Limit cycles near a compound cycle in a near-Hamiltonian system with smooth perturbations.

In this paper, we give a simple relation between the coefficients appearing in the expansions of n + 2 (n ∈ Z + , n ≥ 2) Melnikov functions near a compound cycle C (n) , which can be used to simplify some computations. We further give some conditions for a general near-Hamiltonian system to have limit cycles as many as possible near C (n) . Based on this, for a quintic Hamiltonian system with a compound cycle C (2) we prove that it can produce at least 7 2 (n − 2) + 1 2 (1 + (− 1) n) limit cycles near C (2) under polynomial perturbation of degree n (n ≥ 2). • Relationship among the first order Melnikov functions. • Limit cycle bifurcations near a compound loop. • A sharp lower bound of the maximum number of limit cycles of a polynomial system. [ABSTRACT FROM AUTHOR]