arXiv subjects

Marco Polo García-Rivera

Publications and source records attributed to Marco Polo García-Rivera.

3 recordsLinked to original sources

arXiv2026

Higher Order Structure along Nilpotent Eigendirections in Planar Vector Fields

We study a distinguished one dimensional restriction associated with planar vector fields whose linearization is a nonzero nilpotent matrix of index two. This restriction admits an intrinsic interpretation on the nilpotent eigendirection, while its scalar representation in Jordan coordinates satisfies a simple transformation law under admissible changes of Jordan chain. Its quadratic Taylor coefficient coincides with the intrinsic quadratic Bogdanov--Takens coefficient. % For rational restrictions, we show that the Taylor coefficient sequence satisfies a finite linear recurrence determined by the minimal denominator. Its degree defines a recurrence degree that is invariant under admissible changes of Jordan chain and gives the minimal order of an eventual homogeneous recurrence. Exact cancellations may therefore be detected directly from the restriction before nonlinear normal form transformations are performed. Applications to four planar models exhibit different recurrence degree and cancellation mechanisms, illustrating the complementarity between the distinguished restriction and smooth nilpotent normal form theory.

math.DS↗

arXiv2026

An exact structural identity and its normal form consequences in a modified Leslie--Gower model

Zhao and Zhao (2026) established a codimension-four Bogdanov--Takens singularity in a modified Leslie--Gower predator--prey model through a recursive normal form computation. We show that the nonlinear coefficients governing the restriction of the reduced system to the singularity's distinguished eigendirection are not independent. Instead, a single exact rational identity, already present in the transformed equations, generates the entire family of pure Taylor coefficients and proves that their simultaneous vanishing is an exact all orders property rather than a finite-order coincidence. We further show that this structural identity propagates through the planar Bogdanov--Takens normal form algorithm of Kuznetsov (2005). Although the pure coefficients vanish identically, the corresponding normal form coefficients are generated by mixed quadratic interactions. This leads to an explicit characterization of the degenerate Bogdanov--Takens locus and reduces the search for higher-order Bogdanov--Takens degeneracies to a single explicit algebraic condition.

math.DS↗

arXiv2026

Hopf-Bautin and homoclinic bifurcations in a Saltzman-Maasch model with cubic feedback

This paper investigates a deterministic variant of the Saltzman-Maasch model for Pleistocene glacial cycles, formulated as a three-dimensional dynamical system with cubic feedback in the atmospheric carbon dioxide equation. After reducing the model to a planar system on a critical manifold, we perform a detailed bifurcation analysis and analytically identify both Hopf and Bautin (generalized Hopf) bifurcations, which govern the emergence of stable and unstable limit cycles. To analyze global transitions, we perform a rescaling of time and variables to derive a leading-order Hamiltonian system. This reduction enables the explicit construction of homoclinic orbits and the application of Melnikov's method to assess their persistence under perturbations. The analytical predictions are further validated through numerical continuation and simulations, providing a rigorous foundation for previously reported numerical observations and establishing, in particular, the analytical existence of Bautin bifurcations, homoclinic connections via Melnikov analysis, and a systematic slow--fast geometric reduction of the model.

math.DS↗