Global center of polynomial Newton system and its non-isochronicity

TL;DR

This paper characterizes the monodromy at infinity for polynomial Newton systems using toroidal compactification, linking it to the existence of invariant curves, and proves the non-isochronicity of the global center, solving an open problem.

Contribution

It introduces a new compactification method to analyze monodromy and establishes conditions for global centers, including non-isochronicity, in polynomial Newton systems.

Findings

Complete characterization of monodromy at infinity

Conditions for global centers based on Darboux integrability

Proof of non-isochronicity for the global center

Abstract

Using a new compactification (toroidal compactification) and desingularization, we obtain a complete characterization of monodromy at infinity for polynomial Newton system of arbitrary degree, in which we establish an equivalence between the monodromy and the non-existence of 1/2-fractional formal invariant curves. Combining the complete characterization with either Darboux integrability or algebraic reducibility of local centers, we obtain conditions for all cases of global center. Furthermore, investigating the asymptotic behavior of the period function of orbits near infinity, we prove the non-isochronicity for the global center, which consequently solves an open problem proposed by Conti.