Finite cyclicity of some graphics through a nilpotent point of saddle type inside quadratic systems

TL;DR

This paper proves the finite cyclicity of specific graphics through a nilpotent saddle point in quadratic systems, advancing the understanding of limit cycle bounds in quadratic vector fields.

Contribution

It establishes the finite cyclicity of two particular graphics through a nilpotent saddle point, contributing to the DRR program on limit cycle bounds.

Findings

Finite cyclicity of $(I_{12}^1)$ and $(I_{13}^1)$ graphics proven.

Supports the conjecture of a uniform upper bound for limit cycles in quadratic systems.

Progress in the DRR program on quadratic vector fields.

Abstract

In this paper we show the finite cyclicity of the two graphics $(I_{12}^1)$ and $(I_{13}^1)$ through a triple nilpotent point of saddle type inside quadratic vector fields. These results contribute to the program launched in 1994 by Dumortier, Roussarie and Rousseau (DRR program) to show the existence of a uniform upper bound for the number of limit cycles for planar quadratic vector fields.