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Journal articles on the topic 'System for numerical'

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1

Claus, R. W., A. L. Evans, J. K. Lylte, and L. D. Nichols. "Numerical Propulsion System Simulation." Computing Systems in Engineering 2, no. 4 (1991): 357–64. http://dx.doi.org/10.1016/0956-0521(91)90003-n.

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2

Turoń, Barbara, and Bartosz Miller. "The possibility of DIC system application in numerical models updating." Budownictwo i Architektura 18, no. 4 (2020): 083–102. http://dx.doi.org/10.35784/bud-arch.1443.

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The paper presents the results of updating of numerical models of the rectangular steel plate members in a plane state of stress, the updated parameter was a support length. Three different members loaded in a static or dynamic way were analyzed. The article shows examples of purely numeric updating. The data used to the update of numerical models was obtained from numerical simulations and it corresponds to the data, which can be measured by using the Digital Image Correlation (DIC) system. The main aim of the paper is to check the possibilities of the DIC system application in updating of nu
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3

Kobzev, Kirill, Sergey Vyalov, and Alexander Rybak. "Pumping hydraulic systems and the use of an unloading valve in a hydraulic system." E3S Web of Conferences 175 (2020): 05036. http://dx.doi.org/10.1051/e3sconf/202017505036.

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In order to identify the main functional capabilities of the control system of the pump-accumulator hydraulic power source equipped with a pump unloading machine of the proposed design, a numerical experiment was carried out. The experiment was a numerical solution of the above mathematical model of a power source. During the experiment, the influence of various structural parameters of the hydraulic pump unloading machine and the functional parameters of the power supply system as a whole on its dynamic properties was revealed. The article discusses the control system of a hydraulic power sou
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4

Atendra, Singh Yadav* Ashish Kumar. "NUMERICAL SOLUTION OF SYSTEM OF LINEAR EQUATIONS BY ITERATIVE METHODS." INTERNATIONAL JOURNAL OF ENGINEERING SCIENCES & RESEARCH TECHNOLOGY 6, no. 4 (2017): 203–8. https://doi.org/10.5281/zenodo.546307.

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Numerical method is the important aspects in solving real world problems that are related to mathematics, science, medicine, business are very few examples. Numerical method is the area related to mathematics and computer science which create, analysis and implements algorithm to numerically solve the system of linear equations. Numerical methods commonly involve an iterative method (as to find roots). They are now mostly used as preconditions for the popular iterative solvers. While it is difficult task solve as it takes a lot of time but it is an interesting part of Mathematics. In this pape
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5

Manaa, Saad, Rostam Saeed, and Fadhil Easif. "Numerical Stability of Brusselator System." AL-Rafidain Journal of Computer Sciences and Mathematics 8, no. 2 (2011): 43–52. http://dx.doi.org/10.33899/csmj.2011.163640.

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6

Kashima, Kenji, Shinjiro Ashida, and Yutaka Yamamoto. "SYSTEM THEORY FOR NUMERICAL ANALYSIS." IFAC Proceedings Volumes 38, no. 1 (2005): 480–85. http://dx.doi.org/10.3182/20050703-6-cz-1902.00480.

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7

Erkorkmaz, K., Y. Altintas, and C. H. Yeung. "Virtual Computer Numerical Control System." CIRP Annals 55, no. 1 (2006): 399–402. http://dx.doi.org/10.1016/s0007-8506(07)60444-2.

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8

Chen, Wenliang, Rajesh N. Dave, Robert Pfeffer, and Otis Walton. "Numerical simulation of Mechanofusion system." Powder Technology 146, no. 1-2 (2004): 121–36. http://dx.doi.org/10.1016/j.powtec.2004.07.014.

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9

Kashima, Kenji, and Yutaka Yamamoto. "System theory for numerical analysis." Automatica 43, no. 7 (2007): 1156–64. http://dx.doi.org/10.1016/j.automatica.2006.12.028.

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10

Pankaj, Ram Dayal, and Chiman Lal. "NUMERICAL ELUCIDATION OF KLEIN-GORDON-ZAKHAROV SYSTEM." Jnanabha 51, no. 01 (2021): 207–12. http://dx.doi.org/10.58250/jnanabha.2021.51125.

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We solve the numerically of the coupled 1D Klein-Gordon-Zakharov system (KGZ) equations in short) by PetrovGalerkin method, using linear and cubic B-Spline, as trial functions. The midpoint rule will be functional to advance the solution in time. This scheme is stable to Von Neumann stability analysis. Numerical solution is used to think about the accurateness and show the dynamism of the scheme.
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11

Xiao, Jian, and Chundong Zhu. "The Development of Numerical Control System of Vertical Rotary Forging Machine." International Journal of Materials, Mechanics and Manufacturing 6, no. 1 (2018): 78–81. http://dx.doi.org/10.18178/ijmmm.2018.6.1.351.

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12

Omayu, Yukiyoshi, Nobuatsu Tanaka, and Michitsugu Mori. "ICONE15-10743 NUMERICAL ANALYSIS OF THERMAL-HYDRAULIC BEHAVIORS IN SI SYSTEM." Proceedings of the International Conference on Nuclear Engineering (ICONE) 2007.15 (2007): _ICONE1510. http://dx.doi.org/10.1299/jsmeicone.2007.15._icone1510_392.

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13

Li, Ruo, Tiao Lu, Yanli Wang, and Wenqi Yao. "Numerical Validation for High Order Hyperbolic Moment System of Wigner Equation." Communications in Computational Physics 15, no. 3 (2014): 569–95. http://dx.doi.org/10.4208/cicp.091012.120813a.

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AbstractA globally hyperbolic moment system upto arbitrary order for the Wigner equation was derived in [6]. For numerically solving the high order hyperbolic moment system therein, we in this paper develop a preliminary numerical method for this system following the NRxx method recently proposed in [8], to validate the moment system of the Wigner equation. The method developed can keep both mass and momentum conserved, and the variation of the total energy under control though it is not strictly conservative. We systematically study the numerical convergence of the solution to the moment syst
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14

Zayko, Yuriy N. "Alternative numerical systems." Journal of Nature, Science & Technology 1, no. 3 (2021): 6–10. http://dx.doi.org/10.36937/janset.2021.003.002.

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The article is devoted to the construction of numerical systems, alternative to the system of real numbers and applicable in curvilinear space-time. Examples of such systems are given. Within the framework of a stationary numerical system, it is admissible to sum the diverging series like the Dirichlet series for the Riemann zeta function without resorting to its analytic continuation in the plane of the complex argument. In the framework of a non-stationary numerical system, a description of the Hubble effect is obtained, taking into account the corrections that correspond to the apparently a
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15

Ma, Shichang, Yufeng Xu, and Wei Yue. "Numerical Solutions of a Variable-Order Fractional Financial System." Journal of Applied Mathematics 2012 (2012): 1–14. http://dx.doi.org/10.1155/2012/417942.

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The numerical solution of a variable-order fractional financial system is calculated by using the Adams-Bashforth-Moulton method. The derivative is defined in the Caputo variable-order fractional sense. Numerical examples show that the Adams-Bashforth-Moulton method can be applied to solve such variable-order fractional differential equations simply and effectively. The convergent order of the method is also estimated numerically. Moreover, the stable equilibrium point, quasiperiodic trajectory, and chaotic attractor are found in the variable-order fractional financial system with proper order
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16

Domínguez, Efraín, Felipe Ardila, and Santiago Bustamante. "System Solver: an open source tool for mathematically modelling dynamical systems." Ingeniería e Investigación 30, no. 3 (2010): 157–64. http://dx.doi.org/10.15446/ing.investig.v30n3.18188.

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The following paper presents a freeware modelling tool simulating dynamic systems that can be represented by either an ordinary differential equation (ODE) or a set of differential equations of different orders. The main idea leading to this software development is related to the fact that many physical, biological, ecological, economical, chemical, social and engineering problems can be expressed in this way. Furthermore, the solution to these problems requires some expertise in numerical methods and programming. Such knowledge is uncommon in some of the experts in such scientific domains. A
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17

Su, Yanhui. "A Parallel Spectral Element Method for Fractional Lorenz System." Discrete Dynamics in Nature and Society 2015 (2015): 1–7. http://dx.doi.org/10.1155/2015/682140.

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We provide a parallel spectral element method for the fractional Lorenz system numerically. The detailed construction and implementation of the method are presented. Thanks to the spectral accuracy of the presented method, the storage requirement due to the “global time dependence” can be considerably relaxed. Also, the parareal method combining spectral element method reduces the computing time greatly. Finally, we have tested the chaotic behaviors of fractional Lorenz system. Our numerical results are in excellent agreement with the results from other numerical methods.
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18

YANG, QIGUI, GUANRONG CHEN, and KUIFEI HUANG. "CHAOTIC ATTRACTORS OF THE CONJUGATE LORENZ-TYPE SYSTEM." International Journal of Bifurcation and Chaos 17, no. 11 (2007): 3929–49. http://dx.doi.org/10.1142/s0218127407019792.

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A new conjugate Lorenz-type system is introduced in this paper. The system contains as special cases the conjugate Lorenz system, conjugate Chen system and conjugate Lü system. Chaotic dynamics of the system in the parametric space is numerically and thoroughly investigated. Meanwhile, a set of conditions for possible existence of chaos are derived, which provide some useful guidelines for searching chaos in numerical simulations. Furthermore, some basic dynamical properties such as Lyapunov exponents, bifurcations, routes to chaos, periodic windows, possible chaotic and periodic-window parame
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19

Yu, Bo, and Bo Dong. "A Hybrid Polynomial System Solving Method for Mixed Trigonometric Polynomial Systems." SIAM Journal on Numerical Analysis 46, no. 3 (2008): 1503–18. http://dx.doi.org/10.1137/070681740.

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20

Khin, Thanda Htun, and Kaung Cho Kyaw. "Experimental in Structural Dynamics Base Isolation System Modelling." International Journal of Trend in Scientific Research and Development 3, no. 3 (2019): 326–35. https://doi.org/10.31142/ijtsrd21704.

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This project is to understand and have the ability to perform dynamic test. Furthermore in this project, the author can investigate how the dynamics of a multistory building is modified by base isolation. The dynamic properties are also very important for the dynamics analysis of the structure. The main task in this project is to determine the dynamic behavior of a 5 storey steel structure model with base isolation system and without base isolation system by experimentally and numerically. In this section the author examine the vibration properties, natural modes and the earthquake response of
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21

BENLEKKAM, Mohamed Lamine, Driss NEHARI, and Nassira CHERIET. "Numerical investigation of latent heat thermal energy storage system." Recueil de mécanique 3, no. 1 (2018): 229–35. https://doi.org/10.5281/zenodo.1490505.

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Abstract The present work aims to study numerically the performance of a latent heat thermal energy storage unit. Which is composed of shell and tube. The annular space is filled by a phase change material (PCM), however the water used as a heat transfer fluid (HTF) flows in the inner tube. The computational are based on an iterative numerical procedure that incorporates an enthalpy formulation for the modeling of the solid-liquid phase change. Then our numerical model was validated with experimental and numerical results of the literature, where a good agreement was obtained. A series of nume
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22

Guo, Yu, An Kang Hu, and Hai Tao Dong. "Numerical Simulation of Cathodic Protection System." Advanced Materials Research 548 (July 2012): 682–85. http://dx.doi.org/10.4028/www.scientific.net/amr.548.682.

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Cathodic protection (CP) systems are commonly designed by estimating the overall current demand and then developing an anode configuration sufficient to protect the structure. To a large extent the performance of a CP system is dependent on the skill and experience of the corrosion specialist. As the structures become more complex these traditional approaches may become less reliable. Given the factor it becomes imperative that corrosion engineers are able to predict the electric fields as part of the design process. The difficulty in making reliable estimates for cases can be overcome by usin
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23

Isailovic, V. M., M. M. Nikolic, T. Bibas, et al. "Numerical simulation of human hearing system." EAI Endorsed Transactions on Pervasive Health and Technology 4, no. 13 (2018): 154144. http://dx.doi.org/10.4108/eai.28-2-2018.154144.

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24

Bona, C., and J. Massó. "Hyperbolic evolution system for numerical relativity." Physical Review Letters 68, no. 8 (1992): 1097–99. http://dx.doi.org/10.1103/physrevlett.68.1097.

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25

Zhi-jian, Ren. "Numerical analysis of a chaotic system." Chinese Physics 10, no. 9 (2001): 790–95. http://dx.doi.org/10.1088/1009-1963/10/9/304.

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26

CARR, J., D. B. DUNCAN, and C. H. WALSHAW. "Numerical approximation of a metastable system." IMA Journal of Numerical Analysis 15, no. 4 (1995): 505–21. http://dx.doi.org/10.1093/imanum/15.4.505.

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27

Asady, B., and P. Mansouri. "Numerical solution of fuzzy linear system." International Journal of Computer Mathematics 86, no. 1 (2009): 151–62. http://dx.doi.org/10.1080/00207160701621206.

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28

Wei, Xu, and Chen JiHong. "Research on ARM Numerical Control System." Physics Procedia 25 (2012): 1934–38. http://dx.doi.org/10.1016/j.phpro.2012.03.332.

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29

Wang, Mingyu, and Fengying Su. "Numerical Research on Stochastic Duffing System." Procedia Engineering 29 (2012): 1979–83. http://dx.doi.org/10.1016/j.proeng.2012.01.247.

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30

Kawala, A. M. "Numerical Solutions for Ito Coupled System." Acta Applicandae Mathematicae 106, no. 3 (2008): 325–35. http://dx.doi.org/10.1007/s10440-008-9300-9.

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31

Vajda, Sandor, Keith R. Godfrey, and Peter Valko. "Numerical deconvolution using system identification methods." Journal of Pharmacokinetics and Biopharmaceutics 16, no. 1 (1988): 85–107. http://dx.doi.org/10.1007/bf01061863.

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32

Vasil’eva, Ol’ga Aleksandrovna. "Numerical investigation of the Carleman system." Vestnik MGSU, no. 6 (June 2015): 7–15. http://dx.doi.org/10.22227/1997-0935.2015.6.7-15.

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In the article the Cauchy problem of the Carleman equation is considered. The Carleman system of equations is a model problem of the kinetic theory of gases. It is a discrete kinetic model of one-dimensional gas consisting of identical monatomic molecules. The molecules can have one of two speeds, which have equal values and opposite directions. This system of the equations is quasi-linear hyperbolic system of partial differential equations. There is no analytic solution for this problem in general case. So, the numerical investigation of the Cauchy problem of the Carleman system solution is v
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33

Romeu, Jorge Luis. "Numerical comparison of system reliability bounds." Computers & Industrial Engineering 11, no. 1-4 (1986): 566–70. http://dx.doi.org/10.1016/0360-8352(86)90155-5.

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34

Pickering, Jayne, James S. Adelman, and Matthew Inglis. "Are approximate number system representations numerical?" Journal of Numerical Cognition 9, no. 1 (2023): 129–44. http://dx.doi.org/10.5964/jnc.8553.

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Previous research suggests that the Approximate Number System (ANS) allows people to approximate the cardinality of a set. This ability to discern numerical quantities may explain how meaning becomes associated with number symbols. However, recently it has been argued that ANS representations are not directly numerical, but rather are formed by amalgamating perceptual features confounded with the set’s cardinality. In this paper, we approach the question of whether ANS representations are numerical by studying the properties they have, rather than how they are formed. Across two pre-registered
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35

Soda, Taisuke, Shigeaki Shiotani, and Kenji Sasa. "Basic Study on Numerical Navigation System with Numerical Weather and Ocean." Journal of the Japan Society of Naval Architects and Ocean Engineers 16 (2012): 155–64. http://dx.doi.org/10.2534/jjasnaoe.16.155.

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36

Zhang, Cheng Li, and Yun Zeng. "A Simple Numerical Simulation Method for Lorenz System Families." Applied Mechanics and Materials 444-445 (October 2013): 786–90. http://dx.doi.org/10.4028/www.scientific.net/amm.444-445.786.

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Lorenz system families contain Lorenz system, Chen system and Lu system, their accurate analytical solutions are not yet obtained now. The segmenting recursion method was put forward in this paper, the equations of Lorenz system families were reasonably linearized within small segment, the recursion formulas were obtained by solving the approximate analytical solutions within small segment, and all numerical solutions were got by the recursion formulas. The chaotic motion of Lorenz system families were numerically simulated by means of the segmenting recursion method, the simulation results we
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37

Gao, Xin. "Chaotic Dynamics of Fractional-Order Liu System." Applied Mechanics and Materials 55-57 (May 2011): 1327–31. http://dx.doi.org/10.4028/www.scientific.net/amm.55-57.1327.

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In this paper, we numerically investigate the chaotic behaviors of a new fractional-order system. We find that chaotic behaviors exist in the fractional-order system with order less than 3. The lowest order we find to have chaos is 2.4 in such system. In addition, we numerically simulate the continuances of the chaotic behaviors in the fractional-order system with orders from 2.7 to 3. Our investigations are validated through numerical simulations.
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38

Choi, Cheolyong, Yonmo Sung, Gyung Min Choi, and Duck Jool Kim. "Numerical Analysis of Urea Decomposition with Static Mixers in Marine SCR System." Journal of Clean Energy Technologies 3, no. 1 (2015): 39–42. http://dx.doi.org/10.7763/jocet.2015.v3.165.

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39

Aminuddin, Jamrud, Mukhtar Effendi, Nurhayati Nurhayati, et al. "Numerical Analysis of Energy Converter for Wave Energy Power Generation-Pendulum System." International Journal of Renewable Energy Development 9, no. 2 (2020): 255–61. http://dx.doi.org/10.14710/ijred.9.2.255-261.

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The wave energy power generation-pendulum system (WEPG-PS) is a four-wheeled instrument designed to convert wave power into electric energy. The first wheel is connected to the pendulum by a double freewheel, the second and third are ordinary wheels, while the fourth is a converter component that is axially connected to the electric generator. This design used the Euler-Lagrange formalism and Runge-Kutta method to examine an ideal dimension and determine the numerical solution of the equation of motion related to the rotation speed of the wheels. The result showed that the WEPG-PS' converter s
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40

Xie, Congcong, and Xianliang Hu. "Finite Element Simulations with Adaptively Moving Mesh for the Reaction Diffusion System." Numerical Mathematics: Theory, Methods and Applications 9, no. 4 (2016): 686–704. http://dx.doi.org/10.4208/nmtma.2016.m1229.

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AbstractA moving mesh method is proposed for solving reaction-diffusion equations. The finite element method is used to solving the partial different equation system, and an efficient numerical scheme is applied to implement mesh moving. In the practical calculations, the moving mesh step and the problem equation solver are performed alternatively. Several numerical examples are presented, including the Gray-Scott, the Activator-Inhibitor and a case with a growing domain. It is illustrated numerically that the moving mesh methods costs much lower, compared with the numerical schemes on a fixed
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41

Li, Yin, and Chun Long Zheng. "Application of Synchronization Control Method to Zagzag System." Applied Mechanics and Materials 241-244 (December 2012): 1067–70. http://dx.doi.org/10.4028/www.scientific.net/amm.241-244.1067.

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In this paper, synchronization control method of Zagzag system is discussed both theoretically and numerically. Based on the Lyapunov stability theorem and the chaotic methods, synchronization control is given and illustrated with Zagzag system as example. Numerical simulations are presented to demonstrate the effectiveness of the proposed synchronization scheme.
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42

Zahaykah, Yousef, and Mahmood Jwailes. "Solving first-order systems of linear hyperbolic partial differential equations using fast Fourier transform." General Letters in Mathematics 14, no. 4 (2024): 94–109. https://doi.org/10.31559/glm2024.14.4.2.

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In this paper, we address the exact solution of the general first-order systems of linear hyperbolic partial differential equations using the Fourier transformation technique. This transformation converts the system from the physical domain into a system of first-order ordinary differential equations in the frequency domain. Utilizing this method, we then solve the wave equation system in n dimensions. Following the derivation of the exact solution, we introduce a numerical algorithm based on the fast Fourier transform (FFT) to solve the same system numerically. This approach leverages the exa
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43

Yang, Qigui, Lingbing Yang, and Bin Ou. "Hidden Hyperchaotic Attractors in a New 5D System Based on Chaotic System with Two Stable Node-Foci." International Journal of Bifurcation and Chaos 29, no. 07 (2019): 1950092. http://dx.doi.org/10.1142/s0218127419500925.

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This paper reports some hidden hyperchaotic attractors and complex dynamics in a new five-dimensional (5D) system with only two nonlinear terms. The system is generated by adding two linear controllers to an unusual 3D autonomous quadratic chaotic system with two stable node-foci. In particular, the hyperchaotic system without equilibrium or with only one stable equilibrium can generate two kinds of hidden hyperchaotic attractors with three positive Lyapunov exponents. Numerical methods not only verify the existence of such attractors and hyperchaotic attractors, but also show the dynamical ev
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44

Overmann, Karenleigh A., Thiago Chacon, and Annick Payne. "Desana numerical symbols." Written Language and Literacy 25, no. 2 (2022): 133–58. http://dx.doi.org/10.1075/wll.00064.ove.

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Abstract In 2006, a narrative of the Desana people included a system of graphic symbols reported as a historical Indigenous invention used during intertribal warfare to count the number of enemies and pass warning information. This paper outlines and evaluates the Desana graphic system. The Desana people are described, and their timeline of mythical events is compared to historical accounts of the region. Contemporary Desana spoken numbers are then characterized as a quinary system with a restricted extent that differs significantly from the graphic writing system as presented in the cultural
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45

Verre, Salvatore. "Numerical Strategy for Column Strengthened with FRCM/SRG System." Buildings 12, no. 12 (2022): 2187. http://dx.doi.org/10.3390/buildings12122187.

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The use of fabric-reinforced cementitious mortar (FRCM) or steel-reinforced grout (SRG) is now recognized to be effective in enhancing the axial capacity of masonry columns when confinement is achieved. Numerous experimental tests demonstrated the symbiotic role of the fabric and the inorganic matrix. An open issue is still related to the numerical simulation. In fact, if the compressive behavior by the numerical simulation of the unreinforced and reinforced masonry columns confined by a FRCM/SRG jacket may follow different approaches. The inorganic matrix transfers the stresses from the subst
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46

Liu, Ya Chong, An Kang Hu, and Feng Lei Han. "Numerical Identification of Ship-Roll Chaos Threshold." Applied Mechanics and Materials 556-562 (May 2014): 3078–83. http://dx.doi.org/10.4028/www.scientific.net/amm.556-562.3078.

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Melnikov function is currently the only way to theoretically resolve the chaotic threshold. Considering the calculation difficulties of Melnikov function, Gauss-Legendre numerical method is accepted in this paper to ascertain the chaotic threshold of a nonlinear system. Two forms of numerical technique, namely Lyapunov exponents and phase plan are adopted to validate the computation results. The method is applied to the ship-roll system and the chaos threshold is numerically computed in the end.
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47

Xu Jiancheng, 徐建程, 王飞舟 Wang Feizhou, 邓燕 Deng Yan, and 柴立群 Chai Liqun. "Numerical analysis of system transfer function in interferometric imaging system." High Power Laser and Particle Beams 24, no. 8 (2012): 1811–15. http://dx.doi.org/10.3788/hplpb20122408.1811.

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48

Ramer, Arthur, and Leslie Lander. "SYSTEM PROBLEMS AND COMPUTER TECHNIQUES: NUMERICAL VS. EXPERT SYSTEM APPROACHES." Cybernetics and Systems 21, no. 4 (1990): 445–59. http://dx.doi.org/10.1080/01969729008902252.

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49

FEUDEL, ULRIKE, and WOLFGANG JANSEN. "CANDYS/QA—A SOFTWARE SYSTEM FOR QUALITATIVE ANALYSIS OF NONLINEAR DYNAMICAL SYSTEMS." International Journal of Bifurcation and Chaos 02, no. 04 (1992): 773–94. http://dx.doi.org/10.1142/s0218127492000434.

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Numerical methods are often needed if bifurcation phenomena in nonlinear dynamical systems are studied. In this paper the software system CANDYS/QA for numerical qualitative analysis is presented. A wide class of problems is treated: computation of invariant sets (e.g., steady-states and periodic orbits), path-following (continuation) of such sets, and the related bifurcation phenomena. The following bifurcation situations are detected and the corresponding critical points are calculated during path-following: turning, bifurcation, Hopf bifurcation, period-doubling, torus bifurcation points (o
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50

Shimizu, Kuniyasu. "Experimental Observations of Propagating Waves and Switching Phenomena in a Coupled Bistable Oscillator System." International Journal of Bifurcation and Chaos 24, no. 12 (2014): 1450157. http://dx.doi.org/10.1142/s0218127414501570.

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In this study, we construct a circuit composed of bistable oscillators and we report the experimental observations of quasi-periodic waves propagating in the circuit and compare them with the associated numerical results. Two different types of propagating quasi-periodic waves with identical parameter sets are experimentally verified. The associated numerical results are distinguished by comparing trajectories on the phase planes and by analyzing the one-parameter bifurcation diagrams. Furthermore, the experiments reveal five different types of switching oscillations. The associated numerical
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