Separatrix loop

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 bifurcation of a separatrix loop of a saddle-node was discovered by Andronov and Vitt [14] in their study of the transition phenomena from synchronization to beating modulations in radio-engineering. Specifically, they had studied the periodically forced van der Pol equation... 

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 case of a non-zero saddle value, we present the classical result by Andronov and Leontovich on the birth of a unique limit cycle at the bifurcation of the separatrix loop. Our proof differs from the original proof in [9] where Andronov and Leontovich essentially used the topology of the plane. However, following Andronov and Leontovich we present our proof under a minimal smoothness requirement (C ). 

The case of zero saddle value was considered by E. A. Leontovich in 1951. Her main result is presented in Sec. 13.3, rephrased in somewhat different terms in the case of codimension n (i.e. when exactly the first (n — 1) terms in the Dulac sequence are zero) not more than n limit cycles can bifurcate from a separatrix loop on the plane moreover, this estimate is sharp. 

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. 

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. 8.1.5. The separatrix loop of the saddle 0 is the cj-limit of the separatrix of another saddle O2 in the interior region enclosed by the loop emanating from and terminating at 0. ...

An analogous situation occurs when the system has a separatrix loop to a non-resonant saddle (i.e. its saddle value cr = Ai + A2 0) which is the a -limit of a separatrix of another saddle Oi (see condition (E) and Fig. 8.1.5). In this case, the bifurcation surface is also unattainable from one side, where close nonrough systems may have a heteroclinic connection, as shown in Fig. 8.1.6(b). 

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. 13.1.1. A separatrix loop F is within an annulus consisting of an outer neighborhood V and an inner neighborhood U.

The problem of stability of a separatrix loop on a plane is easily solved when the so-called saddle value... 

It is immediately seen from the above formula that yo < yo for any sufficiently small positive yo, provided i/ > 1 (or, provided that i/ = 1 and A < 1). Therefore, in this case, the iterations of any point by the map T converge to the fixed point yo = 0. The latter is the point of intersection of the separa-trix loop r with 5q. It follows that any trajectory starting from the side of a positive yo must converge to the loop F as t —> -hoc. This means that the separatrix loop is asymptotically one-side stable if cto < 0, or if cto =0 and A < 1. 

A Taylor expansion for the global map Ti 5i —> Sq near the separatrix loop r can be written in the form... 

Theorem 13-3. The stability of a separatrix loop is determined by the first non-zero entry in a Dulac sequence if the first non-zero entry is negative then the loop is stable. Otherwise, if it is positive, then the loop is unstable. 

This theorem in the analytic case is due to Dulac. He had also shown that if the system is analytic and all entries of the sequence (13.1.20) are zero, then the system is integrable (Hamiltonian), and a small neighborhood U of the separatrix loop is filled by periodic orbits. These results had enabled Dulac to show that in the nondegenerate case of polynomial vector fields limit cycles cannot accumulate to a separatrix loop. 

Two-dimensional systems having a separatrix loop to a saddle with non-zero first saddle value ao form a bifurcation set of codimension one. Therefore, we can study such homoclinic bifurcations using one-parameter families. 

Consider a continuous one-parameter family of C -smooth (r > 1) systems on a plane which have a saddle equilibrium state O. Suppose that at /i = 0, the system has a separatrix loop of the saddle i.e. the separatrix T [ coincides with the separatrix at /i = 0. 

If o o < 0, then for sufficiently small p> there exists a unique stable limit cycle L p) in U which as p +0 gets closer to the saddle and becomes the separatrix loop at p = 0 see Fig. 13.2.1). When /a < 0, there are no limit cycles. 

Fig. 13.2.1. Planar bifurcation of a stable separatrix loop of a saddle with (Tq < 0.

Bifurcation of a limit cycle from a separatrix loop... 

The term a[p) governs the splitting of the separatrix loop (see Fig. 13.2.3) it follows from (13.2.2) that a[p) is equal to the difference between the y-coordinates of the points and TiM (the latter is the point where the xmstable separatrix F]" first intersects Sq). By assumption, the separatrix splits inwards for /x > 0, and outwards for p <0. Thus, sign a p) = sign p. 

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