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

In this paper, we consider Liénard systems of the form dxdt=y,dydt=.(x+bx3.x5)+.(.+.x2+.x4)y, where b . ., 0 < |.| . 1, (., ., .) . D . .3 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 is 2 (counting the multiplicity) and this upper bound is a sharp one.