Expansion coefficients and their relation for Melnikov functions near polycycles

Abstract

Under a suitable assumption we obtain some new results on expansion coefficients and their relation for the first order Melnikov functions near any m-polycycle with hyperbolic saddles, $m\in N^{+}$, which establish a general bifurcation theory on limit cycles near the m-polycycles. As an application we consider 2-polycyclic bifurcations for a φ-Laplacian Liénard system and gain the number of limit cycles near the polycycle with two hyperbolic saddles.