Normal forms near a symmetric planar saddle connection

The present paper studies vector fields of the form x˙=(q/2+O(1−x2))(1−x2)+O(y), y˙=(px+O(1−x2))y+O(y2), which contain a separatrix connection between hyperbolic saddles with opposite eigenvalues where the connection is fixed. Smooth semi-local normal forms are provided in vicinity of the connection, both in the resonant and non-resonant case. First, a formal conjugacy is constructed near the separatrix. Then, a smooth change of coordinates is realized by generalizing known local results near the hyperbolic points.