Normal forms near a symmetric planar saddle connection

Abstract

The present paper studies vector fields of the form (x) over dot = (q/2 + O (1 - x(2))) (1 - x(2)) + O (y), (y) over dot =(px + 0 (1 - x(2))) y + O (y(2)), 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. (C) 2016 Elsevier Inc. All rights reserved.