Bifurcation of a homoclinic loop

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. 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. 

Section 13.6 discusses three main cases of codimension-two bifurcations of a homoclinic loop to a saddle. These cases were selected by Shilnikov in [138]... 

Bifurcations of a homoclinic loop to a saddle-node equilibrium state... 

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. 

We remark that a systematic study of the bifurcations of a homoclinic loop which is a limit set for the other separatrix was undertaken in [69]. 

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). 

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. 

A homoclinic bifurcation is a composite construction. Its first stage is based on the local stability analysis for determining whether the equilibrirun state is a saddle or a saddle-focus, as well as what the first and second saddle values are, and so on. On top of that, one deals with the evolution of a -limit sets of separatrices as parameters of the system change. A special consideration should also be given to the dimension of the invariant manifolds of saddle periodic trajectories bifurcating from a homoclinic loop. It directly correlates with the ratio of the local expansion versus contraction near the saddle point, i.e. it depends on the signs of the saddle values. 

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,... 

We have seen that homoclinic bifurcations in symmetric systems have much in common. Let us describe next the universal scenario of the formation of a homoclinic loop to a saddle-focus in a typical system. In particular, this mechanism works adequately in the Rossler model, in the new Lorenz models, in the normal form (C.2.27), and many others. 

What cannot be obtained through local bifurcation analysis however, is that both sides of the one-dimensional unstable manifold of a saddle-type unstable bimodal standing wave connect with the 7C-shift of the standing wave vice versa. This explains the pulsating wave it winds around a homoclinic loop consisting of the bimodal unstable standing waves and their one-dimensional unstable manifolds that connect them with each other. It is remarkable that this connection is a persistent homoclinic loop i.e. it exists for an entire interval in parameter space (131. It is possible to show that such a loop exists, based on the... 

Surprisingly, even non-rough systems of codimension one may have infinitely many moduli. Of course, since the models of nonlinear dynamics are explicitly defined dynamical systems with a finite set of parameters, this creates a new obstacle which the classical bifurcation theory has not nm into. Although the case of homoclinic loops of codimension one does not introduce any principal problem, nevertheless codimensions two and higher are much less trivial as, for example, in the case of a homoclinic or heteroclinic cycle including a saddle-focus where the structure of the bifurcation diagrams is directly determined by the specific values of the corresponding moduli. 

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). 

The bifurcations of periodic orbits from a homoclinic loop of a multidimensional saddle equilibrium state are considered in Sec. 13.4. First, the conditions for the birth of a stable periodic orbit are found. These conditions stipulate that the unstable manifold of the equilibrium state must be one-dimensional and the saddle value must be negative. In fact, the precise theorem (Theorem 13.6) is a direct generalization of the Andronov-Leontovich theorem to the multi-dimensional case. We emphasize again that in comparison with the original proof due to Shilnikov [130], our proof here requires only the -smoothness of the vector field. 

There is no doubt that some subtle aspects of the behavior of homoclinic and heteroclinic trajectories might not be important for nonlinear dynamics since they refiect only fine nuances of the transient process. On the other hand, when we deal with non-wandering trajectories, such as near a homoclinic loop to a saddle-focus with i/ < 1, the associated fi-moduli (i.e. the topological invariants on the non-wandering set) will be of primary importance because they may be employed as parameters governing the bifurcations see [62, 63]. 

As an example, let us consider the codimension-one bifurcation of three-dimensional systems with a homoclinic loop to a saddle-focus with the negative... 

To study such bifurcations one should understand the structure of the limit set into which the periodic orbit transforms when the stability boundary is approached. In particular, such a limit set may be a homoclinic loop to a saddle or to a saddle-node equilibrium state. In another bifurcation scenario (called the blue sky catastrophe ) the periodic orbit approaches a set composed of homoclinic orbits to a saddle-node periodic orbit. In this chapter we consider homoclinic bifurcations associated with the disappearance of the saddle-node equilibrium states and periodic orbits. Note that we do not restrict our attention to the problem on the stability boundaries of periodic orbits but consider also the creation of invariant two-dimensional tori and Klein bottles and discuss briefly their routes to chaos. 

The bifurcation of a limit cycle from the homoclinic loop to the saddle-node was first discovered by Andronov and Vitt in their study of the Van der Pol equation with a small periodic force at a 1 1 resonance ... 

In terms of the original variable (p — — ut, the stationary value of (the equilibrium state of system (12.1.9)) corresponds to an oscillatory regime with the same frequency as that of the external force. The periodic oscillations of (the limit cycle in (12.1.9)) correspond to a two-frequency regime. Hence, the above bifurcation scenario of a limit cycle from a homoclinic loop to a saddle-node characterizes the corresponding route from synchronization to beat modulations in Eq. (12.1.7). 

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). 

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). 

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.

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. 

One can now notice that in the case of a saddle, the above assumptions coincide with the conditions of Theorem 6.1 in Sec. 6.1 of Part I of this book. This theorem constitutes the existence (for the system with a homoclinic loop itself, and for all nearby systems) of a repelling smooth (C ) invariant (m + 1)-dimensional manifold M which contains all orbits staying in a small neighborhood U of r for all positive times. It follows that our problem on the bifurcations of P can be reduced here to that on the bifurcations on A4. 

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.

Case A corresponds to the boundary between positive and negative saddle values. Cases B and C correspond to a violation of the non-degeneracy conditions (1) and (2) of Theorem 13.4.2, respectively (the birth of a saddle periodic orbit from a homoclinic loop with positive saddle value). Condition (3) in the last two cases is necessary to exclude the transition to complex dynamics via these bifurcations (some of the cases with complex dynamics were studied in [44, 70, 78, 96, 79, 71, 72]). 

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. 

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.

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