arXiv subjects

Heli Elorreaga

Publications and source records attributed to Heli Elorreaga.

4 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↗

arXiv2025

$C^{1}$ Smoothness of the Conjugacy in a Delayed Nonautonomous Hartman--Grobman Theorem via $μ$-Dichotomies

We investigate the differentiability of the conjugacy in a nonautonomous version of the Hartman--Grobman Theorem for systems with finite delay, where the linear part satisfies a $μ$-dichotomy. Under suitable conditions on the nonlinear perturbation, the conjugacy is shown to be a $C^{1}$ diffeomorphism.

math.DS↗

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↗

arXiv2023

Exponential dichotomy for a class of asymptotically autonomous delayed differential equations

In this work we give a criterion to have an exponential dichotomy over all $\mathbb{R}$ for delayed systems $x'(t)=L(t)x_t$, where $L_{\pm}=\lim_{t\to\pm\infty}L(t)$, and the systems $x'(t)=L_{\pm}x_t$ are autonomous and hyperbolic. The proof is based in a suitable choice of weighted Sobolev spaces $L^p(\mathbb{R},μ)$ and $W^{1,p}(\mathbb{R},μ)$, where the involves differential operators become Fredholm's operators and this property are strongly exploited.

math.DS↗