Dulac sequence

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. 

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. 

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. 

---