Bifurcations of limit cycles from a quintic Hamiltonian system with a heteroclinic cycle

Abstract

In this paper, we consider Lienard systems of the form $$\frac{{dx}} {{dt}} = y, \frac{{dy}} {{dt}} = - \left( {x + bx^3 - x^5 } \right) + \varepsilon \left( {\alpha + \beta x^2 + \gamma x^4 } \right)y,$$ where b ∈ ℝ, 0 < |∈| ≪ 1, (α, β, γ) ∈ D ∈ ℝ3 and D is bounded. We prove that for |b| ≫ 1 (b < 0) the least upper bound of the number of isolated zeros of the related Abelian integrals Open image in new window is 2 (counting the multiplicity) and this upper bound is a sharp one.