arXiv subjects

Martha Alvarez-Ramírez

Publications and source records attributed to Martha Alvarez-Ramírez.

8 recordsLinked to original sources

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.

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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.

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arXiv2026

Endogenous Cycles in a Keen--Goodwin Model with Minsky Debt

We analyze a three-dimensional Keen--Goodwin model that couples wage--employment dynamics with Minsky-style private debt. At zero real interest the interior equilibrium is nonhyperbolic and organized by a two-dimensional center manifold foliated by neutral Goodwin cycles. Introducing a small positive interest rate unfolds this degeneracy: we derive an explicit Hopf condition, prove persistence of the center manifold as a normally hyperbolic attracting surface, and obtain first-order amplitude and frequency corrections for the emergent limit cycle via phase--amplitude reduction. Numerical simulations support the asymptotic predictions and demonstrate how interest rates determine and modulate endogenous business cycles.

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arXiv2025

Singular bifurcations in a modified Leslie-Gower model

We study a predator-prey system with a generalist Leslie-Gower predator, a functional Holling type II response, and a weak Allee effect on the prey. The prey's population often grows much faster than its predator, allowing us to introduce a small time scale parameter $\varepsilon$ that relates the growth rates of both species, giving rise to a slow-fast system. Zhu and Liu (2022) show that, in the case of the weak Allee effect, Hopf singular bifurcation, slow-fast canard cycles, relaxation oscillations, etc., exist. Our main contribution lies in the rigorous analysis of a degenerate scenario organized by a (degenerate) transcritical bifurcation. The key tool employed is the blow-up method that desingularizes the degenerate singularity. In addition, we determine the criticality of the singular Hopf bifurcation using recent intrinsic techniques that do not require a local normal form. The theoretical analysis is complemented by a numerical bifurcation analysis, in which we numerically identify and analytically confirm the existence of a nearby Takens-Bogdanov point.

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arXiv2024

Global dynamics and integrability of a Leslie-Gower predator-prey model with linear functional response and generalist predator

We deal with a Leslie-Gower predator-prey model with a generalist or alternating food for predator and linear functional response. Using a topological equivalent polynomial system we prove that the system is not Liouvillian (hence also not Darboux) integrable. In order to study the global dynamics of this model, we use the Poincaré compactification of $\mathbb{R}^2$ to characterize all phase portraits in the Poincaré disc, obtaining two different topological phase portraits.

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arXiv2022

Equilibrium points and their linear stability in the planar equilateral restricted four-body problem: A review and new results

In this work, we revisit the planar restricted four-body problem to study the dynamics of an infinitesimal mass under the gravitational force produced by three heavy bodies with unequal masses, forming an equilateral triangle configuration. We unify known results about the existence and linear stability of equilibrium points of this problem which have been obtained earlier, either as relative equilibria or a central configuration of the planar restricted $(3 + 1)$-body problem. It is the first attempt in this direction. A systematic numerical investigation is performed to obtain the resonance curves in the mass space. We use these curves to answer the question about the existing boundary between the domains of linear stability and instability. The characterization of the total number of stable points found inside the stability domain is discussed.

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arXiv2022

On the equilateral pentagonal central configurations

An equilateral pentagon is a polygon in the plane with five sides of equal length. In this paper we classify the central configurations of the $5$-body problem having the five bodies at the vertices of an equilateral pentagon with an axis of symmetry. We prove that there are two unique classes of such equilateral pentagons providing central configurations, one concave equilateral pentagon and one convex equilateral pentagon, the regular one. A key point of our proof is the use of rational parameterizations to transform the corresponding equations, which involve square roots, into polynomial equations.

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arXiv2020

Ejection-collision orbits in two degrees of freedom problems in celestial mechanics

In a general setting of a Hamiltonian system with two degrees of freedom and assuming some properties for the undergoing potential, we study the dynamics close and tending to a singularity of the system which in models of $N$-body problems corresponds to total collision. We restrict to potentials that exhibit two more singularities that can be regarded as two kind of partial collisions when not all the bodies are involved. Regularizing the singularities, the total collision transforms into a 2-dimensional invariant manifold. The goal of this paper is to prove the existence of different types of ejection-collision orbits, that is, orbits that start and end at total collision. Such orbits are regarded as heteroclinic connections between two equilibrium points and are mainly characterized by the partial collisions that the trajectories find on their way. The proof of their existence is based on the transversality of 2-dimensional invariant manifolds and on the behavior of the dynamics on the total collision manifold, both of them are thoroughly described.

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