Bifurcation of a separatrix loop

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... 

Bifurcations of a separatrix loop with zero saddle value... 

The question of the bifurcations of a separatrix loop to a saddle with zero saddle value (Tq was first considered by E. Leontovich. She had proven the following theorem ... 

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. 

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). 

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 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. 

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). 

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. 

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... 

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,... 

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). 

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... 

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. 

Of special consideration are systems with symmetry where both separatrix loops approach together the saddle point. Such a situation is rather trivial namely when the loops split inwards, each gives the birth to a single stable limit cycle, in view of Theorem 13.4.1. When the loops split outwards, the stability migrates to a large-amplitude symmetric stable periodic orbit that bifurcates from the homoclinic-8 as shown in Fig. 13.7.2. And that is it. This is the reason why the theory below focuses primarily on non-symmetric systems. 

In the case of a saddle (the leading characteristic exponent Ai is real), the bifurcation diagram depends on the signs of the separatrix values Ai and A2, as well as on the way the homoclinic loops F1 and F2 enter the saddle at t = -f 00. Let us consider first the case where F1 and F2 enter the saddle tangentially to each other, i.e. bifurcations of the stable homoclinic butterfly. 

Fig. 13.7.22. The bifurcation diagram for the heteroclinic connection in Fig. 13.7.12 when A > 0, A2 < 0, i/i > 1, 1/2 < 1 and 1/11/2 > 1. The system has one simple periodic orbit in regions 1, 2, 3 and 5, two periodic orbits (one simple and one of double period) in region 4, and no periodic orbits elsewhere. The stable periodic orbit loses stability on the curve PD corresponding to a period-doubling (flip) bifurcation. The unstable limit cycle of double period becomes a double-circuit separatrix loop on L. The stable simple limit cycle terminates on Li.

The first example illustrates one of the most typical bifurcations which occur in dissipative systems namely a stable periodic orbit L adheres to the homoclinic loop of a saddle. Denote the unstable separatrices of the saddle by Fi and F2. Let Fi form a homoclinic loop at the bifurcation point. Denote the limit set of the second separatrix by D(F2). In the general case fI(F2) is an attractor for instance, a stable equilibrium state, a stable periodic trajectory, or a stable torus, etc. Since inunediately after bifurcation a representative point will follow closely along F2, it seems likely that fl(F2) will become its new attractor. 

Fig. C.6.12. Plot of the x-coordinate of the equilibrium state versus z at e = 0. The symbols Xmim a max and (x) denote, respectively, the maximal, minimal and averaged values of the x-coordinates of the stable limit cycle which bifurcates from a stable focus at AH and terminates in the separatrix loop to the saddle O (see the next figure) at the point H z cz 2.086.

The point NS. This point is of codimension two as <7 = 0 here. The behavior of trajectories near the homoclinic-8, as well as the structure of the bifurcation set near such a point depends on the separatrix value A (see formula (13.3.8)). Moreover, they do not depend only on whether A is positive (the loops are orientable) or negative (the loops are twisted), but it counts also whether A is smaller or larger than 1. If A < 1, the homoclinic-8 is stable , and unstable otherwise. To find out which case is ours, one can choose an initial point close sufficiently to the homoclinic-8 and follow numerically the trajectory that originates from it. If the figure-eight repels it (and this is the case in Chua s circuit), then A > 1. Observe that a curve of double cycles with multiplier 4-1 must originate from the point NS by virtue of Theorem 13.5. 

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