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Gonzalo Robledo

Publications and source records attributed to Gonzalo Robledo.

At least 19 recordsLinked to original sources
arXiv2026

Conditions for uniform $h$--dichotomy in terms of uniform non criticality, expansiveness and via generalized Floquet theory

In this article, we complete the study of the equivalences between the properties of $h$--dichotomy, $h$--noncriticality and $h$--expansiveness of a linear nonautonomous ODE system which had been initiated in a previous work. Moreover, we extend a result of the generalized Floquet theory developed by T.A. Burton and J.S. Muldowney by providing a necessary and sufficient condition for $h$--dichotomy. It should be noted that all the results have been obtained by using a characterization of the $h$--dichotomy by a group theory approach recently developed by J.F. Peña and S. Rivera--Villagrán.

math.DS↗
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.

math.CA↗
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.

math.FA↗
arXiv2025

Generalizing the Levins metapopulation model to time varying colonization and extinction rates

The metapopulation theory explores the population persistence in fragmented habitats by considering a balance between the extinction of local populations and recolonization of empty sites. In general, the extinction and colonization rates have been considered as constant parameters and the novelty of this paper is to assume that they are subject to deterministic variations. We noticed that an averaging approach proposed by C. Puccia and R. Levins can be adapted to construct the upper and lower averages of the difference between the extinction and colonization rates, whose sign is useful to determine either the permanence or the extinction of the metapopulation. In fact, we use these averages to revisit the classical model introduced by R. Levins. From a mathematical perspective, these averages can be seen as Bohl exponents whereas the corresponding analysis is carried out by using tools of non autonomous dynamics. Last but not least, compared with the Levins model, the resulting dynamics of the time varying model shares the persistence/extinction scenario when the above stated upper and lower averages have the same sign but also raises open questions about metapopulation persistence in the case of the averages have different sign.

q-bio.PE↗
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.

math.DS↗
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.

math.AG↗
arXiv2024

Persistence/extinction scenarios in an almost periodic metapopulation with competition and habitat destruction

We study an almost periodic version of a metapopulation model developed by Tilman \textit{et.al} and Nee \textit{et.al} in the nineties, which generalizes the classical Levins approach by considering several species in competition affected by habitat destruction. The novelty is to assume that the colonization and extinction rates are positive almost periodic functions whereas our main results show that the predominance of either colonization or extinction forces of a specific species is equivalent to the property of exponential dichotomy of a scalar linear differential equation. By using well known results of exponential dichotomy theory, we carry out a recursive and exhaustive description of persistence/extinction scenarios. In addition, we start a preliminary discussion describing a more elusive behavior when the colonization and extinction forces are similar in average.

q-bio.PE↗
arXiv2024

Uniform $h$-dichotomies: noncritical uniformity and expansivity

The property of exponential dichotomy can be seen as a generalization of the hyperbolicity condition for non autonomous linear finite dimensional systems of ordinary differential equations. In 1978 W.A. Coppel proved that the exponential dichotomy on the half line is equivalent to the property of noncritical uniformity provided that a condition of bounded growth is verified. In 2006 K.J. Palmer extended this result by proving that -- also assuming the bounded growth property -- the exponential dichotomy on the half line, noncritical uniformity and the exponential expansiveness are equivalent. The main contribution of this article is to generalize these results for the property of uniform $h$-dichotomy. This has been carried out due to a recent idea: under suitable conditions any $h$-dichotomy can be associated to a totally ordered topological group, which becomes the additive group $(\mathbb{R},+)$ in case of the exponential dichotomy. The properties of this new group make possible such generalization.

math.CA↗
arXiv2024

Controllability and feedback stabilizability in a nonuniform framework

We propose a new controllability property for linear time varying control systems in finite dimension: the nonuniform complete controllability, which is halfway between the classical Kalman's properties of complete controllability and uniform complete controllability. This new concept is described in terms of two gramian inequalities, which have a strong relation; as we prove in our first result; with the property of nonuniform bounded growth for the corresponding plant, also called uncontrolled part. On the other hand, the second result proves that if a control system is nonuniformly completely controllable and its plant has the property of nonuniform bounded growth, then there exist a linear feedback control leading to a nonuniformly exponentially stable closed--loop system.

math.OC↗
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.

math.DS↗
arXiv2023

On stability for generalized linear differential equations and applications to impulsive systems

In this paper, we are interested in investigating notions of stability for generalized linear differential equations (GLDEs). Initially, we propose and revisit several definitions of stability and provide a complete characterisation of them in terms of upper bounds and asymptotic behaviour of the transition matrix. In addition, we illustrate our stability results for GLDEs to linear periodic systems and linear impulsive differential equations. Finally, we prove that the well known definitions of uniform asymptotic stability and variational asymptotic stability are equivalent to the global uniform exponential stability introduced in this article.

math.CA↗
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.

math.CA↗
arXiv2018

Some relations between Bohl Exponents and the Exponential Dichotomy spectrum

We study a liaison between the Bohl's exponents and the exponential dichotomy spectrum of a non autonomous linear system of difference equations on the whole line $\mathbb{Z}$. More specifically, We prove that for any initial condition in an invariant vector bundle, associated to its exponential dichotomy spectrum, its Bohl's exponents are contained in an spectral interval.

math.CA↗
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.

math.DS↗
arXiv2018

Smoothness of Topological Equivalence on the Half Line for Nonautonomous Systems

We study the differentiability properties of the topological equivalence between a uniformly asymptotically stable linear nonautonomous system and a perturbed system with suitable nonlinearities. For this purpose, we construct a uniformly continuous homeomorphism inspired in the Palmer's one restricted to the positive half line, providing sufficient conditions ensuring its $C^{r}$--smoothness. Additionally, we study the preservation of the uniform stability properties by this homeomorphism.

math.CA↗
arXiv2018

Multiple solutions for periodic perturbations of a delayed autonomous system near an equilibrium

Small non-autonomous perturbations around an equilibrium of a nonlinear delayed system are studied. Under appropriate assumptions, it is shown that the number of $T$-periodic solutions lying inside a bounded domain $Ω\subset \R^N$ is, generically, at least $|χ\pm 1|+1$, where $χ$ denotes the Euler characteristic of $Ω$. Moreover, some connections between the associated fixed point operator and the Poincaré operator are explored.

math.CA↗
arXiv2018

A Spectral Dichotomy Version of the Nonautonomous Markus--Yamabe Conjecture

In this article we introduce a nonautonomous version of the Markus--Yamabe conjecture from an exponential dichotomy spectrum point of view. We prove the validity of this conjecture for the scalar and triangular case. Additionally we show that the origin is a global attractor for an autonomous system by using nonautonomous dynamical systems tools.

math.DS↗