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Tsutsumi N, Nakai K, Saiki Y. Constructing low-dimensional ordinary differential equations from chaotic time series of high- or infinite-dimensional systems using radial-function-based regression. Phys Rev E 2023; 108:054220. [PMID: 38115529 DOI: 10.1103/physreve.108.054220] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/08/2023] [Accepted: 10/30/2023] [Indexed: 12/21/2023]
Abstract
In our previous study [N. Tsutsumi, K. Nakai, and Y. Saiki, Chaos 32, 091101 (2022)1054-150010.1063/5.0100166] we proposed a method of constructing a system of ordinary differential equations of chaotic behavior only from observable deterministic time series, which we will call the radial-function-based regression (RfR) method. The RfR method employs a regression using Gaussian radial basis functions together with polynomial terms to facilitate the robust modeling of chaotic behavior. In this paper, we apply the RfR method to several example time series of high- or infinite-dimensional deterministic systems, and we construct a system of relatively low-dimensional ordinary differential equations with a large number of terms. The examples include time series generated from a partial differential equation, a delay differential equation, a turbulence model, and intermittent dynamics. The case when the observation includes noise is also tested. We have effectively constructed a system of differential equations for each of these examples, which is assessed from the point of view of time series forecast, reconstruction of invariant sets, and invariant densities. We find that in some of the models, an appropriate trajectory is realized on the chaotic saddle and is identified by the stagger-and-step method.
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Affiliation(s)
- Natsuki Tsutsumi
- Faculty of Commerce and Management, Hitotsubashi University, Tokyo 186-8601, Japan
| | - Kengo Nakai
- The Graduate School of Environment, Life, Natural Science and Technology, Okayama University, Okayama 700-0082, Japan
| | - Yoshitaka Saiki
- Graduate School of Business Administration, Hitotsubashi University, Tokyo 186-8601, Japan
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Lainscsek C, Mendes EMAM, Salgado GHO, Sejnowski TJ. Transformations that preserve the uniqueness of the differential form for Lorenz-like systems. CHAOS (WOODBURY, N.Y.) 2023; 33:103122. [PMID: 37832517 PMCID: PMC10576629 DOI: 10.1063/5.0156237] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/27/2023] [Accepted: 09/18/2023] [Indexed: 10/15/2023]
Abstract
Differential equations serve as models for many physical systems. But, are these equations unique? We prove here that when a 3D system of ordinary differential equations for a dynamical system is transformed to the jerk or differential form, the jerk form is preserved in relation to a given variable and, therefore, the transformed system shares the time series of that given variable with the original untransformed system. Multiple algebraically different systems of ordinary differential equations can share the same jerk form. They may also share the same time series of the transformed variable depending on the parameters of the jerk form. Here, we studied 17 algebraically different Lorenz-like systems that share the same functional jerk form. There are groups of these systems that share the jerk parameters and, therefore, also have the same time series of the transformed variable.
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Affiliation(s)
| | - Eduardo M. A. M. Mendes
- Laboratório de Modelagem, Análise e Controle de Sistemas Não Lineares, Universidade Federal de Minas Gerais, Av. Antônio Carlos 6627, Belo Horizonte 31270-901, Minas Gerais, Brazil
| | - Gustavo H. O. Salgado
- Universidade Federal de Itajubá, Campus Itabira. Rua Irmã Ivone Drumond, 200 Distrito Industrial II, Itabira 35903-087, Minas Gerais, Brazil
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Shao C, Fang F, Liu Q, Wang T, Wang B, Yin P. Recovering chaotic properties from small data. IEEE TRANSACTIONS ON CYBERNETICS 2014; 44:2545-2556. [PMID: 24686313 DOI: 10.1109/tcyb.2014.2309989] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/03/2023]
Abstract
Physical properties are obviously essential to study a chaotic system that generates discrete-time signals, but recovering chaotic properties of a signal source from small data is a very troublesome work. Existing chaotic models are weak in dealing with such case in that most of them need big data to exploit those properties. In this paper, geometric theory is considered to solve this problem. We build a smooth trajectory from series to implicitly exhibit the chaotic properties with series-nonuniform rational B-spline (S-NURBS) modeling method, which is presented by our team to model slow-changing chaotic time series. As for the part of validation, we reveal how well our model recovers the properties from both the statistical and the chaotic aspects to confirm the effectiveness of the model. Finally a practical chaotic model is built up to recover the chaotic properties contained in the Musa standard dataset, which is used in analyzing software reliability, thereby further proves the high credibility of this model in practical time series. The effectiveness of the S-NURBS modeling leads us to believe that it is really a feasible and worthy research area to study chaotic systems from geometric perspective. For this reason, we reckon that we have opened up a new horizon for chaotic system research.
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Shao C, Liu Q, Wang T, Yin P, Wang B. Series-NonUniform Rational B-Spline (S-NURBS) model: a geometrical interpolation framework for chaotic data. CHAOS (WOODBURY, N.Y.) 2013; 23:033132. [PMID: 24089968 DOI: 10.1063/1.4819479] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/02/2023]
Abstract
Time series is widely exploited to study the innate character of the complex chaotic system. Existing chaotic models are weak in modeling accuracy because of adopting either error minimization strategy or an acceptable error to end the modeling process. Instead, interpolation can be very useful for solving differential equations with a small modeling error, but it is also very difficult to deal with arbitrary-dimensional series. In this paper, geometric theory is considered to reduce the modeling error, and a high-precision framework called Series-NonUniform Rational B-Spline (S-NURBS) model is developed to deal with arbitrary-dimensional series. The capability of the interpolation framework is proved in the validation part. Besides, we verify its reliability by interpolating Musa dataset. The main improvement of the proposed framework is that we are able to reduce the interpolation error by properly adjusting weights series step by step if more information is given. Meanwhile, these experiments also demonstrate that studying the physical system from a geometric perspective is feasible.
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Affiliation(s)
- Chenxi Shao
- Department of Computer Science and Technology, University of Science and Technology of China, Hefei 230027, People's Republic of China
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Lainscsek C, Sejnowski TJ. Electrocardiogram classification using delay differential equations. CHAOS (WOODBURY, N.Y.) 2013; 23:023132. [PMID: 23822497 PMCID: PMC3710263 DOI: 10.1063/1.4811544] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/03/2013] [Accepted: 06/05/2013] [Indexed: 05/22/2023]
Abstract
Time series analysis with nonlinear delay differential equations (DDEs) reveals nonlinear as well as spectral properties of the underlying dynamical system. Here, global DDE models were used to analyze 5 min data segments of electrocardiographic (ECG) recordings in order to capture distinguishing features for different heart conditions such as normal heart beat, congestive heart failure, and atrial fibrillation. The number of terms and delays in the model as well as the order of nonlinearity of the model have to be selected that are the most discriminative. The DDE model form that best separates the three classes of data was chosen by exhaustive search up to third order polynomials. Such an approach can provide deep insight into the nature of the data since linear terms of a DDE correspond to the main time-scales in the signal and the nonlinear terms in the DDE are related to nonlinear couplings between the harmonic signal parts. The DDEs were able to detect atrial fibrillation with an accuracy of 72%, congestive heart failure with an accuracy of 88%, and normal heart beat with an accuracy of 97% from 5 min of ECG, a much shorter time interval than required to achieve comparable performance with other methods.
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Affiliation(s)
- Claudia Lainscsek
- Computational Neurobiology Laboratory, Howard Hughes Medical Institute, Salk Institute for Biological Studies, La Jolla, California 92037, USA
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Gouesbet G. Second modified localized approximation for use in generalized Lorenz-Mie theory and other theories revisited. JOURNAL OF THE OPTICAL SOCIETY OF AMERICA. A, OPTICS, IMAGE SCIENCE, AND VISION 2013; 30:560-564. [PMID: 23595313 DOI: 10.1364/josaa.30.000560] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/02/2023]
Abstract
Arbitrary electromagnetic shaped beams may be described by using expansions over a set of basis functions, with expansion coefficients containing subcoefficients named "beam shape coefficients" (BSCs). When BSCs cannot be obtained in closed form, and/or when the beam description does not exactly satisfy Maxwell's equations, the most efficient method to evaluate the BSCs is to rely on localized approximations. One of them, named the second modified localized approximation, has been presented in a way that may be found ambiguous in some cases. The aim of the present paper is to remove any ambiguity on the use of the second modified localized approximation.
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Affiliation(s)
- Gérard Gouesbet
- Laboratoire d’Electromagnétisme des Systèmes Particulaires (LESP), Département Optique et Lasers (DOL), Unité Mixte de Recherche (UMR) 6614 du Centre National de la Recherche Scientifique (CNRS), Saint-Etienne-du Rouvray 76801, France.
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Lainscsek C. Nonuniqueness of global modeling and time scaling. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 84:046205. [PMID: 22181243 DOI: 10.1103/physreve.84.046205] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/26/2010] [Revised: 09/19/2011] [Indexed: 05/12/2023]
Abstract
Starting from an observed single time series, it is shown how to reconstruct a global model in the original phase space by using the ansatz library approach. This model is then compared to the underlying dynamical system that describes the initial time series, and the nonuniqueness of the reconstructed model is discussed. This framework is extended by taking an additional time scaling factor in the reconstructed model class under consideration.
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Affiliation(s)
- Claudia Lainscsek
- The Salk Institute for Biological Studies, 10010 North Torrey Pines Road, La Jolla, California 92037, USA
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Khanmohamadi O, Xu D. Spatiotemporal system identification on nonperiodic domains using Chebyshev spectral operators and system reduction algorithms. CHAOS (WOODBURY, N.Y.) 2009; 19:033117. [PMID: 19791997 DOI: 10.1063/1.3180843] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/28/2023]
Abstract
A system identification methodology based on Chebyshev spectral operators and an orthogonal system reduction algorithm is proposed, leading to a new approach for data-driven modeling of nonlinear spatiotemporal systems on nonperiodic domains. A continuous model structure is devised allowing for terms of arbitrary derivative order and nonlinearity degree. Chebyshev spectral operators are introduced to realm of inverse problems to discretize that continuous structure and arrive with spectral accuracy at a discrete form. Finally, least squares combined with an orthogonal system reduction algorithm are employed to solve for the parameters and eliminate the redundancies to achieve a parsimonious model. A numerical case study of identifying the Allen-Cahn metastable equation demonstrates the superior accuracy of the proposed Chebyshev spectral identification over its finite difference counterpart.
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Affiliation(s)
- Omid Khanmohamadi
- School of Mechanical and Aerospace Engineering, Nanyang Technological University, Singapore, Singapore
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Xu D, Khanmohamadi O. Spatiotemporal system reconstruction using Fourier spectral operators and structure selection techniques. CHAOS (WOODBURY, N.Y.) 2008; 18:043122. [PMID: 19123632 DOI: 10.1063/1.3030611] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/27/2023]
Abstract
A technique based on trigonometric spectral methods and structure selection is proposed for the reconstruction, from observed time series, of spatiotemporal systems governed by nonlinear partial differential equations of polynomial type with terms of arbitrary derivative order and nonlinearity degree. The system identification using Fourier spectral differentiation operators in conjunction with a structure selection procedure leads to a parsimonious model of the original system by detecting and eliminating the redundant parameters using orthogonal decomposition of the state data. Implementation of the technique is exemplified for a highly stiff reaction-diffusion system governed by the Kuramoto-Sivashinsky equation. Numerical experiments demonstrate the superior performance of the proposed technique in terms of accuracy as well as robustness, even with smaller sets of sampling data.
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Affiliation(s)
- Daolin Xu
- School of Mechanical and Aerospace Engineering, Nanyang Technological University, Singapore 639798
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Xu D, Lu F. Modeling global vector fields of chaotic systems from noisy time series with the aid of structure-selection techniques. CHAOS (WOODBURY, N.Y.) 2006; 16:043109. [PMID: 17199387 DOI: 10.1063/1.2359230] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/13/2023]
Abstract
We address the problem of reconstructing a set of nonlinear differential equations from chaotic time series. A method that combines the implicit Adams integration and the structure-selection technique of an error reduction ratio is proposed for system identification and corresponding parameter estimation of the model. The structure-selection technique identifies the significant terms from a pool of candidates of functional basis and determines the optimal model through orthogonal characteristics on data. The technique with the Adams integration algorithm makes the reconstruction available to data sampled with large time intervals. Numerical experiment on Lorenz and Rossler systems shows that the proposed strategy is effective in global vector field reconstruction from noisy time series.
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Affiliation(s)
- Daolin Xu
- School of Mechanical and Aerospace Engineering, Nanyang Technological University, 639798, Singapore
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Ventura AC, Mindlin GB, Dawson SP. Generic two-variable model of excitability. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 65:046231. [PMID: 12006000 DOI: 10.1103/physreve.65.046231] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/18/2001] [Indexed: 05/23/2023]
Abstract
We present a simple model that displays all classes of two-dimensional excitable regimes. One of the variables of the model displays the usual spikes observed in excitable systems. Since the model is written in terms of a "standard" vector field, it is always possible to fit it to experimental data displaying spikes in an algorithmic way. In fact, we use it to fit a series of membrane potential recordings obtained in the medicinal leech and time series generated with the FitzHugh-Nagumo equations and the excitability model of Eguía et al. [Phys. Rev. E 58, 2636 (1998)]. In each case, we determine the excitability class of the corresponding system.
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Affiliation(s)
- A C Ventura
- Departamento de Física, FCEN, UBA Ciudad Universitaria, Pabellón I (1428), Buenos Aires, Argentina
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Small M, Judd K, Mees A. Modeling continuous processes from data. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 65:046704. [PMID: 12006067 DOI: 10.1103/physreve.65.046704] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/31/2001] [Indexed: 05/23/2023]
Abstract
Experimental and simulated time series are necessarily discretized in time. However, many real and artificial systems are more naturally modeled as continuous-time systems. This paper reviews the major techniques employed to estimate a continuous vector field from a finite discrete time series. We compare the performance of various methods on experimental and artificial time series and explore the connection between continuous (differential) and discrete (difference equation) systems. As part of this process we propose improvements to existing techniques. Our results demonstrate that the continuous-time dynamics of many noisy data sets can be simulated more accurately by modeling the one-step prediction map than by modeling the vector field. We also show that radial basis models provide superior results to global polynomial models.
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Affiliation(s)
- Michael Small
- Department of Electronic and Information Engineering, Hong Kong Polytechnic University, Kowloon, Hong Kong, ROC.
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Laje R, Gardner T, Mindlin GB. Continuous model for vocal fold oscillations to study the effect of feedback. PHYSICAL REVIEW E 2001; 64:056201. [PMID: 11736048 DOI: 10.1103/physreve.64.056201] [Citation(s) in RCA: 34] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/03/2001] [Indexed: 11/07/2022]
Abstract
In this work we study the effects of delayed feedback on vocal fold dynamics. To perform this study, we work with a vocal fold model that is made as simple as possible while retaining the spectral content characteristic of human vocal production. Our results indicate that, even with the simplest explanation for vocal fold oscillation, delayed feedback due to reflected sound in the vocal tract can lead to extremely rich dynamics.
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Affiliation(s)
- R Laje
- Departamento de Física, FCEyN, UBA, Ciudad Universitaria, Pab. I (1428), Buenos Aires, Argentina
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Mininni PD, Gómez DO, Mindlin GB. Stochastic relaxation oscillator model for the solar cycle. PHYSICAL REVIEW LETTERS 2000; 85:5476-5479. [PMID: 11136025 DOI: 10.1103/physrevlett.85.5476] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/25/2000] [Revised: 08/08/2000] [Indexed: 05/23/2023]
Abstract
We perform a detailed analysis of the sunspot number time series to reconstruct the phase space of the underlying dynamical system. The features of this phase space allow us to describe the behavior of the solar cycle in terms of a simple relaxation oscillator in two dimensions. The absence of systematic self-crossings suggests that the complexity of the sunspot time series does not arise as a consequence of chaos. Instead, we show that it can be adequately modeled through the introduction of a stochastic fluctuation in one of the parameters of the dynamic equations.
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Affiliation(s)
- P D Mininni
- Departamento de Física, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria, 1428 Buenos Aires, Argentina
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Díaz-Sierra R, Lozano JB, Fairén V. Deduction of Chemical Mechanisms from the Linear Response around Steady State. J Phys Chem A 1999. [DOI: 10.1021/jp983041e] [Citation(s) in RCA: 13] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Affiliation(s)
- Rubén Díaz-Sierra
- Departamento de Física Fundamental, Universidad Nacional de Educación a Distancia, Apartado 60141, 28080 Madrid, Spain
| | - José Bernardo Lozano
- Departamento de Física Fundamental, Universidad Nacional de Educación a Distancia, Apartado 60141, 28080 Madrid, Spain
| | - Víctor Fairén
- Departamento de Física Fundamental, Universidad Nacional de Educación a Distancia, Apartado 60141, 28080 Madrid, Spain
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Letellier C, Maréchal E, Dutertre P, Maheu B, Gouesbet G, Fei Z, Hudson JL. Global vector field reconstruction from a chaotic experimental signal in copper electrodissolution. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1995; 51:4262-4266. [PMID: 9963137 DOI: 10.1103/physreve.51.4262] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Jinno K, Xu S, Berndtsson R, Kawamura A, Matsumoto M. Prediction of unspots using reconstructed chaotic system equations. ACTA ACUST UNITED AC 1995. [DOI: 10.1029/95ja01167] [Citation(s) in RCA: 21] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/09/2022]
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Gouesbet G, Letellier C. Global vector-field reconstruction by using a multivariate polynomial L2 approximation on nets. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1994; 49:4955-4972. [PMID: 9961817 DOI: 10.1103/physreve.49.4955] [Citation(s) in RCA: 21] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Brown R, Rulkov NF, Tracy ER. Modeling and synchronizing chaotic systems from time-series data. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1994; 49:3784-3800. [PMID: 9961665 DOI: 10.1103/physreve.49.3784] [Citation(s) in RCA: 19] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Gouesbet G. Reconstruction of vector fields: The case of the Lorenz system. PHYSICAL REVIEW. A, ATOMIC, MOLECULAR, AND OPTICAL PHYSICS 1992; 46:1784-1796. [PMID: 9908313 DOI: 10.1103/physreva.46.1784] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Gouesbet G. Reconstruction of standard and inverse vector fields equivalent to a Rössler system. PHYSICAL REVIEW. A, ATOMIC, MOLECULAR, AND OPTICAL PHYSICS 1991; 44:6264-6280. [PMID: 9905758 DOI: 10.1103/physreva.44.6264] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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