion from simple dynamics (p > 1) to complex dynamics (p < 1) were first considered in [29]. 

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... 

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,... 

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