arXiv subjects

Jose S. Cánovas

Publications and source records attributed to Jose S. Cánovas.

6 recordsLinked to original sources

arXiv2026

Chaos and Parrondo's paradox: An overview

Parrondo's paradox (PP) is a fundamental principle in nonlinear science where the alternation of individually losing strategies leads to a winning outcome. In this topical review, we provide the first systematic panorama of the synergy between PP and chaos. We observe a bidirectional connection between the two areas. The first direction is the translation of PP into the interplay between Order and Chaos through either Chaos + Chaos $\to$ Order (CCO) or Order + Order $\to$ Chaos (OOC). In this vein, many quantifiers, such as Lyapunov Exponents, $λ$, and entropic measures, are used. Second, we note that chaos can be used to engineer switching protocols that can lead to nontrivial effects in diverse PP cases. Our review clarifies the universality of PP and highlights its robust theoretical and practical applications across several areas of science and technology. Finally, we delineate key open questions, emphasizing the unresolved theoretical limits, the role of high-dimensional maps and continuous flows, and the critical need for more experimental verification of the dynamic PP in chaotic systems. For completeness, we also provide a full Python code that allows the reader to observe the many facets of the PP.

nlin.CD↗

arXiv2025

Integrating the enveloping technique with the expansion strategy to establish stability

In this paper, we focus on finding one-dimensional maps that detect global stability in multidimensional maps. We consider various local and global stability techniques in discrete-time dynamical systems and discuss their advantages and limitations. Specifically, we navigate through the embedding technique, the expansion strategy, the dominance condition technique, and the enveloping technique to establish a unifying approach to global stability. We introduce the concept of strong local asymptotic stability (SLAS), then integrate what we call the expansion strategy with the enveloping technique to develop the enveloping technique for two-dimensional maps, which allows to give novel global stability results. Our results make it possible to verify global stability geometrically for two-dimensional maps. We provide several illustrative examples to elucidate our concepts, bolster our theory, and demonstrate its application.

math.DS↗

arXiv2025

Increasing delay as a strategy to prove stability

We consider difference equations of the form $x_{n+1}=F_0(x_n,\ldots,x_{n-k+1}),$ and increase the delay through a process of successive substitutions to obtain a sequence of systems $y_{n+1}=F_j(x_{n-j},\ldots,x_{n-k-j+1}),\; j=0,1,\ldots$. We call this process \emph{the expansion strategy} and use it to establish novel results that enable us to prove stability. When the map $F_0$ is sufficiently smooth and has a hyperbolic fixed point, we show the fixed point is locally asymptotically stable if and only if $\|\nabla F_j\|_1<1$ for some finite number $j$. Our local stability results complement recent results obtained on Schur stability, and they can provide an alternative to the highly acclaimed Jury's algorithm. Also, we show the effectiveness of the expansion strategy in obtaining global stability results. Global stability results are obtained by integrating the expansion strategy with the embedding technique. Finally, we give illustrative examples to show the results' practical applicability across various discrete-time dynamical systems.

math.DS↗

arXiv2025

Constrained polynomial roots and a modulated approach to Schur stability

It is common in stability analysis to linearize a system and investigate the spectrum of the Jacobian matrix. This approach faces the challenge of determining the matrix spectrum when the coefficients depend on parameters or when the characteristic polynomial is more than quartic. In this paper, we reverse the classical process and use the authors' work on global stability to find sufficient conditions on the coefficients that ensure the zeros of the characteristic polynomial are in the open unit disk. This leads to an algorithm that begins by testing the $\ell_1$-norm of the polynomial, and if it is not less than two, perform an iteration process that can be implemented with moderate effort. We give examples that show the effectiveness of our method when compared with the Jury's algorithm. Last, we formalize our constructions in terms of semialgebraic sets.

math.DS↗

arXiv2024

Attractors in $k$-dimensional discrete systems of mixed monotonicity

We consider $k$-dimensional discrete-time systems of the form $x_{n+1}=F(x_n,\ldots,x_{n-k+1})$ in which the map $F$ is continuous and monotonic in each one of its arguments. We define a partial order on $\mathbb{R}^{2k}_+$, compatible with the monotonicity of $F$, and then use it to embed the $k$-dimensional system into a $2k$-dimensional system that is monotonic with respect to this poset structure. An analogous construction is given for periodic systems. Using the characteristics of the higher-dimensional monotonic system, global stability results are obtained for the original system. Our results apply to a large class of difference equations that are pertinent in a variety of contexts. As an application of the developed theory, we provide two examples that cover a wide class of difference equations, and in a subsequent paper, we provide additional applications of general interest.

math.DS↗

arXiv2022

Revisiting the dynamic of Q-deformed logistic maps

We consider the logistic family and apply the $q$-deformation $ϕ_q(x)=\frac{1-q^x}{1-q}$. We study the stability regions of the fixed points of the $q$-deformed logistic map and the regions where the dynamic is complex through topological entropy and Lyapunov exponents. Our results show that the dynamic of this deformed family is richer than that of the $q$-deformed family studied in [8].

math.DS↗