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Álvaro Castañeda

Publications and source records attributed to Álvaro Castañeda.

At least 19 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.

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arXiv2026

The weak Markus--Yamabe conjecture fails in dimension 14

We show that the weak Markus--Yamabe conjecture fails in every dimension $n\geq14$. We first prove a chain realization theorem: every polynomial Keller map $F=\I+H$ of $\R^n$ with component degrees $d_1,\ldots,d_n$ yields an explicit polynomial vector field on $\R^N$, with $N=\sum_i\max(d_i,2)-n$, whose Jacobian matrix has spectrum $\{-1\}$ at every point and whose singularities are in bijection with any prescribed fiber of $F$. Applied to the recent counterexample to the Jacobian conjecture, this gives a Hurwitz vector field of degree seven on $\R^{14}$ with three rational singularities. A rank-reduced dehomogenization of the associated cubic stabilization gives, independently, a Hurwitz field of degree three on $\R^{18}$. The dimensions $3\leq n\leq13$ remain open.

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arXiv2026

Nonautonomous Linear Systems: Exponential Dichotomy and its Applications

The first purpose of this work is to provide a friendly introduction to the theory of nonautonomous linear systems of ordinary differential equations, the property of exponential dichotomy and its corresponding spectral theory. The second purpose of this work is disseminate the linearization results carried out by the authors in a nonautonomous framework. The actual structure of this work is a consequence of several elective courses (2014, 2016, 2019, 2021 and 2023) carried out by the authors for undergraduate and graduated students at the Department of Mathematics of the Universidad de Chile. The monography assumes a good knowledge of multivariate calculus, linear algebra and ordinary differential equations.

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arXiv2025

Generalized Evolution Semigroups and $h-$Dichotomies for Evolution Families on Banach Spaces

This paper develops a comprehensive theory generalizing exponential decay patterns for evolution processes in Banach spaces. We replace classical exponential bounds with more flexible decay rates governed by an increasing homeomorphism $h$. The core of our approach lies in constructing particular group structures induced by $h$, which allow us to define generalized semigroups on function spaces. We prove that these $h$-semigroups are equivalent to classical evolution semigroups through a natural transformation. Our main result establishes that three fundamental concepts are equivalent: hyperbolicity of the generalized semigroup, dichotomy of the underlying evolution process, and a spectral condition on the generator. This work extends classical dichotomy theory to encompass a wider class of decay patterns, providing new tools for analyzing asymptotic behavior in dynamical systems.

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arXiv2025

Smoothness of linearization by mixing parameters of dichotomy, bounded growth and perturbation

We study the smoothness properties of a global and nonautonomous topological conjugacy between a linear system and a quasilinear perturbation. The linear system exhibits a nonuniform exponential dichotomy with a nontrivial projector and nonuniform bounded growth property. Additionally, the quasilinear perturbation is dominated by an increasing exponential function. Emphasis is placed on employing a set of parameters to describe the conditions of dichotomy, bounded growth and quasilinear perturbations. Finally, we prove that modifying these conditions enables us to achieve a broader smoothness interval.

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arXiv2025

An application of a nonuniform global stability problem to the study of parametrized polynomial automorphisms

We propose a handful of definitions of injectivity for a parametrized family of maps and study its link with a global nonuniform stability conjecture for nonautonomous differential systems, which has been recently introduced. This relation allow us to address a particular family of parametrized polynomial automorphisms and to prove that they have polynomial inverse for certain parameters, which is reminiscent to the Jacobian Conjecture.

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arXiv2025

Discrete $μ$-dichotomy spectrum: beyond uniformity and new insights

We develop spectral theorems for nonautonomous linear difference systems, considering different types of $μ$-dichotomies, both uniform and nonuniform. In the nonuniform case, intriguing scenarios emerge -- that have been employed but whose consequences have not been thoroughly explored -- which surprisingly exhibit unconventional behavior. These particular cases motivate us to introduce two novel properties of nonautonomous systems (even in the continuous-time framework), which appear to have been overlooked in the existing literature. Additionally, we introduce a new conceptualization of a nonuniform $μ$-dichotomy spectrum, which lies between the traditional nonuniform $μ$-dichotomy spectrum and the slow nonuniform $μ$-dichotomy spectrum. Moreover, and this is particularly noteworthy, we propose a conjecture that enables the derivation of spectral theorems in this new setting. Finally, contrary to what has been believed in recent years, through the lens of optimal ratio maps, we show that the nonuniform exponential dichotomy spectrum is not preserved between systems that are weakly kinematically similar.

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arXiv2023

The dichotomy spectrum approach for a global nonuniform asymptotic stability problem: Triangular case via uniformization

By considering the nonuniform exponential dichotomy spectrum, we introduce a global asymptotic nonuniform stability conjecture for nonautonomous differential systems, whose restriction to the autonomous case is related to the classical Markus--Yamabe Conjecture: we prove that the conjecture is verified for a family of triangular systems of nonautonomous differential equations satisfying boundedness assumptions. An essential tool to carry out the proof is a necessary and sufficient condition ensuring the property of nonuniform exponential dichotomy for upper block triangular linear differential systems. We also obtain some byproducts having interest on itself, such as, the diagonal significance property in terms on the above mentioned spectrum.

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arXiv2023

A nonautonomous $C^r-$topological equivalence involving contractions and unbounded nonlinearities

We study the smoothness of the topological equivalence between a linear equation and its nonlinear perturbation, which is regarded as unbounded. To the best of our knowledge, it has not previously been considered such study in the literature. Therefore, the main result of this work copes with this lack, that is, it is shown, on the positive half line, that such topological equivalence is of class $C^r (r \geq 1)$ when the linear part is a uniform contraction and the nonlinearities considered are unbounded.

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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.

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arXiv2021

A note on the differentiability of Palmer's topological equivalence for discrete systems

A linear system of difference equations and a nonlinear perturbation are considered. We propose sufficient conditions to ensure that the homeomorphism of topological equivalence between them is actually a $C^1$ diffeomorphism. These conditions consider that the linear part satisfies a dichotomy on the positive half-line, while the perturbation satisfies boundness and Lipschitzness conditions, and anothers that are tailored for our goal. The family of dichotomies that we study consider (but are not limited to) the exponential and nonuniform exponential dichotomies. We discuss the possibility of extending this result to a higher class of differentiability.

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arXiv2020

Topological Equivalence of nonautonomous difference equations with a family of dichotomies on the half line

A linear system of difference equations and a nonlinear perturbation are considered, we obtain sufficient conditions to ensure the topological equivalence between them, namely, the linear part satisfies a property of dichotomy on the positive half--line while the nonlinearity has some boundedness and Lipschitzness conditions. As a consequence, we study the asymptotical stability and its preservation by topological equivalence.

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arXiv2019

Nonuniform Almost Reducibility of Nonautonomous Linear Differential Equations

We prove that a linear nonautonomous differential system with nonuniform hyperbolicity on the half line can be expressed as diagonal system with a perturbation which is small enough. Moreover we show that the diagonal terms are contained in the nonuniform exponential dichotomy spectrum. For this purpose we introduce the concepts of \textit{nonuniform almost reducibility} and \textit{nonuniform contractibility} which are generalization of this notions originally defined in a uniform context.

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arXiv2018

Nilpotent Jacobians and Almost Global Stability

In this article we study maps with nilpotent Jacobian in $\mathbb{R}^n$ distinguishing the cases when the rows of $JH$ are linearly dependent over $\mathbb{R}$ and when they are linearly independent over $\mathbb{R}.$ In the linearly dependent case, we show an application of such maps on dynamical systems, in particular, we construct a large family of almost Hurwitz vector fields for which the origin is an almost global attractor. In the linearly independent case, we show explicitly the inverse maps of the counterexamples to Generalized Dependence Problem and proving that this inverse maps also have nilpotent Jacobian with rows linearly independent over $\mathbb{R}.$

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arXiv2018

A new class of Nilpotent Jacobians in any dimension

The classification of the nilpotent Jacobians with some structure has been an object of study because of its relationship with the Jacobian Conjecture. In this paper we classify the polynomial maps in dimension $n$ of the form $H = (u(x,y), u_2(x,y,x_3), \ldots, u_{n-1}(x,y,x_n), h(x,y))$ with $JH$ nilpotent. In addition we prove that the maps $X + H$ are invertible, which shows that for this kind of maps the Jacobian Conjecture is verified.

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arXiv2018

Nonuniform contractions and density stability results via a smooth topological equivalence

We study the smoothness and preserving orientation properties of a global and nonautonomous version of the Hartman--Grobman Theorem when the linear system has a nonuniform contraction on the half line. The nonuniform contraction implies the existence of a density function (\emph{i.e} a dual type of Lyapunov function) for the linear system which combined with the above diffeomorphism allow us to construct a density function for the nonlinear system.

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