Global center of polynomial Newton system and its non-isochronicity

TL;DR

This work develops a toroidal compactification and desingularization framework to study polynomial Newton (Cherkas) systems, proving that monodromy at infinity is equivalent to the nonexistence of $\tfrac{1}{2}$-fractional formal invariant curves at infinity and that monodromy forces the degree parameter to satisfy $m\le 2$. By coupling this global-at-infinity analysis with Darboux integrability or algebraic reducibility of local centers, it delivers a complete global-center classification for Cherkas systems, together with rigorous non-isochronicity results for the global center near infinity. The authors also provide practical computational criteria (via Newton polygons and edge polynomials) and demonstrate applications to homogeneous Kukles systems and polynomial Liénard systems, including solving Conti's open isochronicity problem in this setting. Overall, the paper advances understanding of global centers and period behavior for high-degree polynomial Newton systems and offers broadly applicable techniques for center and isochronicity analysis in planar polynomial dynamics.

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.

Figures (9)

• Figure 1: Rectangular compactification and toroidal compactification.
• Figure 2: Desingularization process of ${\cal X}^{(0)}$.
• Figure 3: Newton polygon ${\cal N}({\cal X}^{(0)})$.
• Figure 4: Newton polygon ${\cal N}({\cal X}^{(i)})$ with $i \ge 1$.
• Figure 5: Transverse sections for vector fields \ref{['vf:seg']} and \ref{['vf:h-saddle']}.

Theorems & Definitions (14)

• Lemma 2.1
• Lemma 2.2
• Theorem 2.1
• Remark 2.1
• Remark 2.2
• Theorem 3.1
• Remark 3.1
• Remark 3.2
• Theorem 3.2
• Lemma 4.1