arXiv subjects

Vladimir García-Morales

Publications and source records attributed to Vladimir García-Morales.

17 recordsLinked to original sources

arXiv2026

Why is Seasonal Climate Predictable Beyond the Limit of Deterministic Predictability set by Chaos?

The Earth's climate is an ensemble of interacting, spatially extended oscillatory media ('climate systems') whose slow-changing averages coexist with chaotic, high-frequency weather fluctuations in quasi-equilibrium. The limit of deterministic predictability (LDP) for any climate system is determined by its fastest-growing errors. However, recent findings show that the Indian Summer Monsoon Rainfall (ISMR) can be predicted up to 18 months in advance-far beyond its LDP. Using a model of two interacting oscillatory media, we show that this extended predictability arises from lag synchronization between ISMR and its predictor, the Global El Nino-Southern Oscillation (G-ENSO), to which it is strongly coupled. We introduce complex order parameters representing the internal dynamics of the two climate systems. Their spatiotemporal evolution is governed by coupled Complex Ginzburg-Landau Equations, producing aperiodic yet strongly correlated time series at long lead times. Our findings have far-reaching consequences in advancing seasonal prediction across climate systems.

physics.ao-ph↗

arXiv2023

Continuum percolation theory of epimorphic regeneration

A biophysical model of epimorphic regeneration based on a continuum percolation process of fully penetrable disks in two dimensions is proposed. All cells within a randomly chosen disk of the regenerating organism are assumed to receive a signal in the form of a circular wave as a result of the action/reconfiguration of neoblasts and neoblast-derived mesenchymal cells in the blastema. These signals trigger the growth of the organism, whose cells read, on a faster time scale, the electric polarization state responsible for their differentiation and the resulting morphology. In the long time limit, the process leads to a morphological attractor that depends on experimentally accessible control parameters governing the blockage of cellular gap junctions and, therefore, the connectivity of the multicellular ensemble. When this connectivity is weakened, positional information is degraded leading to more symmetrical structures. This general theory is applied to the specifics of planaria regeneration. Computations and asymptotic analyses made with the model show that it correctly describes a significant subset of the most prominent experimental observations, notably anterior-posterior polarization (and its loss) or the formation of four-headed planaria.

physics.bio-ph↗

arXiv2021

A constructive theory of shape

We formulate a theory of shape valid for objects of arbitrary dimension whose contours are path connected. We apply this theory to the design and modeling of viable trajectories of complex dynamical systems. Infinite families of qualitatively similar shapes are constructed giving as input a finite ordered set of characteristic points (landmarks) and the value of a continuous parameter $κ\in (0,\infty)$. We prove that all shapes belonging to the same family are located within the convex hull of the landmarks. The theory is constructive in the sense that it provides a systematic means to build a mathematical model for any shape taken from the physical world. We illustrate this with a variety of examples: (chaotic) time series, plane curves, space filling curves, knots and strange attractors.

math.NA↗

arXiv2020

On the nontrivial zeros of the Dirichlet eta function

We construct a two-parameter complex function $η_{κν}:\mathbb{C}\to \mathbb{C}$, $κ\in (0, \infty)$, $ν\in (0,\infty)$ that we call a holomorphic nonlinear embedding and that is given by a double series which is absolutely and uniformly convergent on compact sets in the entire complex plane. The function $η_{κν}$ converges to the Dirichlet eta function $η(s)$ as $κ\to \infty$. We prove the crucial property that, for sufficiently large $κ$, the function $η_{κν}(s)$ can be expressed as a linear combination $η_{κν}(s)=\sum_{n=0}^{\infty}a_n(κ) η(s+2νn)$ of horizontal shifts of the eta function (where $a_{n}(κ) \in \mathbb{R}$ and $a_{0}=1$) and that, indeed, we have the inverse formula $η(s)=\sum_{n=0}^{\infty}b_n(κ) η_{κν}(s+2νn)$ as well (where the coefficients $b_{n}(κ) \in \mathbb{R}$ are obtained from the $a_{n}$'s recursively). By using these results and the functional relationship of the eta function, $η(s)=λ(s)η(1-s)$, we sketch a proof of the Riemann hypothesis which, in our setting, is equivalent to the fact that the nontrivial zeros $s^{*}=σ^{*}+it^{*}$ of $η(s)$ (i.e. those points for which $η(s^{*})=η(1-s^{*})=0)$ are all located on the critical line $σ^{*}=\frac{1}{2}$.

math.GM↗

arXiv2020

A theorem on integration based on the digital expansion

The binary radix expansion of a real number can be used to code the outcome of any series of coin tosses, a fact that provides an intriguing link between number theory, measure theory and statistical physics. Inspired by this fact, a general result is established for the definite integral of a differentiable function of a single variable that allows any such integral to be exactly written in terms of a double series. The theorem can be directly applied to a wide variety of integrals of physical interest and to derive new series expansions of real numbers and real-valued functions. We apply the theorem to the integration of the equation of motion in one dimension of classical Hamiltonian systems, focusing in the analysis of the nonlinear pendulum.

math-ph↗

arXiv2020

Unifying vectors and matrices of different dimensions through nonlinear embeddings

Complex systems may morph between structures with different dimensionality and degrees of freedom. As a tool for their modelling, nonlinear embeddings are introduced that encompass objects with different dimensionality as a continuous parameter $κ\in \mathbb{R}$ is being varied, thus allowing the unification of vectors, matrices and tensors in single mathematical structures. This technique is applied to construct warped models in the passage from supergravity in 10 or 11-dimensional spacetimes to 4-dimensional ones. We also show how nonlinear embeddings can be used to connect cellular automata (CAs) to coupled map lattices (CMLs) and to nonlinear partial differential equations, deriving a class of nonlinear diffusion equations. Finally, by means of nonlinear embeddings we introduce CA connections, a class of CMLs that connect any two arbitrary CAs in the limits $κ\to 0$ and $κ\to \infty$ of the embedding.

nlin.CG↗

arXiv2019

A new approach to fuzzy sets: Application to the design of nonlinear time-series, symmetry-breaking patterns, and non-sinusoidal limit-cycle oscillations

It is shown that characteristic functions of sets can be made fuzzy by means of the $\mathcal{B}_κ$-function, recently introduced by the author, where the fuzziness parameter $κ\in \mathbb{R}$ controls how much a fuzzy set deviates from the crisp set obtained in the limit $κ\to 0$. As applications, we present first a general expression for a switching function that may be of interest in electrical engineering and in the design of nonlinear time-series. We then introduce another general expression that allows wallpaper and frieze patterns for every possible planar symmetry group (besides patterns typical of quasicrystals) to be designed. We show how the fuzziness parameter $κ$ plays an analogous role to temperature in physical applications and may be used to break the symmetry of spatial patterns. As a further, important application, we establish a theorem on the shaping of limit cycle oscillations far from bifurcations in smooth deterministic nonlinear dynamical systems governed by differential equations. Following this application, we briefly discuss a generalization of the Stuart-Landau equation to non-sinusoidal oscillators obtained as a consequence of our theorem.

nlin.CD↗

arXiv2018

Diagrammatic approach to cellular automata and the emergence of form with inner structure

We present a diagrammatic method to build up sophisticated cellular automata (CAs) as models of complex physical systems. The diagrams complement the mathematical approach to CA modeling, whose details are also presented here, and allow CAs in rule space to be classified according to their hierarchy of layers. Since the method is valid for any discrete operator and only depends on the alphabet size, the resulting conclusions, of general validity, apply to CAs in any dimension or order in time, arbitrary neighborhood ranges and topology. We provide several examples of the method, illustrating how it can be applied to the mathematical modeling of the emergence of order out of disorder. Specifically, we show how the the majority CA rule can be used as a building block to construct more complex cellular automata in which separate domains (with substructures having different dynamical properties) are able to emerge out of disorder and coexist in a stable manner.

nlin.CG↗

arXiv2017

Weakly coupled map lattice models for multicellular patterning and collective normalization of abnormal single-cell states

We present a weakly coupled map lattice model for patterning that explores the effects exerted by weakening the local dynamic rules on model biological and artificial networks composed of two-state building blocks (cells). To this end, we use two cellular automata models based on: (i) a smooth majority rule (model I) and (ii) a set of rules similar to those of Conway's Game of Life (model II). The normal and abnormal cell states evolve according with local rules that are modulated by a parameter $κ$. This parameter quantifies the effective weakening of the prescribed rules due to the limited coupling of each cell to its neighborhood and can be experimentally controlled by appropriate external agents. The emergent spatio-temporal maps of single-cell states should be of significance for positional information processes as well as for intercellular communication in tumorigenesis where the collective normalization of abnormal single-cell states by a predominantly normal neighborhood may be crucial.

physics.bio-ph↗

arXiv2017

Nonlinear embeddings: Applications to analysis, fractals and polynomial root finding

We introduce $\mathcal{B}_κ$-embeddings, nonlinear mathematical structures that connect, through smooth paths parameterized by $κ$, a finite or denumerable set of objects at $κ=0$ (e.g. numbers, functions, vectors, coefficients of a generating function...) to their ordinary sum at $κ\to \infty$. We show that $\mathcal{B}_κ$-embeddings can be used to design nonlinear irreversible processes through this connection. A number of examples of increasing complexity are worked out to illustrate the possibilities uncovered by this concept. These include not only smooth functions but also fractals on the real line and on the complex plane. As an application, we use $\mathcal{B}_κ$-embeddings to formulate a robust method for finding all roots of a univariate polynomial without factorizing or deflating the polynomial. We illustrate this method by finding all roots of a polynomial of 19th degree.

math-ph↗

arXiv2016

Digit replacement: A generic map for nonlinear dynamical systems

A simple discontinuous map is proposed as a generic model for nonlinear dynamical systems. The orbit of the map admits exact solutions for wide regions in parameter space and the method employed (digit manipulation) allows the mathematical design of useful signals, such as regular or aperiodic oscillations with specific waveforms, the construction of complex attractors with nontrivial properties as well as the coexistence of different basins of attraction in phase space with different qualitative properties. A detailed analysis of the dynamical behavior of the map suggests how the latter can be used in the modeling of complex nonlinear dynamics including, e.g., aperiodic nonchaotic attractors and the hierarchical deposition of grains of different sizes on a surface.

nlin.CD↗

arXiv2016

On the characteristic function of a collection of sets

The union of a collection of $n$ sets is generally expressed in terms of a characteristic (indicator) function that contains $2^{n}-1$ terms. In this article, a much simpler expression is found that requires the evaluation of $n$ terms only. This leads to a major simplification of any normal form involving characteristic functions of sets. The formula can be useful in recognizing inclusion-exclusion patterns of combinatorial problems.

math.GM↗

arXiv2016

From deterministic cellular automata to coupled map lattices

A general mathematical method is presented for the systematic construction of coupled map lattices (CMLs) out of deterministic cellular automata (CAs). The entire CA rule space is addressed by means of a universal map for CAs that we have recently derived and that is not dependent on any freely adjustable parameters. The CMLs thus constructed are termed real-valued deterministic cellular automata (RDCA) and encompass all deterministic CAs in rule space in the asymptotic limit $κ\to 0$ of a continuous parameter $κ$. Thus, RDCAs generalize CAs in such a way that they constitute CMLs when $κ$ is finite and nonvanishing. In the limit $κ\to \infty$ all RDCAs are shown to exhibit a global homogeneous fixed-point that attracts all initial conditions. A new bifurcation is discovered for RDCAs and its location is exactly determined from the linear stability analysis of the global quiescent state. In this bifurcation, fuzziness gradually begins to intrude in a purely deterministic CA-like dynamics. The mathematical method presented allows to get insight in some highly nontrivial behavior found after the bifurcation.

nlin.CG↗

arXiv2016

Cellular automaton for chimera states

A minimalistic model for chimera states is presented. The model is a cellular automaton (CA) which depends on only one adjustable parameter, the range of the nonlocal coupling, and is built from elementary cellular automata and the majority (voting) rule. This suggests the universality of chimera-like behavior from a new point of view: Already simple CA rules based on the majority rule exhibit this behavior. After a short transient, we find chimera states for arbitrary initial conditions, the system spontaneously splitting into stable domains separated by static boundaries, ones synchronously oscillating and the others incoherent. When the coupling range is local, nontrivial coherent structures with different periodicities are formed.

nlin.PS↗

arXiv2016

Semipredictable dynamical systems

A new class of deterministic dynamical systems, termed semipredictable dynamical systems, is presented. The spatiotemporal evolution of these systems have both predictable and unpredictable traits, as found in natural complex systems. We prove a general result: The dynamics of any deterministic nonlinear cellular automaton (CA) with $p$ possible dynamical states can be decomposed at each instant of time in a superposition of $N$ layers involving $p_{0}$, $p_{1}$,... $p_{N-1}$ dynamical states each, where the $p_{k\in \mathbb{N}}$, $k \in [0, N-1]$ are divisors of $p$. If the divisors coincide with the prime factors of $p$ this decomposition is unique. Conversely, we also prove that $N$ CA working on symbols $p_{0}$, $p_{1}$,... $p_{N-1}$ can be composed to create a graded CA rule with $N$ different layers. We then show that, even when the full spatiotemporal evolution can be unpredictable, certain traits (layers) can exactly be predicted. We present explicit examples of such systems involving compositions of Wolfram's 256 elementary CA and a more complex CA rule acting on a neighborhood of two sites and 12 symbols and whose rule table corresponds to the smallest Moufang loop $M_{12}(S_{3},2)$.

nlin.CD↗

arXiv2014

Unusual synchronization phenomena during electrodissolution of silicon: the role of nonlinear global coupling

The photoelectrodissolution of n-type silicon constitutes a convenient model system to study the nonlinear dynamics of oscillatory media. On the silicon surface, a silicon oxide layer forms. In the lateral direction, the thickness of this layer is not uniform. Rather, several spatio-temporal patterns in the oxide layer emerge spontaneously, ranging from cluster patterns and turbulence to quite peculiar dynamics like chimera states. Introducing a nonlinear global coupling in the complex Ginzburg-Landau equation allows us to identify this nonlinear coupling as the essential ingredient to describe the patterns found in the experiments. The nonlinear global coupling is designed in such a way, as to capture an important, experimentally observed feature: the spatially averaged oxide-layer thickness shows nearly harmonic oscillations. Simulations of the modified complex Ginzburg-Landau equation capture the experimental dynamics very well.

nlin.PS↗

arXiv2013

Coexistence of synchrony and incoherence in oscillatory media under nonlinear global coupling

We report a novel mechanism for the formation of chimera states, a peculiar spatiotemporal pattern with coexisting synchronized and incoherent domains found in ensembles of identical oscillators. Considering Stuart-Landau oscillators we demonstrate that a nonlinear global coupling can induce this symmetry breaking. We find chimera states also in a spatially extended system, a modified complex Ginzburg-Landau equation. This theoretical prediction is validated with an oscillatory electrochemical system, the electrooxidation of silicon, where the spontaneous formation of chimeras is observed without any external feedback control.

nlin.CD↗