Principal part of multi-parameter displacement functions

This paper deals with a perturbation problem from a period annulus, for an analytic Hamiltonian system [J.-P. Françoise, Ergodic Theory Dynam. Systems 16 (1996), no. 1, 87–96 ; L. Gavrilov, Ann. Fac. Sci. Toulouse Math. (6) 14(2005), no. 4, 663–682. The authors consider the planar polynomial multi-parameter deformations and determine the coefficients in the expansion of the displacement function generated on a transversal section to the period annulus. Their first result gives a generalization to the Françoise algorithm for a one-parameter family, following [J.-P. Françoise and M. Pelletier, J. Dyn. Control Syst. 12 (2006), no. 3, 357–369. The second result expresses the principal terms in …