Finite cyclicity of nontrivial double loops

Abstract

Following Roussarie's program (1989) for solving the 16th Hilbert's Problem, it's shown that non-trivial double homoclinic loops with a degenerate hyperbolic saddle are of finite cyclicity in C infinity families of planar vector fields. For this, we study the zeros of the displacement map Delta by blowing up the analytic parameters and the phase variable by the translation parameter.