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Salomón Rebollo-Perdomo

Publications and source records attributed to Salomón Rebollo-Perdomo.

5 recordsLinked to original sources
arXiv2026

A family of Hurwitz polynomial vector fields in $\mathbb{R}^3$ having non-constant periodic orbits

We construct a two-parameter family of polynomial vector fields in $\mathbb{R}^3$ and prove that, on a nonempty open set of parameter values, every member of the family is Hurwitz and possesses two non-constant periodic orbits, one of saddle type and one attracting. This answers Problem 19 in Gasull's list of open problems on low-dimensional dynamical systems concerning the existence of non-constant periodic orbits in Hurwitz smooth vector fields on $\mathbb{R}^3$. This also provides counterexamples to the expected non-existence of non-constant periodic orbits for Hurwitz polynomial vector fields; a question that arose from the first counterexamples to the Markus-Yamabe conjecture in the late 1990s and that was stated by Meisters in his featured MathSciNet review of the Bernat-Llibre counterexample.

math.DS↗
arXiv2025

Abelian integrals for polynomials with trivial global monodromy on $\mathbb{C}^2$

We consider infinitesimal perturbations of Hamiltonian differential equations $dH + \varepsilon ω=0$ on the complex plane $\mathbb{C}^2$, where $H$ is a polynomial of degree $m+1$ and $ω$ is a non-exact polynomial 1-form of degree $n$. In order to study these perturbed differential equations, the associated Abelian integrals $I(c)=\int_{γ(c)} ω$ are valuable tools. We assume that the polynomials $H$ are primitive with trivial global monodromy. For these polynomials, W. D. Neumann and P. Norbury provided a classification in three large families, up to algebraic equivalence. The knowledge of these families allows us to prove as first main result, that the respective Abelian integrals $I(c)$ are polynomial functions of the variable $c$, and to find sharp explicit upper bounds for the number of their zeros. The bounds depend on $m$, $n$ and the number of the generators of the fundamental group of the generic fibers of $H$. These upper bounds works for several new families of infinitesimal perturbations of Hamiltonian differential equations. Under trivial global monodromy, there exist canonical global generators $BC(H)= \{ γ_{\tt i}(c)\}$ of the fundamental groups for all the generic fibers of $H$, which are complex cycles of $dH=0$. As second main result; we compute the number of complex limit cycles of $dH+ \varepsilonω=0$ which originate from complex cycles in $BC(H)$. Several accurate examples are provided.

math.DS↗
arXiv2022

Discrete and continuous dynamics of real $3$-dimensional nilpotent polynomial vector fields

A large class of real $3$-dimensional nilpotent polynomial vector fields of arbitrary degree is considered. The aim of this work is to present general properties of the discrete and continuous dynamical systems induced by these vector fields. In the discrete case, it is proved that each dynamical system has a unique fixed point and no $2$-cycles. Moreover, either the fixed point is a global attractor or there exists a $3$-cycle which is not a repeller. In the continuous setting, it is proved that each dynamical system is polynomially integrable. In addition, for a subclass of the considered vector fields, the system is polynomially completely integrable. Furthermore, for a family of low degree vector fields, it is provided a more precise description about the global dynamics of the trajectories of the induced dynamical system. In particular, it is proved the existence of an invariant surface foliated by periodic orbits. Finally, some remarks and open questions, motivated by our results, the Markus--Yamabe Conjecture and the problem of planar limit cycles, are given.

math.DS↗