1
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Sysoev IV, Bezruchko BP. Noise robust approach to reconstruction of van der Pol-like oscillators and its application to Granger causality. CHAOS (WOODBURY, N.Y.) 2021; 31:083118. [PMID: 34470233 DOI: 10.1063/5.0056901] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/15/2021] [Accepted: 07/28/2021] [Indexed: 06/13/2023]
Abstract
Van der Pol oscillators and their generalizations are known to be a fundamental model in the theory of oscillations and their applications. Many objects of a different nature can be described using van der Pol-like equations under some circumstances; therefore, methods of reconstruction of such equations from experimental data can be of significant importance for tasks of model verification, indirect parameter estimation, coupling analysis, system classification, etc. The previously reported techniques were not applicable to time series with large measurement noise, which is usual in biological, climatological, and many other experiments. Here, we present a new approach based on the use of numerical integration instead of the differentiation and implicit approximation of a nonlinear dissipation function. We show that this new technique can work for noise levels up to 30% by standard deviation from the signal for different types of autonomous van der Pol-like systems and for ensembles of such systems, providing a new approach to the realization of the Granger-causality idea.
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Affiliation(s)
- Ilya V Sysoev
- Institute of Physics, Saratov State University, 83, Astrakhanskaya str., 410012 Saratov, Russia
| | - Boris P Bezruchko
- Institute of Physics, Saratov State University, 83, Astrakhanskaya str., 410012 Saratov, Russia
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2
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Ishbulatov YM, Karavaev AS, Kiselev AR, Simonyan MA, Prokhorov MD, Ponomarenko VI, Mironov SA, Gridnev VI, Bezruchko BP, Shvartz VA. Mathematical modeling of the cardiovascular autonomic control in healthy subjects during a passive head-up tilt test. Sci Rep 2020; 10:16525. [PMID: 33020530 PMCID: PMC7536219 DOI: 10.1038/s41598-020-71532-7] [Citation(s) in RCA: 15] [Impact Index Per Article: 3.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/08/2020] [Accepted: 08/18/2020] [Indexed: 01/10/2023] Open
Abstract
A mathematical model is proposed for the autonomic control of cardiovascular system, which takes into account two separated self-exciting sympathetic control loops of heart rate and peripheral vascular tone. The control loops are represented by self-exciting time-delay systems and their tone depends on activity of the aortic, carotid, and lower-body baroreceptors. The model is used to study the dynamics of the adaptive processes that manifest in a healthy cardiovascular system during the passive head-up tilt test. Computer simulation provides continuous observation of the dynamics of the indexes and variables that cannot be measured in the direct experiment, including the noradrenaline concentration in vessel wall and heart muscle, tone of the sympathetic and parasympathetic control, peripheral vascular resistance, and blood pressure. In the supine and upright positions, we estimated the spectral characteristics of the model variables, especially in the low-frequency band, and the original index of total percent of phase synchronization between the low-frequency oscillations in heart rate and blood pressure signals. The model demonstrates good quantitative agreement with the dynamics of the experimentally observed indexes of cardiovascular system that were averaged for 50 healthy subjects.
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Affiliation(s)
- Yurii M Ishbulatov
- Department of Innovative Cardiological Information Technology, Institute of Cardiological Research, Saratov State Medical University, Saratov, Russia.,Department of Surgical Treatment for Interactive Pathology, Bakulev Scientific Center for Cardiovascular Surgery, Moscow, Russia
| | - Anatoly S Karavaev
- Department of Innovative Cardiological Information Technology, Institute of Cardiological Research, Saratov State Medical University, Saratov, Russia.,Laboratory of Nonlinear Dynamics Modeling, Saratov Branch of the Institute of Radio Engineering and Electronics of Russian Academy of Sciences, Saratov, Russia.,Department of Dynamic Modeling and Biomedical Engineering, Saratov State University, Saratov, Russia
| | - Anton R Kiselev
- Department of Innovative Cardiological Information Technology, Institute of Cardiological Research, Saratov State Medical University, Saratov, Russia. .,Department of Surgical Treatment for Interactive Pathology, Bakulev Scientific Center for Cardiovascular Surgery, Moscow, Russia. .,Department of Dynamic Modeling and Biomedical Engineering, Saratov State University, Saratov, Russia.
| | - Margarita A Simonyan
- Department of Atherocslerosis and Chronic Ischemic Heart Disease, Institute of Cardiological Research, Saratov, Russia
| | - Mikhail D Prokhorov
- Laboratory of Nonlinear Dynamics Modeling, Saratov Branch of the Institute of Radio Engineering and Electronics of Russian Academy of Sciences, Saratov, Russia
| | - Vladimir I Ponomarenko
- Laboratory of Nonlinear Dynamics Modeling, Saratov Branch of the Institute of Radio Engineering and Electronics of Russian Academy of Sciences, Saratov, Russia.,Department of Dynamic Modeling and Biomedical Engineering, Saratov State University, Saratov, Russia
| | - Sergey A Mironov
- Department of Innovative Cardiological Information Technology, Institute of Cardiological Research, Saratov State Medical University, Saratov, Russia
| | - Vladimir I Gridnev
- Department of Innovative Cardiological Information Technology, Institute of Cardiological Research, Saratov State Medical University, Saratov, Russia.,Department of Dynamic Modeling and Biomedical Engineering, Saratov State University, Saratov, Russia
| | - Boris P Bezruchko
- Laboratory of Nonlinear Dynamics Modeling, Saratov Branch of the Institute of Radio Engineering and Electronics of Russian Academy of Sciences, Saratov, Russia.,Department of Dynamic Modeling and Biomedical Engineering, Saratov State University, Saratov, Russia
| | - Vladimir A Shvartz
- Department of Surgical Treatment for Interactive Pathology, Bakulev Scientific Center for Cardiovascular Surgery, Moscow, Russia
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3
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Sysoev IV, Ponomarenko VI, Bezruchko BP, Prokhorov MD. Reconstruction of parameters and unobserved variables of a semiconductor laser with optical feedback from intensity time series. Phys Rev E 2020; 101:042218. [PMID: 32422789 DOI: 10.1103/physreve.101.042218] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/14/2020] [Accepted: 04/09/2020] [Indexed: 11/07/2022]
Abstract
We propose a method for the reconstruction of time-delayed feedback systems having unobserved variables from scalar time series. The method is based on the modified initial condition approach, which allows one to significantly reduce the number of starting guesses for an unobserved variable with a time delay. The proposed method is applied to the reconstruction of the Lang-Kobayashi equations, which describe the dynamics of a single-mode semiconductor laser with external optical feedback. We consider the case where only the time series of laser intensity is observable and the other two variables of the model are hidden. The dependence of the quality of the system reconstruction on the accuracy of assignment of starting guesses for unobserved variables and unknown laser parameters is studied. The method could be used for testing the security of information transmission in laser-based chaotic communication systems.
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Affiliation(s)
- I V Sysoev
- Saratov Branch of Kotelnikov Institute of Radioengineering and Electronics of Russian Academy of Sciences, Zelyonaya Street, 38, Saratov 410019, Russia.,Saratov State University, Astrakhanskaya Street, 83, Saratov, 410012, Russia
| | - V I Ponomarenko
- Saratov Branch of Kotelnikov Institute of Radioengineering and Electronics of Russian Academy of Sciences, Zelyonaya Street, 38, Saratov 410019, Russia.,Saratov State University, Astrakhanskaya Street, 83, Saratov, 410012, Russia
| | - B P Bezruchko
- Saratov Branch of Kotelnikov Institute of Radioengineering and Electronics of Russian Academy of Sciences, Zelyonaya Street, 38, Saratov 410019, Russia.,Saratov State University, Astrakhanskaya Street, 83, Saratov, 410012, Russia
| | - M D Prokhorov
- Saratov Branch of Kotelnikov Institute of Radioengineering and Electronics of Russian Academy of Sciences, Zelyonaya Street, 38, Saratov 410019, Russia
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4
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Chen Y, Xin R, Cheng M, Gao X, Li S, Shao W, Deng L, Zhang M, Fu S, Liu D. Unveil the time delay signature of optical chaos systems with a convolutional neural network. OPTICS EXPRESS 2020; 28:15221-15231. [PMID: 32403553 DOI: 10.1364/oe.388182] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/16/2020] [Accepted: 04/27/2020] [Indexed: 06/11/2023]
Abstract
We propose a time delay signature extraction method for optical chaos systems based on a convolutional neural network. Through transforming the time delay signature of a one-dimensional time series into two-dimensional image features, the excellent ability of convolutional neural networks for image feature recognition is fully utilized. The effectiveness of the method is verified on chaos systems with opto-electronic feedback and all optical feedback. The recognition accuracy of the method is 100% under normal conditions. For the system with extremely strong nonlinearity, the accuracy can be 93.25%, and the amount of data required is less than traditional methods. Moreover, it is verified that the proposed method possesses a strong ability to withstand the effects of noise.
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5
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Zhang Z, Chen Y, Mi Y, Hu G. Reconstruction of dynamic networks with time-delayed interactions in the presence of fast-varying noises. Phys Rev E 2019; 99:042311. [PMID: 31108723 DOI: 10.1103/physreve.99.042311] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/18/2018] [Indexed: 06/09/2023]
Abstract
Most complex social, biological and technological systems can be described by dynamic networks. Reconstructing network structures from measurable data is a fundamental problem in almost all interdisciplinary fields. Network nodes interact with each other and those interactions often have diversely distributed time delays. Accurate reconstruction of any targeted interaction to a node requires measured data of all its neighboring nodes together with information on the time delays of interactions from these neighbors. When networks are large, these data are often not available and time-delay factors are deeply hidden. Here we show that fast-varying noise can be of great help in solving these challenging problems. By computing suitable correlations, we can infer the intensity and time delay of any targeted interaction with the data of two related nodes (driving and driven nodes) only while all other nodes in the network are hidden. This method is analytically derived and fully justified by extensive numerical simulations.
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Affiliation(s)
- Zhaoyang Zhang
- Department of Physics, School of Physical Science and Technology, Ningbo University, Ningbo, Zhejiang 315211, China
- Business School, Ningbo University, Ningbo, Zhejiang 315211, China
| | - Yang Chen
- Brainnetome Center and National Laboratory of Pattern Recognition, Institute of Automation, Chinese Academy of Sciences, Beijing 100190, China
| | - Yuanyuan Mi
- Center for Neurointelligence, Chongqing University, Chongqing 400044, China
| | - Gang Hu
- Department of Physics, Beijing Normal University, 100875 Beijing, China
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6
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Dale R, Bhat HS. Equations of mind: Data science for inferring nonlinear dynamics of socio-cognitive systems. COGN SYST RES 2018. [DOI: 10.1016/j.cogsys.2018.06.020] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/17/2022]
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7
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Letellier C, Mangiarotti S, Sendiña-Nadal I, Rössler OE. Topological characterization versus synchronization for assessing (or not) dynamical equivalence. CHAOS (WOODBURY, N.Y.) 2018; 28:045107. [PMID: 31906632 DOI: 10.1063/1.5011325] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/10/2023]
Abstract
Model validation from experimental data is an important and not trivial topic which is too often reduced to a simple visual inspection of the state portrait spanned by the variables of the system. Synchronization was suggested as a possible technique for model validation. By means of a topological analysis, we revisited this concept with the help of an abstract chemical reaction system and data from two electrodissolution experiments conducted by Jack Hudson's group. The fact that it was possible to synchronize topologically different global models led us to conclude that synchronization is not a recommendable technique for model validation. A short historical preamble evokes Jack Hudson's early career in interaction with Otto E. Rössler.
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Affiliation(s)
- Christophe Letellier
- Normandie University, CORIA, Avenue de l'Université, 76800 Saint-Etienne du Rouvray, France
| | - Sylvain Mangiarotti
- Centre d'Études Spatiales de la Biosphère, UPS-CNRS-CNES-IRD, Observatoire Midi-Pyrénées, 18 Avenue Édouard Belin, 31401 Toulouse, France
| | - Irene Sendiña-Nadal
- Complex Systems Group, Universidad Rey Juan Carlos, 28933 Móstoles, Madrid, Spain
| | - Otto E Rössler
- Faculty of Science, University of Tübingen, Auf der Morgenstelle 8, 72076 Tübingen, Germany
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8
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Uy CH, Rontani D, Breuer S, Sciamanna M. Non-local correlations via chaotic itinerancy in VCSEL with optical feedback. OPTICS EXPRESS 2017; 25:6914-6923. [PMID: 28381033 DOI: 10.1364/oe.25.006914] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/07/2023]
Abstract
Similar to edge-emitting lasers, vertical cavity surface emitting lasers (VCSELs) subjected to optical feedback are known for exhibiting erratic fluctuations of their optical power at slow and fast time scales; these are called low-frequency fluctuations (LFF). Here, we demonstrate both experimentally and numerically that the chaotic itinerancy in phase space associated with LFF has a deep connection with the creation of non-local correlations at multiple time scales between the two linear polarization modes. Our result provides a novel framework to interpret the unknown origin of spectral signatures in the optical power of chaotic lasers with optical feedback, which were observed in the past two decades.
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9
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Blonder B, Moulton DE, Blois J, Enquist BJ, Graae BJ, Macias-Fauria M, McGill B, Nogué S, Ordonez A, Sandel B, Svenning JC. Predictability in community dynamics. Ecol Lett 2017; 20:293-306. [DOI: 10.1111/ele.12736] [Citation(s) in RCA: 53] [Impact Index Per Article: 7.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/04/2016] [Revised: 11/14/2016] [Accepted: 12/22/2016] [Indexed: 02/02/2023]
Affiliation(s)
- Benjamin Blonder
- Environmental Change Institute; School of Geography and the Environment; University of Oxford; Oxford OX1 3QY UK
- Department of Biology; Norwegian University of Science and Technology; Trondheim N-7491 Norway
| | | | - Jessica Blois
- School of Natural Sciences; University of California - Merced; Merced CA 95343 USA
| | - Brian J. Enquist
- Department of Ecology and Evolutionary Biology; University of Arizona; Tucson Arizona 85721 USA
| | - Bente J. Graae
- Department of Biology; Norwegian University of Science and Technology; Trondheim N-7491 Norway
| | - Marc Macias-Fauria
- School of Geography and the Environment; University of Oxford; Oxford OX1 3QY UK
| | - Brian McGill
- School of Biology and Ecology; University of Maine; Orono ME 04469 USA
| | - Sandra Nogué
- Department of Geography and Environment; University of Southampton; Southampton SO17 1BJ UK
| | - Alejandro Ordonez
- Section for Biodiversity & Ecoinformatics; Department of Bioscience; Aarhus University; Aarhus C DK-8000 Denmark
| | - Brody Sandel
- Section for Biodiversity & Ecoinformatics; Department of Bioscience; Aarhus University; Aarhus C DK-8000 Denmark
| | - Jens-Christian Svenning
- Section for Biodiversity & Ecoinformatics; Department of Bioscience; Aarhus University; Aarhus C DK-8000 Denmark
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10
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Zhu S, Gan L. Incomplete phase-space method to reveal time delay from scalar time series. Phys Rev E 2016; 94:052210. [PMID: 27967148 DOI: 10.1103/physreve.94.052210] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/10/2016] [Indexed: 11/07/2022]
Abstract
A computationally quick and conceptually simple method to recover time delay of the chaotic system from scalar time series is developed in this paper. We show that the orbits in the incomplete two-dimensional reconstructed phase-space will show local clustering phenomenon after the component reordering procedure proposed in this work. We find that information captured by the incomplete two-dimensional reconstructed phase-space is related to the time delay τ_{0} present in the system, and will be transferred to the reordered component by the procedure of component reordering. We then propose the segmented mean variance (SMV) from the reordered component to identify the time delay τ_{0} of the system. The proposed SMV shows clear maximum when the embedding delay τ of the incomplete reconstruction matches the time delay τ_{0} of the chaotic system. Numerical data generated by a time-delay system based on the Mackey-Glass equation operating in the chaotic regime are used to illustrate the effectiveness of the proposed SMV. Experimental results show that the proposed SMV is robust to additive observational noise and is able to recover the time delay of the chaotic system even though the amount of data is relatively small and the feedback strength is weak. Moreover, the time complexity of the proposed method is quite low.
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Affiliation(s)
- Shengli Zhu
- Center for Cyber Security, School of Electronic Engineering, University of Electronic Science and Technology of China, Chengdu 611731, China
| | - Lu Gan
- Center for Cyber Security, School of Electronic Engineering, University of Electronic Science and Technology of China, Chengdu 611731, China
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11
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Sysoev IV, Ponomarenko VI, Kulminskiy DD, Prokhorov MD. Recovery of couplings and parameters of elements in networks of time-delay systems from time series. Phys Rev E 2016; 94:052207. [PMID: 27967060 DOI: 10.1103/physreve.94.052207] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/22/2016] [Indexed: 06/06/2023]
Abstract
We propose a method for the recovery of coupling architecture and the parameters of elements in networks consisting of coupled oscillators described by delay-differential equations. For each oscillator in the network, we introduce an objective function characterizing the distance between the points of the reconstructed nonlinear function. The proposed method is based on the minimization of this objective function and the separation of the recovered coupling coefficients into significant and insignificant coefficients. The efficiency of the method is shown for chaotic time series generated by model equations of diffusively coupled time-delay systems and for experimental chaotic time series gained from coupled electronic oscillators with time-delayed feedback.
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Affiliation(s)
- I V Sysoev
- Saratov Branch of Kotel'nikov Institute of Radio Engineering and Electronics of Russian Academy of Sciences, Zelyonaya Street, 38, Saratov 410019, Russia
- Saratov State University, Astrakhanskaya Street, 83, Saratov 410012, Russia
| | - V I Ponomarenko
- Saratov Branch of Kotel'nikov Institute of Radio Engineering and Electronics of Russian Academy of Sciences, Zelyonaya Street, 38, Saratov 410019, Russia
- Saratov State University, Astrakhanskaya Street, 83, Saratov 410012, Russia
| | - D D Kulminskiy
- Saratov Branch of Kotel'nikov Institute of Radio Engineering and Electronics of Russian Academy of Sciences, Zelyonaya Street, 38, Saratov 410019, Russia
- Saratov State University, Astrakhanskaya Street, 83, Saratov 410012, Russia
| | - M D Prokhorov
- Saratov Branch of Kotel'nikov Institute of Radio Engineering and Electronics of Russian Academy of Sciences, Zelyonaya Street, 38, Saratov 410019, Russia
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12
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Xue C, Jiang N, Lv Y, Wang C, Li G, Lin S, Qiu K. Security-enhanced chaos communication with time-delay signature suppression and phase encryption. OPTICS LETTERS 2016; 41:3690-3693. [PMID: 27519064 DOI: 10.1364/ol.41.003690] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/06/2023]
Abstract
A security-enhanced chaos communication scheme with time delay signature (TDS) suppression and phase-encrypted feedback light is proposed, in virtue of dual-loop feedback with independent high-speed phase modulation. We numerically investigate the property of TDS suppression in the intensity and phase space and quantitatively discuss security of the proposed system by calculating the bit error rate of eavesdroppers who try to crack the system by directly filtering the detected signal or by using a similar semiconductor laser to synchronize the link signal and extract the data. The results show that TDS embedded in the chaotic carrier can be well suppressed by properly setting the modulation frequency, which can keep the time delay a secret from the eavesdropper. Moreover, because the feedback light is encrypted, without the accurate time delay and key, the eavesdropper cannot reconstruct the symmetric operation conditions and decode the correct data.
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13
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Impact of Noise on a Dynamical System: Prediction and Uncertainties from a Swarm-Optimized Neural Network. COMPUTATIONAL INTELLIGENCE AND NEUROSCIENCE 2015; 2015:145874. [PMID: 26351449 PMCID: PMC4553171 DOI: 10.1155/2015/145874] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 04/27/2015] [Revised: 07/15/2015] [Accepted: 07/27/2015] [Indexed: 12/01/2022]
Abstract
An artificial neural network (ANN) based on particle swarm optimization (PSO) was developed for the time series prediction. The hybrid ANN+PSO algorithm was applied on Mackey-Glass chaotic time series in the short-term x(t + 6). The performance prediction was evaluated and compared with other studies available in the literature. Also, we presented properties of the dynamical system via the study of chaotic behaviour obtained from the predicted time series. Next, the hybrid ANN+PSO algorithm was complemented with a Gaussian stochastic procedure (called stochastic hybrid ANN+PSO) in order to obtain a new estimator of the predictions, which also allowed us to compute the uncertainties of predictions for noisy Mackey-Glass chaotic time series. Thus, we studied the impact of noise for several cases with a white noise level (σN) from 0.01 to 0.1.
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14
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Used J, Martín JC. Validating the physical model of a chaotic system by topological analysis. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 87:052921. [PMID: 23767612 DOI: 10.1103/physreve.87.052921] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/25/2013] [Indexed: 06/02/2023]
Abstract
Topological analysis is employed for the first time to our knowledge as a method of validation for a physical model describing a chaotic system. Topological analysis theory provides both a way to characterize the topological structure of chaotic attractors by means of a set of integer numbers and a method to obtain this set departing from a time series generated by the chaotic system. The validation method proposed here consists of comparing the topological structure of chaotic attractors obtained from time series generated on the one hand by an experimental system and on the other hand by the numerical model under test. This procedure has been applied to an erbium-doped fiber laser subject to pump power sine-wave modulation.
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Affiliation(s)
- Javier Used
- Nonlinear Dynamics, Chaos and Complex Systems Group, Departamento de Física Universidad Rey Juan Carlos, Tulipán s/n, 28933 Móstoles, Madrid, Spain.
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15
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Nguimdo RM, Colet P, Larger L, Pesquera L. Digital key for chaos communication performing time delay concealment. PHYSICAL REVIEW LETTERS 2011; 107:034103. [PMID: 21838363 DOI: 10.1103/physrevlett.107.034103] [Citation(s) in RCA: 28] [Impact Index Per Article: 2.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/25/2011] [Indexed: 05/31/2023]
Abstract
We introduce a scheme that integrates a digital key in a phase-chaos electro-optical delay system for optical chaos communications. A pseudorandom binary sequence (PRBS) is mixed within the chaotic dynamics in a way that a mutual concealment is performed; e.g., the time delay is hidden by the binary sequence, and the PRBS is also masked by the chaos. In addition to bridging the gap between algorithmic symmetric key cryptography and chaos-based analog encoding, the proposed approach is intended to benefit from the complex algebra mixing between a (pseudorandom) Boolean variable, and another continuous time (chaotic) variable. The scheme also provides a large flexibility allowing for easy reconfigurations to communicate securely at a high bit rate between different systems.
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Affiliation(s)
- Romain Modeste Nguimdo
- Instituto de Física Interdisciplinar y Sistemas Complejos, IFISC (CSIC-UIB), Campus Universitat de les Illes Balears, E-07122 Palma de Mallorca, Spain
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16
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Zunino L, Soriano MC, Fischer I, Rosso OA, Mirasso CR. Permutation-information-theory approach to unveil delay dynamics from time-series analysis. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 82:046212. [PMID: 21230370 DOI: 10.1103/physreve.82.046212] [Citation(s) in RCA: 75] [Impact Index Per Article: 5.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/21/2010] [Revised: 08/09/2010] [Indexed: 05/30/2023]
Abstract
In this paper an approach to identify delay phenomena from time series is developed. We show that it is possible to perform a reliable time delay identification by using quantifiers derived from information theory, more precisely, permutation entropy and permutation statistical complexity. These quantifiers show clear extrema when the embedding delay τ of the symbolic reconstruction matches the characteristic time delay τ(S) of the system. Numerical data originating from a time delay system based on the well-known Mackey-Glass equations operating in the chaotic regime were used as test beds. We show that our method is straightforward to apply and robust to additive observational and dynamical noise. Moreover, we find that the identification of the time delay is even more efficient in a noise environment. Our permutation approach is also able to recover the time delay in systems with low feedback rate or high nonlinearity.
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Affiliation(s)
- L Zunino
- Instituto de Física Interdisciplinar y Sistemas Complejos, CSIC-UIB, Campus Universitat de les Illes Balears, E-07122 Palma de Mallorca, Spain.
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17
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Tao C, Liu X, Du G. Detection of dynamical structure from short and noisy chaotic series. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 81:046209. [PMID: 20481809 DOI: 10.1103/physreve.81.046209] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/18/2009] [Revised: 02/02/2010] [Indexed: 05/29/2023]
Abstract
A method is proposed to detect the dynamical structure hiding behind complex chaotic series by comparing prediction performance of trial functions. This method is valid even when the original system is contaminated with noise or only a relatively short data series is recorded. Using this method, the dynamical structure of the Gray-Scott model is detected from its rich spatiotemporal patterns. Finally, this method is successfully applied to the experimental data of Chua's circuit, which promises its potential of the application to realistic systems.
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Affiliation(s)
- Chao Tao
- Key Laboratory of Modern Acoustics, Ministry of Education, Nanjing University, Nanjing 210093, People's Republic of China.
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18
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Prokhorov MD, Ponomarenko VI. Reconstruction of time-delay systems using small impulsive disturbances. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 80:066206. [PMID: 20365253 DOI: 10.1103/physreve.80.066206] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/11/2009] [Revised: 09/30/2009] [Indexed: 05/29/2023]
Abstract
We propose a method for the reconstruction of time-delayed feedback systems from time series. The method is based on the analysis of the system response to a weak external disturbance having the form of rectangular pulses. To apply the method one must have access to the state variable of the system in order to perturb it and the time series of the driving signal and the system response having at least about one hundred points on the time interval equal to the delay time. The method is intended to recover delays in low-order time-delay systems performing periodic oscillations, but can also be applied to systems in chaotic regimes in the presence of high level of noise. We verify the method by applying it to both numerical and experimental data.
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Affiliation(s)
- M D Prokhorov
- Saratov Branch of the Institute of Radio Engineering and Electronics, Russian Academy of Sciences, Zelyonaya Street, Saratov, Russia
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Yu D, Boccaletti S. Real-time estimation of interaction delays. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 80:036203. [PMID: 19905196 DOI: 10.1103/physreve.80.036203] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/07/2009] [Indexed: 05/28/2023]
Abstract
We introduce a real-time method for the identification of time-varying interaction delays in coupled systems with and without unknown structural parameters and analyze the convergence of the delay identification process. The identification method can be applied to monitor the change in interaction delays, as well as to decode the encoding of interaction delays. Several examples are presented to illustrate the reliability and robustness of the suggested strategy.
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Affiliation(s)
- Dongchuan Yu
- Key Laboratory for NeuroInformation of Ministry of Education, School of Automation Engineering, University of Electronic Science and Technology of China, Chengdu 610054, China
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20
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Ponomarenko VI, Prokhorov MD. Recovery of systems with a linear filter and nonlinear delay feedback in periodic regimes. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2008; 78:066207. [PMID: 19256925 DOI: 10.1103/physreve.78.066207] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/11/2008] [Indexed: 05/27/2023]
Abstract
We propose a set of methods for the estimation of the parameters of time-delay systems with a linear filter and nonlinear delay feedback performing periodic oscillations. The methods are based on an analysis of the system response to regular external perturbations and are valid only for systems whose dynamics can be perturbed. The efficiency of the methods is illustrated using both numerical and experimental data.
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Affiliation(s)
- V I Ponomarenko
- Saratov Branch of the Institute of Radio Engineering and Electronics of Russian Academy of Sciences, Zelyonaya Street, 38, Saratov, 410019, Russia
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21
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Yu D, Frasca M, Liu F. Control-based method to identify underlying delays of a nonlinear dynamical system. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2008; 78:046209. [PMID: 18999510 DOI: 10.1103/physreve.78.046209] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/06/2008] [Revised: 09/17/2008] [Indexed: 05/27/2023]
Abstract
We suggest several stationary state control-based delay identification methods which do not require any structural information about the controlled systems and are applicable to systems described by delayed ordinary differential equations. This proposed technique includes three steps: (i) driving a system to a steady state; (ii) perturbing the control signal for shifting the steady state; and (iii) identifying all delays by detecting the time that the system is abruptly drawn out of stationarity. Some aspects especially important for applications are discussed as well, including interaction delay identification, stationary state convergence speed, performance comparison, and the influence of noise on delay identification. Several examples are presented to illustrate the reliability and robustness of all delay identification methods suggested.
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Affiliation(s)
- Dongchuan Yu
- College of Automation Engineering, Qingdao University, Qingdao, Shandong 266071, China.
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22
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Siefert M. Practical criterion for delay estimation using random perturbations. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 76:026215. [PMID: 17930129 DOI: 10.1103/physreve.76.026215] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/15/2007] [Indexed: 05/25/2023]
Abstract
Many systems contain an internal time delay, which significantly influences their dynamical properties. Methods to estimate this delay from times series in the presence of dynamical noise are not systematically studied. Addressing this problem, we demonstrate that it is sufficient to analyze the system's response to short-correlated external disturbances or internal noise. Following this idea, it is shown for linear and nonlinear systems, as well as for periodic dynamics, that the delay can be estimated by analyzing the correlation function. This method covers the case of strong noise and multiple delays.
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Affiliation(s)
- M Siefert
- Institute of Physics, University of Potsdam, Potsdam, Germany.
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23
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Aguirre LA, Furtado EC, Tôrres LAB. Evaluation of dynamical models: dissipative synchronization and other techniques. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 74:066203. [PMID: 17280135 DOI: 10.1103/physreve.74.066203] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/15/2005] [Revised: 08/28/2006] [Indexed: 05/13/2023]
Abstract
Some recent developments for the validation of nonlinear models built from data are reviewed. Besides giving an overall view of the field, a procedure is proposed and investigated based on the concept of dissipative synchronization between the data and the model, which is very useful in validating models that should reproduce dominant dynamical features, like bifurcations, of the original system. In order to assess the discriminating power of the procedure, four well-known benchmarks have been used: namely, Duffing-Ueda, Duffing-Holmes, and van der Pol oscillators, plus the Hénon map. The procedure, developed for discrete-time systems, is focused on the dynamical properties of the model, rather than on statistical issues. For all the systems investigated, it is shown that the discriminating power of the procedure is similar to that of bifurcation diagrams--which in turn is much greater than, say, that of correlation dimension--but at a much lower computational cost.
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Affiliation(s)
- Luis Antonio Aguirre
- Laboratório de Modelagem, Análise e Controle de Sistemas Não-Lineares, Programa de Pós-Graduação em Engenharia Elétrica, Universidade Federal de Minas Gerais, Brazil.
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24
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Prokhorov MD, Ponomarenko VI. Estimation of coupling between time-delay systems from time series. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2005; 72:016210. [PMID: 16090069 DOI: 10.1103/physreve.72.016210] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/09/2004] [Revised: 01/03/2005] [Indexed: 05/03/2023]
Abstract
We propose a method for estimation of coupling between the systems governed by scalar time-delay differential equations of the Mackey-Glass type from the observed time series data. The method allows one to detect the presence of certain types of linear coupling between two time-delay systems, to define the type, strength, and direction of coupling, and to recover the model equations of coupled time-delay systems from chaotic time series corrupted by noise. We verify our method using both numerical and experimental data.
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Affiliation(s)
- M D Prokhorov
- Saratov Department of the Institute of RadioEngineering and Electronics of Russian Academy of Sciences, Zelyonaya Street, 38, Saratov, 410019, Russia
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25
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Kye WH, Choi M, Kim CM, Park YJ. Encryption with synchronized time-delayed systems. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2005; 71:045202. [PMID: 15903715 DOI: 10.1103/physreve.71.045202] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/12/2004] [Revised: 10/26/2004] [Indexed: 05/02/2023]
Abstract
We propose a new communication scheme that uses time-delayed chaotic systems with delay time modulation. In this method, the transmitter encodes a message as an additional modulation of the delay time and then the receiver decodes the message by tracking the delay time. We demonstrate our communication scheme in a system of coupled logistic maps. Also we discuss the error of the transferred message due to an external noise and present its correction method.
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Affiliation(s)
- Won-Ho Kye
- National Creative Research Initiative Center for Controlling Optical Chaos, Pai-Chai University, Daejeon 302-735, Korea
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26
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Cimponeriu L, Rosenblum M, Pikovsky A. Estimation of delay in coupling from time series. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2004; 70:046213. [PMID: 15600501 DOI: 10.1103/physreve.70.046213] [Citation(s) in RCA: 19] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/20/2004] [Indexed: 05/24/2023]
Abstract
We demonstrate that a time delay in weak coupling between two self-sustained oscillators can be estimated from the observed time series data. We present two methods which are based on the analysis of interrelations between the phases of the signals. We show analytically and numerically that irregularity of the phase dynamics (due to the intrinsic noise or chaos) is essential for determination of the delay. We compare and contrast both methods to the standard cross-correlation analysis.
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Affiliation(s)
- Laura Cimponeriu
- Department of Medical Physics, University of Patras, 26 500 Rion Patras, Greece
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27
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Ryu JW, Kye WH, Lee SY, Kim MW, Choi M, Rim S, Park YJ, Kim CM. Effects of time-delayed feedback on chaotic oscillators. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2004; 70:036220. [PMID: 15524625 DOI: 10.1103/physreve.70.036220] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/22/2004] [Revised: 06/11/2004] [Indexed: 05/24/2023]
Abstract
We study the effects of time-delayed feedback on chaotic systems where the delay time is both fixed (static case) and varying (dynamic case) in time. For the static case, typical phase coherent and incoherent chaotic oscillators are investigated. Detailed phase diagrams are investigated in the parameter space of feedback gain ( K ) and delay time ( tau ). Linear stability analysis, by assuming the time-delayed perturbation, varies as e(lambdat) where lambda is the eigenvalue, gives the boundaries of the stability islands and critical feedback gains ( K(c) ) for both Rössler oscillators and Lorenz oscillators. We also found that the stability island are found when the delay time is about tau= (n+ 1 / 2 ) T , where n is an integer and T is the average period of the chaotic oscillator. It is shown that these analytical predictions agree well with the numerical results. For the dynamic case, we investigate Rössler oscillator with periodically modulated delay time. Stability regimes are found for parameter space of feedback gain and modulation frequency in which it was impossible to be stabilized for a fixed delay time. We also trace the detailed routes to the stability near the island boundaries for both cases by investigating bifurcation diagrams.
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Affiliation(s)
- Jung-Wan Ryu
- National Creative Research Initiative Center for Controlling Optical Chaos, Pai-Chai University, Daejeon 302-735, Korea
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28
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Kye WH, Choi M, Rim S, Kurdoglyan MS, Kim CM, Park YJ. Characteristics of a delayed system with time-dependent delay time. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2004; 69:055202. [PMID: 15244871 DOI: 10.1103/physreve.69.055202] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/04/2003] [Indexed: 05/24/2023]
Abstract
The characteristics of a time-delayed system with time-dependent delay time is investigated. We demonstrate that the nonlinearity characteristics of the time-delayed system are significantly changed depending on the properties of time-dependent delay time and especially that the reconstructed phase trajectory of the system is not collapsed into simple manifold, differently from the delayed system with fixed delay time. We discuss the possibility of a phase space reconstruction and its applications.
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Affiliation(s)
- Won-Ho Kye
- National Creative Research Initiative Center for Controlling Optical Chaos, Pai-Chai University, Daejeon 302-735, Korea.
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29
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Ponomarenko VI, Prokhorov MD. Extracting information masked by the chaotic signal of a time-delay system. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 66:026215. [PMID: 12241275 DOI: 10.1103/physreve.66.026215] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/05/2002] [Indexed: 05/23/2023]
Abstract
We further develop the method proposed by Bezruchko et al. [Phys. Rev. E 64, 056216 (2001)] for the estimation of the parameters of time-delay systems from time series. Using this method we demonstrate a possibility of message extraction for a communication system with nonlinear mixing of information signal and chaotic signal of the time-delay system. The message extraction procedure is illustrated using both numerical and experimental data and different kinds of information signals.
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Affiliation(s)
- V I Ponomarenko
- Saratov Department of the Institute of RadioEngineering and Electronics of Russian Academy of Sciences, Zelyonaya Street 38, Saratov 410019, Russia
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