arXiv subjects

Martha Alvarez Ramírez

Publications and source records attributed to Martha Alvarez Ramírez.

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.

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arXiv2026

On distinguishing genuine from spurious chaos in planar singular and nonsmooth systems: A diagnostic approach

We present a rigorous reassessment of chaotic behavior in two-dimensional autonomous systems with singular or nonsmooth dynamics. For the Cummings-Dixon-Kaus (CDK) model, we show that blow-up regularization restores smoothness and renders the hypotheses of the Poincaré-Bendixson theorem applicable, thereby excluding chaotic attractors away from the singular set. We prove topological equivalence between the original and regularized flows on annular domains, ensuring that no spurious invariant sets are introduced by desingularization. In contrast, for a nonsmooth system with a $|x|$ term, we recompute the entire period-doubling cascade, obtain a seven-term sequence of bifurcation values converging to Feigenbaum's constant, and confirm robust chaos through positive Lyapunov exponents, broadband spectra, and fractal dimension estimates. As a main outcome, we propose a diagnostic protocol integrating regularization, numerical refinement, and invariant-set criteria. This protocol provides a reproducible standard for distinguishing genuine planar chaos from artifacts caused by singularities or discretization, and offers a benchmark for future studies of low-dimensional nonsmooth systems.

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arXiv2024

Hopf and Bautin bifurcations in a 3D model for pest leafhopper with stage structure and generalist predatory mite

In a recent paper of Yuan and Zhu (J. Differential Equations 321(2022) 99-129), the nature of dynamics of a generalist predator and prey with stage structure is modeled as a three-dimensional coupled nonlinear differential system. A detailed analysis of the bifurcation shows that the model exhibits a high complexity in its dynamics, which arises from the use of predatory mites as agent for controlling stage structures of tea green leafhopper pest. Unfortunately, there is a mistake in Proposition 2.2, item (3), also the hypothesis in Proposition 3.3 is incorrect. In this paper, we revisit the model and straighten those mentioned errors as in item (3), changed the existing hypothesis by a suitable one and give a corrected proof of Proposition 3.3 in its new form. Also, we show that the model does undergo both Hopf bifurcation and Bautin bifurcation, and numerical examples are given to support the analytic results.

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