arXiv subjects

R. Aguilar-Sanchez

Publications and source records attributed to R. Aguilar-Sanchez.

13 recordsLinked to original sources

arXiv2024

Dissipative fractional standard maps: Riemann-Liouville and Caputo

In this study, given the inherent nature of dissipation in realistic dynamical systems, we explore the effects of dissipation within the context of fractional dynamics. Specifically, we consider the dissipative versions of two well known fractional maps: the Riemann-Liouville (RL) and the Caputo (C) fractional standard maps (fSMs). Both fSMs are two-dimensional nonlinear maps with memory given in action-angle variables $(I_n,θ_n)$; $n$ being the discrete iteration time of the maps. In the dissipative versions these fSMs are parameterized by the strength of nonlinearity $K$, the fractional order of the derivative $α\in(1,2]$, and the dissipation strength $γ\in(0,1]$. In this work we focus on the average action $\left< I_n \right>$ and the average squared action $\left< I_n^2 \right>$ when~$K\gg1$, i.e. along strongly chaotic orbits. We first demonstrate, for $|I_0|>K$, that dissipation produces the exponential decay of the average action $\left< I_n \right> \approx I_0\exp(-γn)$ in both dissipative fSMs. Then, we show that while $\left< I_n^2 \right>_{RL-fSM}$ barely depends on $α$ (effects are visible only when $α\to 1$), any $α< 2$ strongly influences the behavior of $\left< I_n^2 \right>_{C-fSM}$. We also derive an analytical expression able to describe $\left< I_n^2 \right>_{RL-fSM}(K,α,γ)$.

nlin.CD↗

arXiv2024

Singular-value statistics of directed random graphs

Singular-value statistics (SVS) has been recently presented as a random matrix theory tool able to properly characterize non-Hermitian random matrix ensembles [PRX Quantum {\bf 4}, 040312 (2023)]. Here, we perform a numerical study of the SVS of the non-Hermitian adjacency matrices $\mathbf{A}$ of directed random graphs, where $\mathbf{A}$ are members of diluted real Ginibre ensembles. We consider two models of directed random graphs: Erdös-Rényi graphs and random regular graphs. Specifically, we focus on the ratio $r$ between nearest neighbor singular values and the minimum singular value $λ_{min}$. We show that $\langle r \rangle$ (where $\langle \cdot \rangle$ represents ensemble average) can effectively characterize the transition between mostly isolated vertices to almost complete graphs, while the probability density function of $λ_{min}$ can clearly distinguish between different graph models.

stat.AP↗

arXiv2024

Tunable subdiffusion in the Caputo fractional standard map

The Caputo fractional standard map (C-fSM) is a two-dimensional nonlinear map with memory given in action-angle variables $(I,θ)$. It is parameterized by $K$ and $α\in(1,2]$ which control the strength of nonlinearity and the fractional order of the Caputo derivative, respectively. In this work we perform a scaling study of the average squared action $\left< I^2 \right>$ along strongly chaotic orbits, i.e. when $K\gg1$. We numerically prove that $\left< I^2 \right>\propto n^μ$ with $0\leμ(α)\le1$, for large enough discrete times $n$. That is, we demonstrate that the C-fSM displays subdiffusion for $1<α<2$. Specifically, we show that diffusion is suppressed for $α\to1$ since $μ(1)=0$, while standard diffusion is recovered for $α=2$ where $μ(2)=1$. We describe our numerical results with a phenomenological analytical estimation. We also contrast the C-fSM with the Riemann-Liouville fSM and Chirikov's standard map.

nlin.CD↗

arXiv2024

Scaling properties of the action in the Riemann-Liouville fractional standard map

The Riemann-Liouville fractional standard map (RL-fSM) is a two-dimensional nonlinear map with memory given in action-angle variables $(I,θ)$. The RL-fSM is parameterized by $K$ and $α\in(1,2]$ which control the strength of nonlinearity and the fractional order of the Riemann-Liouville derivative, respectively. In this work, we present a scaling study of the average squared action $\left< I^2 \right>$ of the RL-fSM along strongly chaotic orbits, i.e. for $K\gg1$. We observe two scenarios depending on the initial action $I_0$, $I_0\ll K$ or $I_0\gg K$. However, we can show that $\left< I^2 \right>/I_0^2$ is a universal function of the scaled discrete time $nK^2/I_0^2$ ($n$ being the $n$th iteration of the RL-fSM). In addition, we note that $\left< I^2 \right>$ is independent of $α$ for $K\gg1$. Analytical estimations support our numerical results.

nlin.CD↗

arXiv2023

Multiplicative topological indices: Analytical properties and application to random networks

We make use of multiplicative degree-based topological indices $X_Π(G)$ to perform a detailed analytical and statistical study of random networks $G=(V(G),E(G))$. We consider two classes of indices: $X_Π(G) = \prod_{u \in V(G)} F_V(d_u)$ and $X_Π(G) = \prod_{uv \in E(G)} F_E(d_u,d_v)$, where $uv$ denotes the edge of $G$ connecting the vertices $u$ and $v$, $d_u$ is the degree of the vertex $u$, and $F_V(x)$ and $F_E(x,y)$ are functions of the vertex degrees. Specifically, we find analytical inequalities involving these multiplicative indices. Also, we apply $X_Π(G)$ on three models of random networks: Erdös-Rényi networks, random geometric graphs, and bipartite random networks. We show that $\left< \ln X_Π(G) \right>$, normalized to the order of the network, scale with the corresponding average degree; here $\left< \cdot \right>$ denotes the average over an ensemble of random networks.

math.CO↗

arXiv2022

Revan-degree indices on random graphs

Given a simple connected non-directed graph $G=(V(G),E(G))$, we consider two families of graph invariants: $RX_Σ(G) = \sum_{uv \in E(G)} F(r_u,r_v)$ (which has gained interest recently) and $RX_Π(G) = \prod_{uv \in E(G)} F(r_u,r_v)$ (that we introduce in this work); where $uv$ denotes the edge of $G$ connecting the vertices $u$ and $v$, $r_u$ is the Revan degree of the vertex $u$, and $F$ is a function of the Revan vertex degrees. Here, $r_u = Δ+ δ- d_u$ with $Δ$ and $δ$ the maximum and minimum degrees among the vertices of $G$ and $d_u$ is the degree of the vertex $u$. Particularly, we apply both $RX_Σ(G)$ and R$X_Π(G)$ on two models of random graphs: Erdös-Rényi graphs and random geometric graphs. By a thorough computational study we show that $\left< RX_Σ(G) \right>$ and $\left< \ln RX_Π(G) \right>$, normalized to the order of the graph, scale with the average Revan degree $\left< r \right>$; here $\left< \cdot \right>$ denotes the average over an ensemble of random graphs. Moreover, we provide analytical expressions for several graph invariants of both families in the dense graph limit.

math.CO↗

arXiv2021

Mean Sombor index

We introduce a degree-based variable topological index inspired on the power (or generalized) mean. We name this new index as the mean Sombor index: $mSO_α(G) = \sum_{uv \in E(G)} \left[\left( d_u^α+d_v^α\right) /2 \right]^{1/α}$. Here, $uv$ denotes the edge of the graph $G$ connecting the vertices $u$ and $v$, $d_u$ is the degree of the vertex $u$, and $α\in \mathbb{R} \backslash \{0\}$. We also consider the limit cases $mSO_{α\to 0}(G)$ and $mSO_{α\to\pm\infty}(G)$. Indeed, for given values of $α$, the mean Sombor index is related to well-known topological indices such as the inverse sum indeg index, the reciprocal Randic index, the first Zagreb index, the Stolarsky--Puebla index and several Sombor indices. Moreover, through a quantitative structure property relationship (QSPR) analysis we show that $mSO_α(G)$ correlates well with several physicochemical properties of octane isomers. Some mathematical properties of mean Sombor indices as well as bounds and new relationships with known topological indices are also discussed.

math.CO↗

arXiv2021

Stolarsky-Puebla index

We introduce a degree-based variable topological index inspired on the Stolarsky mean (known as the generalization of the logarithmic mean). We name this new index as the Stolarsky-Puebla index: $SP_α(G) = \sum_{uv \in E(G)} d_u$, if $d_u=d_v$, and $SP_α(G) = \sum_{uv \in E(G)} \left[\left( d_u^α-d_v^α\right)/\left( α(d_u-d_v\right)\right]^{1/(α-1)}$, otherwise. Here, $uv$ denotes the edge of the network $G$ connecting the vertices $u$ and $v$, $d_u$ is the degree of the vertex $u$, and $α\in \mathbb{R} \backslash \{0,1\}$. Indeed, for given values of $α$, the Stolarsky-Puebla index reproduces well-known topological indices such as the reciprocal Randic index, the first Zagreb index, and several mean Sombor indices. Moreover, we apply these indices to random networks and demonstrate that $\left< SP_α(G) \right>$, normalized to the order of the network, scale with the corresponding average degree $\left< d \right>$.

math.CO↗

arXiv2021

Normalized Sombor indices as complexity measures of random graphs

We perform a detailed computational study of the recently introduced Sombor indices on random graphs. Specifically, we apply Sombor indices on three models of random graphs: Erdös-Rényi graphs, random geometric graphs, and bipartite random graphs. Within a statistical random matrix theory approach, we show that the average values of Sombor indices, normalized to the order of the graph, scale with the graph average degree. Moreover, we discuss the application of average Sombor indices as complexity measures of random graphs and, as a consequence, we show that selected normalized Sombor indices are highly correlated with the Shannon entropy of the eigenvectors of the graph adjacency matrix.

math.CO↗

arXiv2021

Analytical and computational properties of the variable symmetric division deg index

The aim of this work is to obtain new inequalities for the variable symmetric division deg index $SDD_α(G) = \sum_{uv \in E(G)} (d_u^α/d_v^α+d_v^α/d_u^α)$, and to characterize graphs extremal with respect to them. Here, $uv$ denotes the edge of the graph $G$ connecting the vertices $u$ and $v$, $d_u$ is the degree of the vertex $u$, and $α\in \mathbb{R}$. Some of these inequalities generalize and improve previous results for the symmetric division deg index. In addition, we computationally apply the $SDD_α(G)$ index on random graphs and show that the ratio $\left\langle SDD_α(G) \right\rangle/n$ ($n$ being the order of the graph) depends only on the average degree $\left\langle d \right\rangle$.

math.CO↗

arXiv2020

Topological versus spectral properties of random geometric graphs

In this work we perform a detailed statistical analysis of topological and spectral properties of random geometric graphs (RGGs); a graph model used to study the structure and dynamics of complex systems embedded in a two dimensional space. RGGs, $G(n,\ell)$, consist of $n$ vertices uniformly and independently distributed on the unit square, where two vertices are connected by an edge if their Euclidian distance is less or equal than the connection radius $\ell \in [0,\sqrt{2}]$. To evaluate the topological properties of RGGs we chose two well-known topological indices, the Randić index $R(G)$ and the harmonic index $H(G)$. While we characterize the spectral and eigenvector properties of the corresponding randomly-weighted adjacency matrices by the use of random matrix theory measures: the ratio between consecutive eigenvalue spacings, the inverse participation ratios and the information or Shannon entropies $S(G)$. First, we review the scaling properties of the averaged measures, topological and spectral, on RGGs. Then we show that: (i) the averaged--scaled indices, $\left\langle R(G) \right\rangle$ and $\left\langle H(G) \right\rangle$, are highly correlated with the average number of non-isolated vertices $\left\langle V_\times(G) \right\rangle$; and (ii) surprisingly, the averaged--scaled Shannon entropy $\left\langle S(G) \right\rangle$ is also highly correlated with $\left\langle V_\times(G) \right\rangle$. Therefore, we suggest that very reliable predictions of eigenvector properties of RGGs could be made by computing topological indices.

cond-mat.dis-nn↗

arXiv2013

Dynamical properties of a dissipative discontinuous map: A scaling investigation

The effects of dissipation on the scaling properties of nonlinear discontinuous maps are investigated by analyzing the behavior of the average squared action $\left< I^2 \right>$ as a function of the $n$-th iteration of the map as well as the parameters $K$ and $γ$, controlling nonlinearity and dissipation, respectively. We concentrate our efforts to study the case where the nonlinearity is large; i.e., $K\gg 1$. In this regime and for large initial action $I_0\gg K$, we prove that dissipation produces an exponential decay for the average action $\left< I \right>$. Also, for $I_0\cong 0$, we describe the behavior of $\left< I^2 \right>$ using a scaling function and analytically obtain critical exponents which are used to overlap different curves of $\left< I^2 \right>$ onto an universal plot. We complete our study with the analysis of the scaling properties of the deviation around the average action $ω$.

nlin.CD↗

arXiv2012

Scaling properties of discontinuous maps

We study the scaling properties of discontinuous maps by analyzing the average value of the squared action variable $I^2$. We focus our study on two dynamical regimes separated by the critical value $K_c$ of the control parameter $K$: the slow diffusion ($K K_c$) regimes. We found that the scaling of $I^2$ for discontinuous maps when $K\ll K_c$ and $K\gg K_c$ obeys the same scaling laws, in the appropriate limits, than Chirikov's standard map in the regimes of weak and strong nonlinearity, respectively. However, due to absence of KAM tori, we observed in both regimes that $I^2\propto nK^β$ for $n\gg 1$ (being $n$ the $n$-th iteration of the map) with $β\approx 5/2$ when $K\ll K_c$ and $β\approx 2$ for $K\gg K_c$.

nlin.CD↗