Chaos: An Interdisciplinary Journal of Nonlinear Science: Most Cited articles
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Most cited articles from Chaos: An Interdisciplinary Journal of Nonlinear Scienceen-usSat, 23 Oct 2021 01:26:05 GMTAtypon® Literatum™http://validator.w3.org/feed/docs/rss2.html10080Chaos: An Interdisciplinary Journal of Nonlinear Science: Most Cited articleshttps://aip.scitation.org/na101/home/literatum/publisher/aip/journals/covergifs/cha/cover.jpg
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Quantification of scaling exponents and crossover phenomena in nonstationary heartbeat time series
https://aip.scitation.org/doi/10.1063/1.166141?feed=most-cited
https://aip.scitation.org/doi/10.1063/1.166141?feed=most-citedThe healthy heartbeat is traditionally thought to be regulated according to the classical principle of homeostasis whereby physiologic systems operate to reduce variability and achieve an equilibrium‐like state [Physiol. Rev. 9, 399–431 (1929)]. However, recent studies [Phys. Rev. Lett. 70, 1343–1346 (1993); Fractals in Biology and Medicine (Birkhauser‐Verlag, Basel, 1994), pp. 55–65] reveal that under normal conditions, beat‐to‐beat fluctuations in heart rate display the kind of long‐range correlations typically exhibited by dynamical systems far from equilibrium [Phys. Rev. Lett. 59, 381–384 (1987)]. In contrast, heart rate time series from patients with severe congestive heart failure show a breakdown of thisC.‐K. Peng, Shlomo Havlin, H. Eugene Stanley, and Ary L. GoldbergerFri, 14 Aug 1998 07:00:00 GMTVORO++: A three-dimensional Voronoi cell library in C++
https://aip.scitation.org/doi/10.1063/1.3215722?feed=most-cited
https://aip.scitation.org/doi/10.1063/1.3215722?feed=most-citedChris H. RycroftTue, 27 Oct 2009 07:00:00 GMTLow dimensional behavior of large systems of globally coupled oscillators
https://aip.scitation.org/doi/10.1063/1.2930766?feed=most-cited
https://aip.scitation.org/doi/10.1063/1.2930766?feed=most-citedIt is shown that, in the infinite size limit, certain systems of globally coupled phase oscillators display low dimensional dynamics. In particular, we derive an explicit finite set of nonlinear ordinary differential equations for the macroscopic evolution of the systems considered. For example, an exact, closed form solution for the nonlinear time evolution of the Kuramoto problem with a Lorentzian oscillator frequency distribution function is obtained. Low dimensional behavior is also demonstrated for several prototypical extensions of the Kuramoto model, and time-delayed coupling is also considered.Edward Ott and Thomas M. AntonsenMon, 22 Sep 2008 07:00:00 GMTApproximate entropy (ApEn) as a complexity measure
https://aip.scitation.org/doi/10.1063/1.166092?feed=most-cited
https://aip.scitation.org/doi/10.1063/1.166092?feed=most-citedApproximate entropy (ApEn) is a recently developed statistic quantifying regularity and complexity, which appears to have potential application to a wide variety of relatively short (greater than 100 points) and noisy time‐series data. The development of ApEn was motivated by data length constraints commonly encountered, e.g., in heart rate, EEG, and endocrine hormone secretion data sets. We describe ApEn implementation and interpretation, indicating its utility to distinguish correlated stochastic processes, and composite deterministic/ stochastic models. We discuss the key technical idea that motivates ApEn, that one need not fully reconstruct an attractor to discriminate in a statistically valid manner—marginal probabilitySteve PincusFri, 14 Aug 1998 07:00:00 GMTPractical implementation of nonlinear time series methods: The TISEAN package
https://aip.scitation.org/doi/10.1063/1.166424?feed=most-cited
https://aip.scitation.org/doi/10.1063/1.166424?feed=most-citedWe describe the implementation of methods of nonlinear time series analysis which are based on the paradigm of deterministic chaos. A variety of algorithms for data representation, prediction, noise reduction, dimension and Lyapunov estimation, and nonlinearity testing are discussed with particular emphasis on issues of implementation and choice of parameters. Computer programs that implement the resulting strategies are publicly available as the TISEAN software package. The use of each algorithm will be illustrated with a typical application. As to the theoretical background, we will essentially give pointers to the literature.Rainer Hegger, Holger Kantz, and Thomas SchreiberWed, 26 May 1999 07:00:00 GMTApplied Koopmanism
https://aip.scitation.org/doi/10.1063/1.4772195?feed=most-cited
https://aip.scitation.org/doi/10.1063/1.4772195?feed=most-citedA majority of methods from dynamical system analysis, especially those in applied settings, rely on Poincaré's geometric picture that focuses on “dynamics of states.” While this picture has fueled our field for a century, it has shown difficulties in handling high-dimensional, ill-described, and uncertain systems, which are more and more common in engineered systems design and analysis of “big data” measurements. This overview article presents an alternative framework for dynamical systems, based on the “dynamics of observables” picture. The central object is the Koopman operator: an infinite-dimensional, linear operator that is nonetheless capable of capturing the full nonlinear dynamics. TheMarko Budišić, Ryan Mohr, and Igor MezićFri, 21 Dec 2012 08:00:00 GMTUsing machine learning to replicate chaotic attractors and calculate Lyapunov exponents from data
https://aip.scitation.org/doi/10.1063/1.5010300?feed=most-cited
https://aip.scitation.org/doi/10.1063/1.5010300?feed=most-citedWe use recent advances in the machine learning area known as “reservoir computing” to formulate a method for model-free estimation from data of the Lyapunov exponents of a chaotic process. The technique uses a limited time series of measurements as input to a high-dimensional dynamical system called a “reservoir.” After the reservoir's response to the data is recorded, linear regression is used to learn a large set of parameters, called the “output weights.” The learned output weights are then used to form a modified autonomous reservoir designed to be capable of producing an arbitrarily long time series whose ergodic propertiesJaideep Pathak, Zhixin Lu, Brian R. Hunt, Michelle Girvan, and Edward OttWed, 06 Dec 2017 03:30:27 GMTGait dynamics in Parkinson’s disease: Common and distinct behavior among stride length, gait variability, and fractal-like scaling
https://aip.scitation.org/doi/10.1063/1.3147408?feed=most-cited
https://aip.scitation.org/doi/10.1063/1.3147408?feed=most-citedParkinson’s disease (PD) is a common, debilitating neurodegenerative disease. Gait disturbances are a frequent cause of disability and impairment for patients with PD. This article provides a brief introduction to PD and describes the gait changes typically seen in patients with this disease. A major focus of this report is an update on the study of the fractal properties of gait in PD, the relationship between this feature of gait and stride length and gait variability, and the effects of different experimental conditions on these three gait properties. Implications of these findings are also briefly described. This update highlights theJeffrey M. HausdorffMon, 29 Jun 2009 07:00:00 GMTComplex systems analysis of series of blackouts: Cascading failure, critical points, and self-organization
https://aip.scitation.org/doi/10.1063/1.2737822?feed=most-cited
https://aip.scitation.org/doi/10.1063/1.2737822?feed=most-citedWe give an overview of a complex systems approach to large blackouts of electric power transmission systems caused by cascading failure. Instead of looking at the details of particular blackouts, we study the statistics and dynamics of series of blackouts with approximate global models. Blackout data from several countries suggest that the frequency of large blackouts is governed by a power law. The power law makes the risk of large blackouts consequential and is consistent with the power system being a complex system designed and operated near a critical point. Power system overall loading or stress relative to operating limitsIan Dobson, Benjamin A. Carreras, Vickie E. Lynch, and David E. NewmanThu, 28 Jun 2007 07:00:00 GMTChimera states in a multilayer network of coupled and uncoupled neurons
https://aip.scitation.org/doi/10.1063/1.4993836?feed=most-cited
https://aip.scitation.org/doi/10.1063/1.4993836?feed=most-citedWe study the emergence of chimera states in a multilayer neuronal network, where one layer is composed of coupled and the other layer of uncoupled neurons. Through the multilayer structure, the layer with coupled neurons acts as the medium by means of which neurons in the uncoupled layer share information in spite of the absence of physical connections among them. Neurons in the coupled layer are connected with electrical synapses, while across the two layers, neurons are connected through chemical synapses. In both layers, the dynamics of each neuron is described by the Hindmarsh-Rose square wave bursting dynamics. We showSoumen Majhi, Matjaž Perc, and Dibakar GhoshMon, 17 Jul 2017 03:18:22 GMTAttractor reconstruction by machine learning
https://aip.scitation.org/doi/10.1063/1.5039508?feed=most-cited
https://aip.scitation.org/doi/10.1063/1.5039508?feed=most-citedA machine-learning approach called “reservoir computing” has been used successfully for short-term prediction and attractor reconstruction of chaotic dynamical systems from time series data. We present a theoretical framework that describes conditions under which reservoir computing can create an empirical model capable of skillful short-term forecasts and accurate long-term ergodic behavior. We illustrate this theory through numerical experiments. We also argue that the theory applies to certain other machine learning methods for time series prediction.Zhixin Lu, Brian R. Hunt, and Edward OttFri, 22 Jun 2018 03:16:30 GMTHybrid forecasting of chaotic processes: Using machine learning in conjunction with a knowledge-based model
https://aip.scitation.org/doi/10.1063/1.5028373?feed=most-cited
https://aip.scitation.org/doi/10.1063/1.5028373?feed=most-citedA model-based approach to forecasting chaotic dynamical systems utilizes knowledge of the mechanistic processes governing the dynamics to build an approximate mathematical model of the system. In contrast, machine learning techniques have demonstrated promising results for forecasting chaotic systems purely from past time series measurements of system state variables (training data), without prior knowledge of the system dynamics. The motivation for this paper is the potential of machine learning for filling in the gaps in our underlying mechanistic knowledge that cause widely-used knowledge-based models to be inaccurate. Thus, we here propose a general method that leverages the advantages of theseJaideep Pathak, Alexander Wikner, Rebeckah Fussell, Sarthak Chandra, Brian R. Hunt, Michelle Girvan, and Edward OttWed, 18 Apr 2018 04:17:34 GMTFractional kinetic equations: solutions and applications
https://aip.scitation.org/doi/10.1063/1.166272?feed=most-cited
https://aip.scitation.org/doi/10.1063/1.166272?feed=most-citedFractional generalization of the diffusion equation includes fractional derivatives with respect to time and coordinate. It had been introduced to describe anomalous kinetics of simple dynamical systems with chaotic motion. We consider a symmetrized fractional diffusion equation with a source and find different asymptotic solutions applying a method which is similar to the method of separation of variables. The method has a clear physical interpretation presenting the solution in a form of decomposition of the process of fractal Brownian motion and Lévy-type process. Fractional generalization of the Kolmogorov–Feller equation is introduced and its solutions are analyzed.Alexander I. Saichev and George M. ZaslavskyThu, 04 Jun 1998 07:00:00 GMTFractional modeling of blood ethanol concentration system with real data application
https://aip.scitation.org/doi/10.1063/1.5082907?feed=most-cited
https://aip.scitation.org/doi/10.1063/1.5082907?feed=most-citedIn this study, a physical system called the blood ethanol concentration model has been investigated in its fractional (non-integer) order version. The three most commonly used fractional operators with singular (Caputo) and non-singular (Atangana-Baleanu fractional derivative in the Caputo sense—ABC and the Caputo-Fabrizio—CF) kernels have been used to fractionalize the model, whereas during the process of fractionalization, the dimensional consistency for each of the equations in the model has been maintained. The Laplace transform technique is used to determine the exact solution of the model in all three cases, whereas its parameters are fitted through the least-squares error minimization technique.Sania Qureshi, Abdullahi Yusuf, Asif Ali Shaikh, Mustafa Inc, and Dumitru BaleanuThu, 31 Jan 2019 05:26:24 GMTLong time evolution of phase oscillator systems
https://aip.scitation.org/doi/10.1063/1.3136851?feed=most-cited
https://aip.scitation.org/doi/10.1063/1.3136851?feed=most-citedIt is shown, under weak conditions, that the dynamical evolution of large systems of globally coupled phase oscillators with Lorentzian distributed oscillation frequencies is, in an appropriate physical sense, time-asymptotically attracted toward a reduced manifold of the system states. This manifold was previously known and used to facilitate the discovery of attractors and bifurcations of such systems. The result of this paper establishes that attractors for the order parameter dynamics obtained by restriction to this reduced manifold are, in fact, the only such attractors of the full system. Thus all long time dynamical behaviors of the order parameters of theseEdward Ott and Thomas M. AntonsenTue, 19 May 2009 07:00:00 GMTReservoir observers: Model-free inference of unmeasured variables in chaotic systems
https://aip.scitation.org/doi/10.1063/1.4979665?feed=most-cited
https://aip.scitation.org/doi/10.1063/1.4979665?feed=most-citedDeducing the state of a dynamical system as a function of time from a limited number of concurrent system state measurements is an important problem of great practical utility. A scheme that accomplishes this is called an “observer.” We consider the case in which a model of the system is unavailable or insufficiently accurate, but “training” time series data of the desired state variables are available for a short period of time, and a limited number of other system variables are continually measured. We propose a solution to this problem using networks of neuron-like units known as “reservoir computers.” TheZhixin Lu, Jaideep Pathak, Brian Hunt, Michelle Girvan, Roger Brockett, and Edward OttWed, 05 Apr 2017 03:33:26 GMTNonlinear time-series analysis revisited
https://aip.scitation.org/doi/10.1063/1.4917289?feed=most-cited
https://aip.scitation.org/doi/10.1063/1.4917289?feed=most-citedIn 1980 and 1981, two pioneering papers laid the foundation for what became known as nonlinear time-series analysis: the analysis of observed data—typically univariate—via dynamical systems theory. Based on the concept of state-space reconstruction, this set of methods allows us to compute characteristic quantities such as Lyapunov exponents and fractal dimensions, to predict the future course of the time series, and even to reconstruct the equations of motion in some cases. In practice, however, there are a number of issues that restrict the power of this approach: whether the signal accurately and thoroughly samples the dynamics, for instance, and whetherElizabeth Bradley and Holger KantzMon, 13 Apr 2015 07:00:00 GMTRobust detection of dynamic community structure in networks
https://aip.scitation.org/doi/10.1063/1.4790830?feed=most-cited
https://aip.scitation.org/doi/10.1063/1.4790830?feed=most-citedWe describe techniques for the robust detection of community structure in some classes of time-dependent networks. Specifically, we consider the use of statistical null models for facilitating the principled identification of structural modules in semi-decomposable systems. Null models play an important role both in the optimization of quality functions such as modularity and in the subsequent assessment of the statistical validity of identified community structure. We examine the sensitivity of such methods to model parameters and show how comparisons to null models can help identify system scales. By considering a large number of optimizations, we quantify the variance of networkDanielle S. Bassett, Mason A. Porter, Nicholas F. Wymbs, Scott T. Grafton, Jean M. Carlson, and Peter J. MuchaMon, 18 Mar 2013 07:00:00 GMTOn the formulation of Adams-Bashforth scheme with Atangana-Baleanu-Caputo fractional derivative to model chaotic problems
https://aip.scitation.org/doi/10.1063/1.5085490?feed=most-cited
https://aip.scitation.org/doi/10.1063/1.5085490?feed=most-citedMathematical analysis with the numerical simulation of the newly formulated fractional version of the Adams-Bashforth method using the Atangana-Baleanu operator which has both nonlocal and nonsingular properties is considered in this paper. We adopt the fixed point theory and approximation method to prove the existence and uniqueness of the solution via general two-component time fractional differential equations. The method is tested with three nonlinear chaotic dynamical systems in which the integer-order derivative is modeled with the proposed fractional-order case. The simulation result for different values in is presented.Kolade M. Owolabi and Abdon AtanganaThu, 07 Feb 2019 06:42:53 GMTFrom diffusion to anomalous diffusion: A century after Einstein’s Brownian motion
https://aip.scitation.org/doi/10.1063/1.1860472?feed=most-cited
https://aip.scitation.org/doi/10.1063/1.1860472?feed=most-citedEinstein’s explanation of Brownian motion provided one of the cornerstones which underlie the modern approaches to stochastic processes. His approach is based on a random walk picture and is valid for Markovian processes lacking long-term memory. The coarse-grained behavior of such processes is described by the diffusion equation. However, many natural processes do not possess the Markovian property and exhibit anomalous diffusion. We consider here the case of subdiffusive processes, which correspond to continuous-time random walks in which the waiting time for a step is given by a probability distribution with a diverging mean value. Such a process can beI. M. Sokolov and J. KlafterFri, 17 Jun 2005 07:00:00 GMT