Perturbation of a Period Annulus with a Unique Two-Saddle Cycle in Higher Order Hamiltonian

Abstract EN

In this paper, we study the number of limit cycles emerging from the period annulus by perturbing the Hamiltonian system x ˙ = y , y ˙ = x ( x 2 - 1 ) ( x 2 + 1 ) ( x 2 + 2 ) .

The period annulus has a heteroclinic cycle connecting two hyperbolic saddles as the outer boundary.

It is proved that there exist at most 4 and at least 3 limit cycles emerging from the period annulus, and 3 limit cycles are near the boundaries.