Parallel synchronization and a RBF neuro-fuzzy system to synchronization of chaotic systems

[1] M. A. Abdelkawy, A. Z. M. Amin, A. H. Bhrawy, J. A. T. Machado, A. M. Lopes, Jacobi collocation approximation
for solving multi-dimensional Volterra integral equations, International Journal of Nonlinear Sciences and
Numerical Simulation, 18(5) (2017), 411-425. https://doi.org/10.1515/ijnsns-2016-0160

[2] M. T. Akter, A. Tarammim, S. Hussen, Chaos control and synchronization of modified Lorenz system using active
control and backstepping scheme, Waves in Random and Complex Media, (2023). https://doi.org/10.1080/
17455030.2023.2205529

[3] G. V. Badalyan, Generalization of Legendre polynomials and some of their applications, Russian, Armenian
Summary, 9(3a) (1956), 22. https://doi.org/10.1007/978-3-0348-8685-7-13

[4] F. Behloul, B. P. F. Lelieveldt, A. Boudraa, J. H. C. Reiber, Optimal design of radial basis function neural
networks for fuzzy-rule extraction in high dimensional data, Pattern Recognition, 35 (2002), 659-675. https:
//doi.org/10.1016/S0031-3203(01)00033-4

[5] A. H. Bhrawy, A Jacobi spectral collocation method for solving multi-dimensional nonlinear fractional sub-diffusion
equations, Numerical Algorithms, 73(1) (2016), 91-113. https://doi.org/10.1007/s11075-015-0087-2

[6] S. Boccaletti, J. Kurths, G. Osipov, D. L. Valladares, C. S. Zhou, The synchronization of chaotic systems, Physics
reports, 366(1-2) (2002), 1-101. https://doi.org/10.1016/S0370-1573(02)00137-0

[7] M. D. Buhmann, Radial basis functions: Theory and implementations, 12 , Cambridge University Press, 2003.

[8] T. L. Carroll, L. M. Pecora, Synchronizing nonautonomous chaotic circuits, IEEE Transactions on Circuits and
Systems II: Analog and Digital Signal Processing, 40(10) (1993), 646-650. https://doi.org/10.1109/82.24616

[9] G. W. Chang, C. I. Chen, Y. F. Teng, Radial-Basis-Function-Based neural network for harmonic detection, IEEE
Transactions on Industrial Electronics, 57 (2010), 2171-2179. https://doi.org/10.1109/TIE.2009.2034681

[10] Y. Y. Chen, Y. T. Chang, B. S. Chen, Fuzzy solutions to partial differential equations: Adaptive approach, IEEE
Transactions on Fuzzy Systems, 17(1) (2009), 116-127. https://doi.org/10.1109/TFUZZ.2008.2005010

[11] J. Chen, C. Li, X. Yang, Chaos synchronization of the distributed-order Lorenz system via active control and
applications in chaotic masking, International Journal of Bifurcation and Chaos, 28(10) (2018), 1850121. https:
//doi.org/10.1142/S0218127418501213

[12] S. Chen, J. Lu, Synchronization of an uncertain unified chaotic system via adaptive control, Chaos, Solitons and
Fractals, 14(4) (2002), 643-647. https://doi.org/10.1016/S0960-0779(02)00006-1

[13] Y. Cui, H. G. Zhang, Y. Wang, W. Gao, Adaptive control for a class of uncertain strict-feedback nonlinear systems
based on a generalized fuzzy hyperbolic model, Fuzzy Sets and Systems, 302 (2016), 52-64. https://doi.org/10.
1016/j.fss.2015.11.015

[14] D. Ding, X. Qian, N. Wang, D. Liang, Synchronization and anti-synchronization of a fractional order delayed
memristor-based chaotic system using active control, Modern Physics Letters B, 32(14) (2018), 1850142. https:
//doi.org/10.1142/S0217984918501427

[15] L. G. Dominguez, R. A. Wennberg, W. Gaetz, D. Cheyne, O. C. Snead, J. L. P. Velazquez, Enhanced synchrony
in epileptiform activity local versus distant phase synchronization in generalized seizures, Journal of Neuroscience,
25(35) (2005), 8077-8084. https://doi.org/10.1523/JNEUROSCI.1046-05.2005

[16] A. Durdu, Y. Uyaroglu, Comparison of synchronization of chaotic Burke-Shaw attractor with active control and
integer-order and fractional-order P-C method, Chaos, Solitons and Fractals, 164 (2022), 112646. https://doi.
org/10.1016/j.chaos.2022.112646

[17] S. Effati, A. V. Kamyad, R. A. Kamyabi-Gol, On infinite-horizon optimal control problems, Zeitschrift f¨ur Analysis
und ihre Anwendungen, 19(1) (2000), 269-278. https://doi.org/10.4171/ZAA/950

[18] S. Effati, A. Nazemi, A new approach for asymptotic stability of the nonlinear ordinary differential equations,
Journal of Applied Mathematics and Computing, 25 (2007), 231-244. https://doi.org/10.1007/BF02832349

[19] I. Famelis, A. Donas, G. Galanis, Comparative study of feed forward and radial basis function neural networks for
solving an environmental boundary value problem, Results in Applied Mathematics, 16 (2022), 100344. https:
//doi.org/10.1016/j.rinam.2022.100344

[20] C. F. Feng, Projective synchronization between two different time-delayed chaotic systems using active control
approach, Nonlinear Dynamics, 62(1-2) (2010), 453-459. https://doi.org/10.1007/s11071-010-9733-3

[21] L. Glass, M. C. Mackey, From clocks to chaos: The rhythms of life, Princeton University Press, 1988.

[22] J. M. Gonzalez-Miranda, Amplitude envelope synchronization in coupled chaotic oscillators, Physical Review E,
65(3) (2002), 036232. https://doi.org/10.1103/PhysRevE.65.036232

[23] A. Hadian Rasanan, N. Bajalan, K. Parand, J. Amani Rad, Simulation of nonlinear fractional dynamics arising
in the modeling of cognitive decision making using a new fractional neural network, Mathematical Methods in the
Applied Sciences, 43 (2020), 1-30. https://doi.org/10.1002/mma.5981

[24] M. Hajipour, A. Jajarmi, Numerical solution of the state-delayed optimal control problems by a fast and accurate
finite difference θ-method, Communications in Nonlinear Science and Numerical Simulation, 55 (2018), 265-276.
https://doi.org/10.1016/j.cnsns.2017.07.017

[25] C. Hammond, H. Bergman, P. Brown, Pathological synchronization in Parkinson¸as disease: Networks, models and
treatments, Trends in Neurosciencesm, 30(7) (2007), 357-364. https://doi.org/10.1016/j.tins.2007.05.004

[26] R. Hermann, A. Krener, Nonlinear controllability and observability, IEEE Transactions on Automatic Control,
22(5) (1977), 728-740. https://doi.org/10.1109/TAC.1977.1101601

[27] K. Hirosawa, S. Kittaka, Y. Oishi, F. Kannari, T. Yanagisawa, Phase locking in a Nd: YVO 4 waveguide laser array
using Talbot cavity, Optics Express, 21(21) (2013), 24952-24961. https://doi.org/10.1364/OE.21.024952

[28] M. C. Ho, Y. C. Hung, Synchronization of two different systems by using generalized active control, Physics Letters
A, 301(5-6) (2002), 424-428. https://doi.org/10.1016/S0375-9601(02)00987-8

[29] K. Hornick, M. Stinchcombe, H. White, Multilayer feedforward networks are universal approximators, Neural
Networks, 2(5) (1989), 359-366. https://doi.org/10.1016/0893-6080(89)90020-8

[30] L. Huang, R. Feng, M.Wang, Synchronization of chaotic systems, via nonlinear control, Physics Letters A, 320(4)
(2004), 271-275. https://doi.org/10.1016/j.physleta.2003.11.027

[31] L. Jianyu, L. Siwei, Q. Yingjian, H. Yaping, Numerical solution of elliptic partial differential equation using
radial basis function neural networks, Neural Networks, 16(5-6) (2003), 729-734. https://doi.org/10.1016/
S0893-6080(03)00083-2

[32] S. Karimi, S. Effati, F. Ghane, The synchronization of chaotic systems applying the parallel synchronization
method, Physica Scripta, 94 (2016), 1-20. https://doi.org/10.1088/1402-4896/ab2679

[33] F. Kheyrinataj, A. Nazemi, Fractional Chebyshev functional link neural network-optimization method for solving
delay fractional optimal control problems with Atangana–Baleanu derivative, Optimal Control Applications and
Methods, 41(3) (2020), 808-832. https://doi.org/10.1002/oca.2572

[34] F. Kheyrinataj, A. Nazemi, Muntz-Legendre neural network construction for solving delay optimal control problems
of fractional order with equality and inequality constraints, Soft Computing, 24 (2020), 9575-9594. https://doi.
org/10.1007/s00500-019-04465-7

[35] R. Kiruthika, R. Krishnasamy, S. Lakshmanan, M. Prakash, A. Manivannan, Non-fragile sampled-data control
for synchronization of chaotic fractional-order delayed neural networks via LMI approach, Chaos, Solitons and
Fractals, 169 (2023), 113252. https://doi.org/10.1016/j.chaos.2023.113252

[36] S. Kopeikin, M. Efroimsky, G. Kaplan, Relativistic celestial mechanics of the solar system, John Wiley Sons, 2011.
https://doi.org/10.1002/9783527634569

[37] M. Kumar, N. Yadav, Multilayer perceptrons and radial basis function neural network methods for the solution of
differential equations, A Survey, Computers and Mathematics with Applications, 62 (2011), 3796-3811. https:
//doi.org/10.1016/j.camwa.2011.09.028

[38] C. Liu, L. Liu, T. Liu, A novel three-dimensional autonomous chaos system, Chaos, Solitons and Fractals, 39(4)
(2009), 1950-1958. https://doi.org/10.1016/j.chaos.2007.06.079

[39] M. Liu, W. Peng, M. Hou, Z. Tian, Radial basis function neural network with extreme learning machine algorithm
for solving ordinary differential equations, Soft Computing, 27 (2023), 3955-3964. https://doi.org/10.21203/
rs.3.rs-1416056/v1

[40] E. N. Lorenz, Deterministic nonperiodic flow, Journal of The Atmospheric Sciences, 20(2) (1963), 130-141. https:
//doi.org/10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2

[41] N. Mai-Duy, T. Tran-Cong, Numerical solution of differential equations using multiquadric radial basis function
networks, Neural Networks, 14(2) (2001), 185-199. https://doi.org/10.1016/S0893-6080(00)00095-2

[42] S. Mall, S. Chakraverty, Chebyshev neural network based model for solving Lane¸a Emden type equations, Applied
Mathematics and Computation, 247 (2014), 100-114. https://doi.org/10.1016/j.amc.2014.08.085

[43] S. Mall, S. Chakraverty, Single layer Chebyshev neural network model for solving elliptic partial differential equations,
Neural Processing Letters, 45(3) (2017), 825-840. https://doi.org/10.1007/s11063-016-9551-9

[44] M. Mansoori, A. Nazemi, Solving infinite-horizon optimal control problems of the time-delayed systems by Haar
wavelet collocation method, Network: Computation in Neural Systems, 35 (2016), 97-117. https://doi.org/10.
1080/0954898X.2021.1884762

[45] A. K. Mittal, Error analysis and approximation of Jacobi pseudospectral method for the integer and fractional
order integro-differential equation, Applied Numerical Mathematics, 171 (2022), 249-268. https://doi.org/10.
1016/j.apnum

[46] J. J. Mor´e, The Levenberg-Marquardt algorithm: Implementation and theory, In: Numerical Analysis, Springer,
(1978), 105-116. https://doi.org/10.1007/BFb0067700

[47] M. Mortezaee, M. Ghovatmand, A. Nazemi, An application of a fuzzy system for solving time delay fractional
optimal control problems with Atangana–Baleanu derivative, Optimal Control Applications and Methods, 43(6)
(2022), 1753-1777. https://doi.org/10.1002/oca.2924

[48] M. Mortezaee, M. Ghovatmand, A. Nazemi, Solving variable-order fractional delay differential algebraic equations
via fuzzy systems with application in delay optimal control problems, Iranian Journal of Fuzzy Systems, 21(2)
(2024), 67-85. https://doi.org/10.22111/IJFS.2024.43498.7646

[49] F. Mostajeran, S. M. Hosseini, Radial basis function neural network (RBFNN) approximation of Cauchy inverse
problems of the Laplace equation, Computers and Mathematics with Applications, 141 (2023), 129-144. https:
//doi.org/10.1016/j.camwa.2023.04.026

[50] U. Parlitz, L. Junge, August synchronization of chaotic systems, In 1999 European Control Conference (ECC)
(4637-4642), IEEE, (1999). https://doi.org/10.1002/3527607455.CH11

[51] L. Perko, Differential equations and dynamical systems, Springer Science Business Media, 7 (2013).

[52] M. Raissi, P. Perdikarism, G. E. Karniadakis, Physics-informed neural networks: A deep learning framework for
solving forward and inverse problems involving nonlinear partial differential equations, Journal of Computational
Physics, 378 (2019), 686-707. https://doi.org/10.1016/j.jcp.2018.10.045

[53] T. Rezazadeh, E. Najafi, Jacobi collocation method and smoothing transformation for numerical solution of neutral
nonlinear weakly singular Fredholm integro-differential equations, Applied Numerical Mathematics, 181 (2022),
135-150. https://doi.org/10.21203/rs.3.rs-4186485/v1

[54] R. Roy, Jr. K. Scott Thornburg, Experimental synchronization of chaotic lasers, Physical Review Letters, 72
(2009). https://doi.org/10.1103/PhysRevLett.72.2009

[55] M. G. Rosenblum, A. S. Pikovsky, J. Kurths, Phase synchronization of chaotic oscillators, Physical Review
Letters, 76(11) (1996), 1804. https://doi.org/10.1103/PhysRevLett.76.1804

[56] O. E. Rossler, Continuous chaos-four prototype equations, Annals of the New York Academy of Sciences, 316(1)
(1979), 376-392. https://doi.org/10.1111/j.1749-6632.1979.tb29482.x

[57] N. F. Rulkov, M. M. Sushchik, L. S. Tsimring, H. D. Abarbanel, Generalized synchronization of chaos in directionally
coupled chaotic systems, Physical Review E, 51(2) (1995), 980. https://doi.org/10.1103/PhysRevE.
51.980

[58] S. J. Russell, P. Norvig, Artificial intelligence - A modern approach, 3rd Edition, Prentice Hall, 2010.

[59] H. Saberi-Nik, P. Rebelo, M. Shamsyeh Zahedi, Solution of infinite horizon nonlinear optimal control problems by
piecewise adomian decomposition method, Mathematical Modelling and Analysis, 18(4) (2013), 543-560. https:
//doi.org/10.3846/13926292.2013.841598

[60] F. Setoudeh, M. M. Dezhdar, M. Najafi, Nonlinear analysis and chaos synchronization of a memristive-based
chaotic system using adaptive control technique in noisy environments, Chaos, Solitons and Fractals, 164 (2022),
112710. https://doi.org/10.1016/j.chaos.2022.112710

[61] L. Shi, X. Yang, Y. Li, Z. Feng, Finite-time synchronization of nonidentical chaotic systems with multiple timevarying
delays and bounded perturbations, Nonlinear Dynamics, 83(1-2) (2016), 75-87. https://doi.org/10.
1007/s11071-015-2310-z

[62] G. Simone, F. C. Morabito, RBFNN-based hole identification system in conducting plates, IEEE Transactions on
Neural Networks, 12(6) (2001), 1445-1454. https://doi.org/10.1109/72.963779

[63] S. Singh, A. T. Azar, Q. Zhu, Multi-switching Master¸a Slave synchronization of nonidentical chaotic systems,
In Innovative Techniques and Applications of Modelling, Identification and Control, (321-330) (2018), Springer,
Singapore.

[64] M. Srivastava, S. P. Ansari, S. K. Agrawal, S. Das, A. Y. T. Leung, Anti-synchronization between identical and
non-identical fractional-order chaotic systems using active control method, Nonlinear Dynamics, 76(2) (2014),
905-914. https://doi.org/10.1007/s11071-013-1177-0

[65] A. Staiano, R. Tagliaferri, W. Pedrycz, Improving RBF networks performance in regression tasks by means of a
supervised fuzzy clustering, Neurocomputing, 69 (2006), 1570-1581. https://doi.org/10.1016/j.neucom.2005.
06.014

[66] H. Su, R. Luo, M. Huang, J. Fu, Practical fixed time active control scheme for synchronization of a class of
chaotic neural systems with external disturbances, Chaos, Solitons and Fractals, 157 (2022), 111917. https:
//doi.org/10.1016/j.chaos.2022.111917

[67] Z. Wang, M. Chen, J. Chen, Solving multiscale elliptic problems by sparse radial basis function neural networks,
Journal of Computational Physics, 492 (2023), 112452. https://doi.org/10.1016/j.jcp.2023.112452

[68] L. X. Wang, J. M. Mendel, Fuzzy basis functions, universal approximation, and orthogonal least-squares learning,
IEEE Transactions on Neural Networks, 3(5) (1992), 807-814. https://doi.org/10.1109/72.159070

[69] Y. Wang, C. Sun, Z. Wang, J. Sun, Projection synchronization of three-dimensional chaotic systems with active
control based on DNA strand displacement, IEEE Transactions on NanoBioscience, 22 (2023), 836-844. https:
//doi.org/10.1109/TNB.2023.3241652

[70] E. Wu, X. Yang, Adaptive synchronization of coupled nonidentical chaotic systems with complex variables
and stochastic perturbations, Nonlinear Dynamics, 84(1) (2016), 261-269. https://doi.org/10.1007/
S11071-015-2433-2

[71] L. Xiao, Y. Zhang, L. Jia, Y. He, A ZNN-based sliding mode control strategy for solving synchronization problems
of chaotic systems with application in memristor oscillator, Journal of the Franklin Institute, 360 (2023), 12101-
12124. https://doi.org/10.1016/j.jfranklin.2023.09.019

[72] X. Yang, Y. Feng, K. F. C. Yiu, Q. Song, F. E. Alsaadi, Synchronization of coupled neural networks with infinitetime
distributed delays via quantized intermittent pinning control, Nonlinear Dynamics, 94(3) (2018), 2289-2303.
https://doi.org/10.1007/s11071-018-4449-x

[73] M. Yavari, A. Nazemi, An efficient numerical scheme for solving fractional infinite-horizon optimal control problems,
ISA Transactions, 94 (2019), 108-118. https://doi.org/10.1016/j.isatra.2019.04.016