On the Defense of the Recent Solution to Hilbert's Sixteenth Problem: Clarifying Misinterpretations and the Incorrect Conclusions in Buzzi and Novaes's note

TL;DR

This paper defends a recent geometric bifurcation theory approach to Hilbert's sixteenth problem, clarifies misconceptions, and corrects errors in a prior critique by Buzzi and Novaes, affirming the validity of the new solutions.

Contribution

It clarifies and defends the covariant geometric bifurcation theory's application to Hilbert's sixteenth problem and corrects misconceptions in previous critiques.

Findings

The theory correctly predicts the maximum number of limit cycles for polynomial systems.

Misinterpretations in prior critique lead to incorrect conclusions.

The approach provides a valid solution to Hilbert's sixteenth problem.

Abstract

Recently, the covariant formulation of the geometric bifurcation theory, developed in a previous paper, has been applied to two elementary problems: the study of limit cycles of dynamical systems and the second part of Hilbert's sixteenth problem. First, it has been shown that dynamical systems with more than one limit cycle are understood to be those in which the scalar curvature $\mbox{R}$ is positive and its magnitude diverges to infinity at different singular points. In the second, it has been demonstrated that $n$th-degree polynomial systems have the maximum number of $2(n-1)(4(n-1)-2)$ limit cycles with their relative positions determined by the singularities of the magnitude of $\mbox{R}$, thus providing a successful response to the original Hilbert's challenge. It is the purpose of this letter to point out that Buzzi and Novaes's note is incorrect and leads to erroneous results as an immediate consequence of their superficial reading of our work, the omission of critical aspects of the GBT to induce doubt, and reliance on incorrect assumptions or interpretations that deviate from those established in the framework of GBT.