Homoclinic loops

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... 

The multi-dimensional extension of two-dimensional rough systems is the Morse-Smale systems discussed in Sec. 7.4. The list of limit sets of such a system includes equilibrium states and periodic orbits only furthermore, such systems may only have a finite number of them. Morse-Smale systems do not admit homoclinic trajectories. Homoclinic loops to equilibrium states may not exist here because they are non-rough — the intersection of the stable and unstable invariant manifolds of an equilibrium state along a homoclinic loop cannot be transverse. Rough Poincare homoclinic orbits (homoclinics to periodic orbits) may not exist either because they imply the existence of infinitely many periodic orbits. The Morse-Smale systems have properties similar to two-dimensional ones, and it was presumed (before and in the early sixties) that they are dense in the space of all smooth dynamical systems. The discovery of dynamical chaos destroyed this idealistic picture. 

The spiral-like shape of this attractor follows from the shape of homoclinic loops to a saddle-focus (2, 1) which appear to form its skeleton. Its wildness is due to the simultaneous existence of saddle periodic orbits of different topological type and both rough and non-rough Poincare homoclinic orbits. 

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. 

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). 

Another typical codimension-one bifurcation (left untouched in this book) within the class of Morse-Smale systems includes the so-called saddle-saddle bifurcations, where a non-rough saddle equilibrium state with one zero characteristic exponent (the others lie in both left and right half-planes) coalesces with another saddle having a different topological type. If, in addition, the stable and unstable manifolds of the saddle-saddle point intersect each other transversely along some homoclinic orbits, then as the bifurcating point disappears, saddle periodic orbits are born from the homoclinic loops. If there is only one homoclinic loop, then only one periodic orbit is born from it, and respectively, this bifurcation does not lead the system out of the Morse-Smale class. However, if there are more than one homoclinic loops, a hyperbolic limit set with infinitely many saddle periodic orbits will appear after the saddle-saddle vanishes [135]. 

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. 

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. 

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. 

We end this section with a consideration of the homoclinic loop to a saddle-focus whose unstable manifold is one-dimensional. It is shown that when the saddle value is positive, infinitely many saddle periodic orbits coexist near such a homoclinic loop of the saddle-focus (Theorem 13.8). 

The existence of complex dynamics near a homoclinic loop to a saddle-focus was discovered by L. Shilnikov for the three-dimensional case in [131]. Subsequently, the four-dimensional case was considered in [132] and the general case in [136]. 

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). 

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]... 

Similarly, if there were a separatrix loop to a saddle at // = 0, it would be split for some non-zero /i, as shown in Fig. 7.1.2. We see that an arbitrarily small smooth perturbation of the vector field will modify the phase portrait of a system with a homoclinic loop or a heteroclinic connection this obviously means that such a system is non-rough. 

Fig 7.1.2. (a) A homoclinic loop to a saddle is structurally unstable. A separatrix behavior (b) prior and (c) after the loop. 

Fig. 7.2.1. The homoclinic loop to saddle is an tJ-limit for a trajectory from the interior region.

E) there exists no separatrix which tends to a homoclinic loop of a saddle (in forward or backward time), as depicted in Fig. 8.1.5. 

Fig. 8.1.6. (a) A structurally unstable saddle connection after the disappearance of a saddle-node cycle in Pig. 8.1.4 (b) Phase plane after the splitting of the homoclinic loop in Fig. 8.1.5. 

The simplest example of such a situation is a homoclinic loop to a saddle-focus in the three-dimensional system... 

Another example is the system shown in Fig. 8.2.1 containing a homoclinic loop r to a saddle-focus. If the saddle index... 

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... 

The boundaries of the second type correspond to the merging of a periodic orbit into an equilibrium state (Sec. 11.5) or to a homoclinic loop, or a blue-sky catastrophe (Chaps. 12 and 13). 

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. 

It follows from our assumption that F tends to O bi-asymptotically, i.e. as t —> 00. In other words, the trajectory F is homoclinic to O. The imion FUO is a closed curve which is called a homoclinic loop of the saddle-node. 

We will show below that one can choose a small neighborhood V of the homoclinic loop so that for all small p the forward trajectories of points in V remain in it forever. Moreover, at p 0, all trajectories in V come into the... 

Theorem 12.1. The disappearance of the saddle-node equilibrium with the homoclinic loop results in the appearance of a stable periodic orbit of period... 

Fig. 12.1.2. Two cross-sections 5o and Si to the homoclinic loop F are chosen near the saddle-node equilibrium O.

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). 

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... 

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