On the stability of double homoclinic and heteroclinic cycles

The Global Bifurcation of a Cubic System

“…For fixed a 4 > 0 and small, according to Lemma 2, when a 1 < K 1 (ε, a 4 ), and 0 < |a 1 − K 1 (ε, a 4 )| a 4 , the double-homoclinic loop L * and homoclinic loop L * 1 , L * 2 become unstable [2,4,5] , then there exists a large stable limit cycle Γ 1 which is near but outside the double-homoclinic loop L * , and two small stable limit cycles L 1 1 , L 1 2 near and inside L * 1 , L * 2 appeare. For a 4 , a 1 fixed and small, according to Lemma 1, when 0 < a 3 − K 3 (ε, a 1 , a 4 ) |a 1 − K 1 (ε, a 4 )| a 4 , then the homoclinic loop L * 2 breaks up and a small unstable limit cycle L 2 2 appears near L * 2 and outside L 1 2 .…”

Bifurcations of limit cycles in a perturbed quintic Hamiltonian system with six double homoclinic loops

“…The bifurcation of limit cycles in a Z 8 -equivariant planar vector field of degree 7 is studied in [14] by Li and 49 limit cycles are obtained. In [6,9], the method of double homoclinic loops bifurcation is posed to study the the bifurcation of limit cycles. In this paper, using the method above and the Hopf bifurcation method, we study the bifurcation of limit cycles in the following Z 6 -quintic Hamiltonian perturbed by seven order polynomials.…”

On the limit cycles of a quintic planar vector field

“…When the double homoclinic loops of (1.1) are isolated, we can define the inner stability and the outer stability of them(see [13,19] for details). Now we give the proof of Theorem 1.1.…”