Numerical Attractors Research Articles

Predictive Artificial Intelligence Models for Energy Efficiency in Hybrid and Electric Vehicles: Analysis for Enna, Sicily

Developments in artificial intelligence techniques allow for an improvement in sustainable mobility strategies with particular reference to energy consumption estimates of electric vehicles (EVs). This research proposes a vehicle energy model developed on the basis of deep neural network (DNN) technology. This study also explores the potential application of the model developed for the movement data of new vehicles in the province of Enna, Sicily, Italy, which are characterized by numerous attractors and the increasing number of hybrid and electric cars circulating. The energy model for electric vehicles shows high accuracy and versatility, requiring vehicle velocity and acceleration as input data to predict energy consumption. This research article also provides recommendations for the energy modeling of electric vehicles and outlines additional steps for model development. The implemented methodological approach and its results can be used by transport decision-makers to plan new transport policies in Italian cities aimed at optimizing vehicle charging infrastructure. They can also help vehicle users accurately estimate energy consumption, generate maps, and identify locations with the highest energy consumption.

Numerical dynamics for discrete nonlinear damping Korteweg–de Vries equations

We study the numerical scheme of both solution and attractor for the time-space discrete nonlinear damping Korteweg–de Vries (KdV) equation, which is neither conservative nor coercive. First, we establish a new Taylor expansion as well as a global attractor for the KdV lattice system. Second, we prove the unique existence of numerical solution as well as numerical attractor for the discrete-time KdV lattice system via the implicit Euler scheme. Third, we estimate the discretization error and interpolation error between continuous-time and discrete-time solutions, and then establish the upper semi-convergence from numerical attractors to the global attractor as the time-size tends to zero. Fourth, we establish the finitely dimensional approximation of numerical attractors. Finally, we establish the upper bound as well as the lower semi-convergence of numerical attractors with respect to the external force and the damping constant.

Numerical Attractors for Rough Differential Equations

Numerical Attractors for Rough Differential Equations

Enlarged Numerical Attractors for Lattice Systems with Porous Media Degeneracies

Enlarged Numerical Attractors for Lattice Systems with Porous Media Degeneracies

New Physical–Mathematical Analysis of Cardiac Dynamics and Temperature for the Diagnosis of Infectious Disease

Background: Physical and mathematical theories have made it possible to generate methods for the characterization and diagnosis of physiological variables such as cardiac dynamics. Therefore, it would be useful to implement them to evaluate the dynamic changes in human physiology during the development of COVID-19, which causes disease, severe respiratory and death. Objective: to establish a method for detecting possible alterations associated with COVID-19 through simulations of adult cardiac dynamics and body temperature using dynamic systems theory, probability, entropy and set theory. Methodology: simulations of cardiac dynamics were generated in subjects with 10 temperature ranges between 32 °C and 42 °C via numerical attractors after their evaluation using entropy proportions. Results: differences were observed in the proportions of entropy that differentiate normal cardiac dynamics and acute myocardial infarction towards progression to fever. Conclusion: the physical mathematical analysis of cardiac behavior in relation to body temperature in people with COVID-19 allowed the establishment of a possible surveillance method for detecting minor alterations.

Numerical Attractors via Discrete Rough Paths

Numerical Attractors via Discrete Rough Paths

Optimization and Convergence of Numerical Attractors for Discrete-Time Quasi-Linear Lattice System

Optimization and Convergence of Numerical Attractors for Discrete-Time Quasi-Linear Lattice System

Numerical attractors and approximations for stochastic or deterministic sine-Gordon lattice equations

First, we apply the implicit Euler scheme to discretize the sine-Gordon lattice equation (possessing a global attractor) and prove the existence of a numerical attractor for the time-discrete sine-Gordon lattice system with small step sizes. Second, we establish the upper semi-convergence from the numerical attractor towards the global attractor when the step size tends to zero. Third, we establish the upper semi-convergence from the random attractor of the stochastic sine-Gordon lattice equation to the global attractor when the intensity of noise goes to zero. Fourth, we show the finitely dimensional approximations of the three (numerical, random and global) attractors as the dimension of the state space goes to infinity. In a word, we establish four paths of convergence of finitely dimensional (numerical and random) attractors towards the global attractor.

BSTAB: an open-source software for computing the basin stability of multi-stable dynamical systems

The pervasiveness of multi-stability in nonlinear dynamical systems calls for novel concepts of stability and a consistent quantification of long-term behavior. The basin stability is a global stability metric that builds on estimating the basin of attraction volumes by Monte Carlo sampling. The computation involves extensive numerical time integrations, attractor characterization, and clustering of trajectories. We introduce bSTAB, an open-source software project that aims at enabling researchers to efficiently compute the basin stability of their dynamical systems with minimal efforts and in a highly automated manner. The source code, available at https://github.com/TUHH-DYN/bSTAB/, is available for the programming language Matlab featuring parallelization for distributed computing, automated sensitivity and bifurcation analysis as well as plotting functionalities. We illustrate the versatility and robustness of bSTAB for four canonical dynamical systems from several fields of nonlinear dynamics featuring periodic and chaotic dynamics, complicated multi-stability, non-smooth dynamics, and fractal basins of attraction. The bSTAB projects aims at fostering interdisciplinary scientific collaborations in the field of nonlinear dynamics and is driven by the interaction and contribution of the community to the software package.

Attractor as a convex combination of a set of attractors

This paper presents an effective approach to constructing numerical attractors of a general class of continuous homogenous dynamical systems: decomposing an attractor as a convex combination of a set of other existing attractors. For this purpose, the convergent Parameter Switching (PS) numerical method is used to integrate the underlying dynamical system. The method is built on a convergent fixed step-size numerical method for ODEs. The paper shows that the PS algorithm, incorporating two binary operations, can be used to approximate any numerical attractor via a convex combination of some existing attractors. Several examples are presented to show the effectiveness of the proposed method.

Application of a Diagnostic Methodology of Cardiac Systems Based on the Proportions of Entropy in Normal Patients and with Different Pathologies

Introduction: Dynamical systems theory, probability, and entropy were the substrate for the development of the diagnostic and predictive methodology of adult heart dynamics. Objective: To apply a previously developed methodology from dynamical systems, probability, and entropy in both normal and pathological subjects. Methods: Electrocardiographic records were selected from 30 healthy subjects and 200 with different pathologies, with a length of least 18 h. Numerical attractors from dynamical attractors and the probability of occurrence of ordered pairs of consecutive heart rates were built. A calculation of entropy and its proportions was performed and statistical analysis was conducted. Results: The normal patients’ heart dynamics were evaluated according to the methodology of entropy proportions, highlighting that there are differences in normal patients with different pathologies. There was maximal level of sensitivity, specificity, and diagnostic agreement. Conclusion: Proportional entropy constitutes a diagnostic and predictive method of heart systems, and may be useful as a tool to objectively diagnose and perform the follow-up of normal and pathological cases.

Discretisation of Global Attractors for Lattice Dynamical Systems

The existence of numerical attractors for lattice dynamical systems is established, where the implicit Euler scheme is used for time discretisation. Infinite dimensional discrete lattice systems as well as their finite dimensional truncations are considered. It is shown that the finite dimensional numerical attractors converge upper semicontinuously to the global attractor of the original lattice model as the discretisation step size tends to zero.

A novel heart rate attractor for the prediction of cardiovascular disease

Abstract Rationale The cardiac dynamic has been studied with a new methodology based on probability and proportions of entropy of frequencies of appearance, by means of heart rate (HR) in a new kind of attractor. This methodology has established numerical predictions of cardiac dynamics, including for patients from the coronary care unit. Objective To validate a novel method with heart rate attractor in the prediction of cardiovascular disease, applied at the coronary care unit. Methods and results An application of this methodology in a blind study with 800 Holter registries was done. With the maximum and minimum values of HR, and the number of heartbeats for each hour, a numerical attractor was built for each Holter monitor. Occurrence probabilities of HR ordered pairs were calculated as well as the Boltzmann Gibb's entropy. The S/k value and entropy proportions were found, quantifying the dynamic state. Finally, the mathematical evaluation was compared with the gold standard by means of sensitivity, specificity and Kappa coefficient. The proportion's values were within the expected ranges, both normal and pathological Holter registries, and the diagnostic concordance measurements showed the maximum possible values; the sensitivity and specificity were 100% and the Kappa coefficient was 1. Conclusions The diagnostic capacity of the methodology was confirmed with 800 Holter registries, showing that this methodology quantitatively differentiates normality, illness and evolution between both states, which enables quicker assessments to be made.

Prediction of Solutions of the Duffing System with Tuned Mass Damper

Abstract In this work we analyze the behavior of a nonlinear dynamical system using a probabilistic approach. We focus on the coexistence of solutions and we check how the changes in the parameters of excitation influence the dynamics of the system. For the demonstration we use the Duffing oscillator with the tuned mass absorber. We mention the numerous attractors present in such a system and describe how they were found with the method based on the basin stability concept.

On the fractal basins of convergence of the libration points in the axisymmetric five-body problem: The convex configuration

In the present work, the Newton–Raphson basins of convergence, corresponding to the coplanar libration points (which act as numerical attractors), are unveiled in the axisymmetric five-body problem, where convex configuration is considered. In particular, the four primaries are set in axisymmetric central configuration, where the motion is governed only by mutual gravitational attractions. It is observed that the total number libration points are either eleven, thirteen or fifteen for different combinations of the angle parameters. Moreover, the stability analysis revealed that all the libration points are linearly stable for all the studied combination of angle parameters. The multivariate version of the Newton–Raphson iterative scheme is used to reveal the structures of the basins of convergence, associated with the libration points, on various types of two-dimensional configuration planes. In addition, we present how the basins of convergence are related with the corresponding number of required iterations. It is unveiled that in almost every case, the basins of convergence corresponding to the collinear libration point L2 have infinite extent. Moreover, for some combination of the angle parameters, the other collinear libration points L1,2 also have infinite extent. In addition, it can be observed that the domains of convergence, associated with the collinear libration point L1, look like exotic bugs with many legs and antennas whereas the domains of convergence, associated with L4,5 look like butterfly wings for some combinations of angle parameters. Particularly, our numerical investigation suggests that the evolution of the attracting domains in this dynamical system is very complicated, yet a worth studying problem.

Revealing the Newton–Raphson basins of convergence in the circular pseudo-Newtonian Sitnikov problem

In this paper we numerically explore the convergence properties of the pseudo-Newtonian circular restricted problem of three and four primary bodies. The classical Newton–Raphson iterative scheme is used for revealing the basins of convergence and their respective fractal basin boundaries on the complex plane. A thorough and systematic analysis is conducted in an attempt to determine the influence of the transition parameter on the convergence properties of the system. Additionally, the roots (numerical attractors) of the system and the basin entropy of the convergence diagrams are monitored as a function of the transition parameter, thus allowing us to extract useful conclusions. The probability distributions, as well as the distributions of the required number of iterations are also correlated with the corresponding basins of convergence.

Comparing the Geometry of the Basins of Attraction, the Speed and the Efficiency of Several Numerical Methods

We use simple equations in order to compare the basins of attraction on the complex plane, corresponding to a large collection of numerical methods, of several order. Two cases are considered, regarding the total number of the roots, which act as numerical attractors. For both cases we use the iterative schemes for performing a thorough and systematic classification of the nodes on the complex plane. The distributions of the required iterations as well as the probability and their correlations with the corresponding basins of convergence are also discussed. Our numerical calculations suggest that most of the iterative schemes provide relatively similar convergence structures on the complex plane. In addition, several aspects of the numerical methods are compared in an attempt to obtain general conclusions regarding their speed and efficiency. Moreover, we try to determine how the complexity of the each case influences the main characteristics of the numerical methods.

Proportional entropy of normal cardiac dynamic and with arrhythmia in the Intensive Care Unit

Based on the theory of probability and not equiprobable entropy ratios in the context of dynamical systems theory, a methodology for cardiac dynamic evaluation was developed. To confirm the clinical applicability of the diagnostic methodology based on probability and entropy for patients with different types of arrhythmia assessing the degree of intensification in the ICU through its characteristic proportions. 80 Holter records from ICU diagnosed with arrhythmia and 20 normal were analyzed. Holter clinical findings were masked and were taken from them the maximum and minimum values of heart rate and total number of beats per minute to build the numerical attractor for each dynamics. Probability, entropy, S/k ratio, and proportions of entropy were evaluated, establishing its mathematical diagnosis and quantifying its severity level. It was shown that the methodology difference in all cases normal of abnormal dynamics, obtaining a sensitivity and specificity of 100 % and Kappa coefficient of 1, showing that it is also possible to establish quantitatively the degree of exacerbation. The methodology developed is useful in clinical practice to correlate the severity of arrhythmias with physical and mathematical predictions of exacerbation.

Dissipativity of the linearly implicit Euler scheme for Navier‐Stokes equations with delay

In this article, we study the dissipativity of the linearly implicit Euler scheme for the 2D Navier‐Stokes equations with time delay volume forces (NSD). This scheme can be viewed as an application of the implicit Euler scheme to linearized NSD. Therefore, only a linear system is needed to solve at each time step. The main results we obtain are that this scheme is L2 dissipative for any time step size and H1 dissipative under a time‐step constraint. As a consequence, the existence of a numerical attractor of the discrete dynamical system is established. A by‐product of the dissipativity analysis of the linearly implicit Euler scheme for NSD is that the dissipativity of an implicit‐explicit scheme for the celebrated Navier‐Stokes equations that treats the volume forces term explicitly is obtained.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 2114–2140, 2017

Initial-state dependence of the route to chaos of an external-cavity laser

External-cavity semiconductor lasers (ECLs), consisting of a laser diode in front of a mirror to reflect light back into the laser diode, are among the most important dynamical systems because of their ultrafast dynamics, their tunability, and the numerous existing applications. The dynamics of an ECL is greatly influenced by the existence and stability of numerous modes of the external cavity. In such high-dimensional nonlinear systems, numerous attractors, located around various modes of the external cavity, can coexist in phase space for a given set of parameters, a phenomenon called generalized multistability. In this work, we propose a procedure that allows one to select experimentally different modes of the external cavity as different initial states. We use this procedure to reveal experimentally generalized multistability in an ECL through the demonstration that different routes to chaos exist in an ECL, depending on the initial state selected. In particular, we show that the famous quasiperiodic route to chaos is only observed for specific choices of initial conditions.