A note on a generalization of Françoise's algorithm for calculating higher order Melnikov functions

Chen, 1971, Algebras of iterated path integrals and fundamental groups, Trans. Amer. Math. Soc, 156, 359, 10.1090/S0002-9947-1971-0275312-1 Françoise, 1996, Successive derivatives of a first return map, application to the study of quadratic vector fields, Ergodic Theory Dynam. Systems, 16, 87, 10.1017/S0143385700008725 Gavrilov Iliev, 1996, The cyclicity of the period annulus of the quadratic Hamiltonian triangle, JDE, 128, 309, 10.1006/jdeq.1996.0097 Jebrane, 2002, A generalization of Françoise's algorithm for calculating higher order Melnikov functions, Bull. Sci. Math, 126, 705, 10.1016/S0007-4497(02)01138-7 Yakovenko, 1995, A geometric proof of the Bautin theorem, AMS Transl. (2), 165, 203 Zoladek, 1995, The cyclicity of triangles and segments in quadratic systems, JDE, 122, 137, 10.1006/jdeq.1995.1142