Cyclicity of a fake saddle inside the quadratic vector fields

Abstract

This paper concerns the study of small-amplitude limit cycles that appear in the phase portrait near an unfolded fake saddle singularity. This degenerate singularity is also known as a impassable grain. The normal form of the unperturbed vector field is like a degenerate flow box. Near the singularity,the phase portrait consists of parallel fibers, all of which but one have no singular points, and at the singular fiber, there is one node. We study different techniques in order to show that the cyclicity is bigger or equal than two when the normal form is quadratic.

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

“…From (15) and the result of Proposition 3.1, we find that the transformation in each step can be expressed as a C ∞ function in (x, y, w n log(1 + x), w n log(1 − x)). Using Borel's theorem, there exists a function ψ(x, y, w n log(1 + x), w n log(1 − x)), formally equal to w ∞ , so thatψ is formally identically zero.…”