Asymptotic expansion of the heteroclinic bifurcation for the planar normal form of the 1:2 resonance

Abstract

Agraïments: The first author is partially supported by CAPES and FAPESP. The second author is partially supported by CAPES, CNPq-Brazil, and FAPESP.

We consider the family of planar differential systems depending on two real parameters \[ x =y, y = _1 x _2 y x^3-x^2y.\] This system corresponds to the normal form for the 1:2 resonance which exhibits a heteroclinic connection. The phase portrait of the system has a limit cycle which disappears in the heteroclinic connection for the parameter values on the curve _2=c(_1)=-15_1 O(_1^2), _1<0. We significantly improve the knowledge of this curve in a neighborhood of the origin.