Normal linearization and transition map near a saddle connection with symmetric resonances

We consider a heteroclinic connection in a planar system, between two symmetric hyperbolic saddles of which the eigenvalues are resonant. Starting with a C∞ normal form, defined globally near the connection, we normally linearize the vector field by using finitely smooth tags of logarithmic form. We furthermore define an asymptotic entry-exit relation, and we discuss on two examples how to deal with counting limit cycles near a limit periodic set involving such a connection.