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Carlos F. Álvarez

Publications and source records attributed to Carlos F. Álvarez.

10 recordsLinked to original sources
arXiv2026

Limit Cycles in a Discontinuous Generalized Liénard Systems with Bivariate Perturbation

We study the number of limit cycles of the piecewise smooth differential system \[ \dot{x}=y, \quad \dot{y}=-x-\varepsilon\bigl(f(x)y+\operatorname{sgn}(x)g(x,y)\bigr),\] where $f(x)$ is a polynomial of degree $n\geq 1$, $g(x,y)$ is a bivariate polynomial of degree $m\geq 2$, and $\varepsilon$ is a sufficiently small parameter. Using the first-order averaging theorem for discontinuous systems, we obtain that the number $H_{m,n}=\lfloor n/2\rfloor+\lfloor m/2\rfloor$ is a lower bound for the number of limit cycles bifurcating from the linear center $\dot{x}=y, \ \dot{y}=-x$. The bound is sharp, and we provide an explicit example that attains $H_{m,n}$ limit cycles.

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arXiv2026

Topological Detection of Hopf Bifurcations via Persistent Homology: A Functional Criterion from Time Series

We propose a topological framework for detecting Hopf-type dynamical transitions directly from scalar time series. The method combines delay-coordinate reconstruction with persistent homology and uses the maximum persistence of one-dimensional homology classes as a scalar descriptor of cyclic structure. For the supercritical Hopf setting, we derive finite-resolution persistence bounds that relate detectability of the reconstructed periodic orbit to its geometry, sampling quality, and finite-data perturbations. A derivative-based estimator is then introduced to localize the critical parameter from the sampled topological functional. The approach is evaluated on the Hopf normal form, the Lorenz system, and a reduced Belousov--Zhabotinsky model. The numerical experiments show accurate finite-resolution localization of the corresponding transitions and illustrate the effects of embedding parameters, temporal sampling, smoothing, observational noise, and transient removal. These results support persistent homology as an interpretable data-driven tool for detecting geometric reorganizations in nonlinear time series.

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arXiv2026

Expansivity and shadowing for composition operators on Paley-Wiener spaces

In this article, we study composition operators $C_ϕf=f\circ ϕ$ on the full range of Paley-Wiener spaces $B^2_a$ for $a>0.$ Here we compute the spectrum and the spectral radius of $C_ϕ$ on $B^2_a.$ The spectrum of $C_ϕ$ and the location of the fixed point of $ϕ$ play a fundamental role in our analysis to show that no Paley-Wiener space $B^2_a$ supports composition operators possessing the positive shadowing property. We also provide a complete classification of the composition operators on $B^2_a$ that are positively expansive and absolutely Cesàro bounded. As a consequence of these results we show that no Paley-Wiener space $B^2_a$ supports Li-Yorke chaotic composition operators.

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arXiv2026

On the dynamics of a semigroup and its relation with the Riemann Hypothesis

The semigroup of weighted composition operators $(W_n)_{n\in \mathbb{N}}$, defined by $$W_nf(z)=(1+z+\cdots +z^n)f(z^n),$$ acts on the classical Hardy-Hilbert space $H^{2}(\mathbb{D})$, and exhibits intriguing connections with both the Riemann Hypothesis (RH) and the Invariant Subspace Problem (ISP). In this paper, we prove that the adjoint operators $W^{\ast}_{n}$, for $n\geq 2$, are Devaney chaotic, frequently hypercyclic and mixing. In particular, these operators are hypercyclic and discuss connections with the RH and invariant subspaces.

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arXiv2026

Li-Yorke chaotic weighted composition operators on Hardy and Bergman spaces over the unit disk

We study Li--Yorke and mean Li--Yorke chaos for weighted composition operators $C_{w,φ}$ on Banach spaces of analytic functions on the unit disk $\mathbb{D}$. Under natural conditions on the space, we show that $C_{w,φ}$ is (densely) Li--Yorke chaotic if and only if it is not power-bounded, and (densely) mean Li--Yorke chaotic if and only if it is not absolutely Cesàro bounded. These results are applied to Hardy spaces $H^p(\mathbb{D})$, $1 \le p \le \infty$, and weighted Bergman spaces $A^p_β(\mathbb{D})$, $-1 < β< \infty$ and $1 < p < \infty$.

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arXiv2025

Dynamical properties for composition operators on $H^{2}(\mathbb{C}_{+})$

Expansivity, Li-Yorke chaos and shadowing are popular and well-studied notions of dynamical systems. Several simple and useful characterizations of these notions within the setting of linear dynamics were obtained recently. We explore these three dynamical properties for composition operators $C_ϕf = f \circ ϕ$ induced by affine self-maps $ϕ$ of the right half-plane $\mathbb{C}_{+}$ on the Hardy-Hilbert space $H^{2}(\mathbb{C_{+}})$.

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arXiv2025

On the composition operator with variable integrability

In this note, we consider a class of composition operators on Lebesgue spaces with variable exponents over metric measure spaces. Taking advantage of the compatibility between the metric-measurable structure and the regularity properties of the variable exponent, we provide necessary and sufficient conditions for this class of operators to be bounded and compact, respectively. In addition, we show the usefulness of the variable change to study weak compactness properties in the framework of non-standard spaces.

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arXiv2023

Equilibrium states for partially hyperbolic maps with one-dimensional center

We prove the existence of equilibrium states for partially hyperbolic endomorphisms with one-dimensional center bundle. We also prove, regarding a class of potentials, the uniqueness of such measures for endomorphisms defined on the 2-torus that: have a linear model as a factor; and with the condition that this measure gives zero weight to the set where the conjugacy with the linear model fails to be invertible. In particular, we obtain the uniqueness of the measure of maximal entropy. For the n-torus, the uniqueness in the case with one-dimensional center holds for absolutely partially hyperbolic maps with additional hypotheses on the invariant leaves, namely, dynamical coherence and quasi-isometry.

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arXiv2020

Equilibrium states for maps isotopic to Anosov

In this work we address the problem of existence and uniqueness (finiteness) of ergodic equilibrium states for partially hyperbolic diffeomorphisms isotopic to Anosov on $\mathbb{T}^4$, with 2-dimensional center foliation. To do so we propose to study the disintegration of measures along 1-dimensional subfoliations of the center bundle. Moreover, we obtain a more general result characterizing the disintegration of ergodic measures in our context.

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