Abstract
Without decomposing the complex-valued systems into two real-valued subsystems, this paper investigates quasi-projective synchronization (QPS) problem for Caputo type fractional-order complex-valued neural networks (FOCVNNs) with mixed delays by choosing suitable controllers. To realize QPS, the linear feedback controller and adaptive feedback controller are designed, by constructing suitable Lyapunov function, utilizing the fractional Razumikhin theorem and the properties of Mittag-Leffler function and inequality technique, and several sufficient criteria for QPS of FOCVNNs with mixed delays are derived. In addition, the upper bound of the error of QPS is estimated. Finally, two numerical examples are simulated to verify the effectiveness and feasibility of the proposed results.
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R. L. Bagley and P. J. Torvik, “A theoretical basis for the application of fractional calculus to viscoelasticity,” Journal of Rheology, vol. 27, no. 3, pp. 201–210, 1983.
X. Li and X. Tian, “Fractional order thermo-viscoelastic theory of biological tissue with dual phase lag heat conduction model,” Applied Mathematical Modelling, vol. 95, pp. 612–622, 2021.
R. L. Magin, “Fractional calculus models of complex dynamics in biological tissues,” Computers Mathematics with Applications, vol. 59, no. 5, pp. 1586–1593, 2010.
F. B. Yousef, A. Yousef, and C. Maji, “Effects of fear in a fractional-order predator-prey system with predator density-dependent prey mortality,” Chaos, Solitons and Fractals, vol. 145, p. 110711, 2021.
M. Das and G. P. Samanta, “A delayed fractional order food chain model with fear effect and prey refuge,” Mathematics and Computers in Simulation, vol. 178, pp. 218–245, 2020.
E. K. Lenzi, M. dos Santos, M. K. Lenzi, D. S. Vieira, and L. R. da Silva, “Solutions for a fractional diffusion equation: Anomalous diffusion and adsorption-desorption processes,” Journal of King Saud University-Science, vol. 28, no. 1, pp. 3–6, 2016.
J. Teng, H. Zhang, C. Tang, and H. Lin, “Novel molecular level insights into forward osmosis membrane fouling affected by reverse diffusion of draw solutions based on thermodynamic mechanisms,” Journal of Membrane Science, vol. 620, p. 118815, 2021.
P. Ghamisi, M. S. Couceiro, J. A. Benediktsson, and N. Ferreira, “An efficient method for segmentation of images based on fractional calculus and natural selection,” Expert Systems with Applications, vol. 39, no. 16, pp. 12407–12417, 2012.
A. Gomez-Echavarrla, J. P. Ugarte, and C. Tobon, “The fractional fourier transform as a biomedical signal and image processing tool: A review,” Biocybernetics and Biomedical Engineering, vol. 40, pp. 1081–1093, 2020.
V. E. Bondarenko, “Information processing, memories, and synchronization in chaotic neural network with the time delay,” Complexity, vol. 11, pp. 39–52, 2005.
G. A. Anastassiou, “Fractional neural network approximation,” Computers Mathematics with Applications, vol. 64, pp. 1655–1676, 2012.
P. Muthukumar and P. Balasubramaniam, “Feedback synchronization of the fractional order reverse butterfly-shaped chaotic system and its application to digital cryptography,” Nonlinear Dynamics, vol. 74, no. 4, pp. 1169–1181, 2013.
L. Chen, R. Wu, J. Cao, and J. B. Liu, “Stability and synchronization of memristor-based fractional-order delayed neural networks,” Neural Networks, vol. 71, pp. 37–44, 2015.
X. Wu and H. Bao, “Finite time complete synchronization for fractional-order multiplex networks,” Applied Mathematics and Computation, vol. 377, p. 125188, 2020.
H. Zhang, X. Y. Wang, and X. Lin, “Topology identification and module phase synchronization of neural network with time delay,” IEEE Transactions on Systems, Man, and Cybernetics: Systems, vol. 47, pp. 885–892, 2017.
Z. Yao, P. Zhou, Z. Zhu, and J. Ma, “Phase synchronization between a light-dependent neuron and a thermosensitive neuron,” Neurocomputing, vol. 423, pp. 518–534, 2021.
Q. Gan, X. Rui, and X. Kang, “Synchronization of chaotic neural networks with mixed time delays,” Communications in Nonlinear Science Numerical Simulation, vol. 16, pp. 966–974, 2011.
L. Yang and J. Jiang, “The role of coupling-frequency weighting exponent on synchronization of a power network,” Physica A, vol. 464, pp. 115–122, 2016.
G. Arthi and N. Brindha, “On finite-time stability of nonlinear fractional-order systems with impulses and multistate time delays,” Results in Control and Optimization, vol. 2, p. 100010, 2021.
S. Cai and M. Hou, “Quasi-synchronization of fractional-order heterogeneous dynamical networks via aperiodic intermittent pinning control,” Chaos, Solitons and Fractals, vol. 146, p. 110901, 2021.
Y. Y. Chen, R. Huang, Y. Ge, and Y. Zhang, “Spherical formation tracking control of nonlinear second-order agents with adaptive neural flow estimate,” IEEE Transactions on Neural Networks and Learning Systems, pp. 1–12, 2021. DOI: https://doi.org/10.1109/TNNLS.2021.3071317
Y. Y. Chen, K. Chen, and A. Astolfi, “Adaptive formation tracking control for first-order agents in a time-varying flowfield,” IEEE Transactions on Automatic Control, p. 1, 2021. DOI: https://doi.org/10.1109/TAC.2021.3074900
Y. Y. Chen, K. Chen, and A. Astolfi, “Adaptive formation tracking control for directed networked vehicless in a time-varying flowfield,” Journal of Guidance, Control, and Dynamics, vol. 44, no. 10, 2021.
Y. Zhang and S. Deng, “Finite-time projective synchronization of fractional-order complex-valued memristor-based neural networks with delay,” Chaos, Solitons and Fractals, vol. 128, pp. 176–190, 2019.
J. Yu, C. Hu, H. Jiang, and X. Fan, “Projective synchronization for fractional neural networks,” Neural Networks, vol. 49, pp. 87–95, 2014.
W. S. Mcculloch and W. Pitts, “A logical calculus of the ideas immanent in nervous activity,” The Bulletin of Mathematical Biophysics, vol. 5, no. 4, pp. 115–133, 1943.
M. A. Islas, J. J. Rubio, S. Muñiz, G. Ochoa, J. Pacheco, J. A. Meda-Campaña, D. Mujica-Varga, C. Aguilar-Ibañez, G. J. Gutierrez, and A. Zacarias, “A fuzzy logic model for hourly electrical power demand modeling,” Electronics, vol. 10, no. 4, p. 448, 2021.
J. J. Rubio, “SOFMLS: Online self-organizing fuzzy modified least-squares network,” IEEE Transactions on Fuzzy Systems, vol. 17, no. 6, pp. 1296–1309, 2009.
H. S. Chiang, M. Y. Chen, and Y. J. Huang, “Wavelet-based EEG processing for epilepsy detection using fuzzy entropy and associative petri net,” IEEE Access, vol. 7, pp. 103255–103262, 2019.
J. J. Rubio, “Stability analysis of the modified Levenberg-Marquardt Algorithm for the artificial neural network training,” IEEE Transactions on Neural Networks and Learning Systems, vol. 32, no. 8, pp. 3510–3524, 2021.
L. A. Soriano, E. Zamora, J. M. Vazquez-Nicolas, G. Hernández, J. A. B. Madrigal, and D. Balderas, “PD control compensation based on a cascade neural network applied to a robot manipulator, “Frontiers in Neurorobotics, vol. 14, 2020.
F. Furlán, E. Rubio, H. Sossa, and V. Ponce, “CNN based detectors on planetary environments: A performance evaluation,” Frontiers in Neurorobotics, vol. 14, p. 85, 2020.
Z. T. Huang, Q. G. Yang, and X. S. Luo, “Exponential stability of impulsive neural networks with time-varying delays,” Chaos Solitons and Fractals, vol. 35, no. 4, pp. 770–780, 2008.
Y. Li, Y. Q. Chen, and I. Podlubny, “Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized mittag-leffler stability,” Computers Mathematics with Applications, vol. 59, no. 5, pp. 1810–1821, 2010.
H. Zhang, M. Ye, and J. Cao, and A. Alsaedi, “Synchronization control of Riemann-Liouville fractional competitive network systems with time-varying delay and different time scales,” International Journal of Control, Automation, and Systems, vol. 16, pp. 1404–1414, 2018.
M. Zarefard and S. Effati, “Adaptive synchronization between two non-identical BAM neural networks with unknown parameters and time-varying delays,” International Journal of Control, Automation, and Systems, vol. 15, pp. 1877–1887, 2017.
S. B. Stojanovic, D. L. Debeljkovic, and M. A. Misic, “Finite-time stability for a linear discrete-time delay systems by using discrete convolution: An LMI approach,” International Journal of Control, Automation, and Systems, vol. 14, no. 4, pp. 1144–1151, 2016.
E. Kaslik, M. Neamu, and L. F. Vesa, “Global stability analysis of an unemployment model with distributed delay,” Mathematics and Computers in Simulation, vol. 185, no. 4, pp. 535–546, 2021.
W. Zhang, H. Zhang, J. Cao, H. M. Zhang, and D. Chen, “Synchronization of delayed fractional-order complex-valued neural networks with leakage delay,” Physica A: Statistical Mechanics and its Applications, vol. 556, p. 124710, 2020.
C. Xu, M. Liao, P. Li, and S. Yuan, “Impact of leakage delay on bifurcation in fractional-order complex-valued neural networks,” Chaos, Solitons and Fractals, vol. 142, p. 110535, 2021.
X. You, S. Dian, R. Guo, and S. Li, “Exponential stability analysis for discrete-time quaternion-valued neural networks with leakage delay and discrete time-varying delays,” Neurocomputing, vol. 430, pp. 71–81, 2021.
S. Yang, J. Yu, C. Hu, and H. Jiang, “Quasi-projective synchronization of fractional-order complex-valued recurrent neural networks,” Neural Networks, vol. 104, pp. 104–113, 2018.
X. You, Q. Song, and Z. Zhao, “Existence and finite-time stability of discrete fractional-order complex-valued neural networks with time delays,” Neural Networks, vol. 123, pp. 248–260, 2020.
Y. Xu and W. Li, “Finite-time synchronization of fractional-order complex-valued coupled systems,” Physica A, vol. 549, p. 123903, 2020.
H. Zhang, M. Ye, and R. Ye, and J. Cao, “Synchronization stability of Riemann-Liouville fractional delay-coupled complex neural networks,” Physica A: Statistical Mechanics and its Applications, vol. 508, pp. 155–165, 2018.
E. Arslan, G. Narayanan, M. S. Ali, S. Arik, and S. Saroha, “Controller design for finite-time and fixed-time stabilization of fractional-order memristive complex-valued BAM neural networks with uncertain parameters and time-varying delays,” Neural Networks, vol. 130, pp. 60–74, 2020.
X. Yang, C. Li, T. Huang, Q. Song, and J. Huang, “Synchronization of fractional-order memristor-based complex-valued neural networks with uncertain parameters and time delays,” Chaos, Solitons and Fractals, vol. 110, pp. 105–123, 2018.
S. Khorashadizadeh and M. H. Majidi, “Synchronization of two different chaotic systems using Legendre polynomials with applications in secure communications,” Frontiers of Information Technology and Electronic Engineering, vol. 19, pp. 1180–1190, 2018.
M. Samimi, M. H. Majidi, and S. Khorashadizadeh, “Secure communication based on chaos synchronization using brain emotional learning,” International Journal of Electronics and Communications, vol. 127, p. 153424, 2020.
A. A. Kilbas, and H. M. Srivastava, Theory and Applications of Fractional Differential Equations, Elsevier Science B.V. Amsterdam, vol. 204, no. 49–52, pp. 2453–2461, 2006.
H. L. Li, C. Hu, J. Cao, H. Jiang, and A. Alsaedi, “Quasiprojective and complete synchronization of fractional-order complex-valued neural networks with time delays,” Neural Networks, vol. 118, pp. 102–109, 2019.
D. Baleanu, S. J. Sadati, R. Ghaderi, A. Ranjbar, and F. Jarad, “Razumikhin stability theorem for fractional systems with delay,” Abstract and Applied Analysis, vol. 2010, p. 124812, 2010.
H. L. Li, J. Cao, H. Jiang, and A. Alsaedi, “Finite-time synchronization of fractional-order complex networks via hybrid feedback control,” Neurocomputing, vol. 320, pp. 69–75, 2018.
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This work was jointly supported by the Natural Science Foundation of Anhui Province (1908085MA01), the Natural Science Foundation of the Higher Education Institutions of Anhui Province (KJ2019A0573, KJ2019A0557), and the Top Young Talents Program of Higher Learning Institutions of Anhui Province (gxyq2019048).
Jingshun Cheng received her Bachelor’s degree in mathematics and applied mathematics from the School of Information Engineering, Fuyang Normal University, China, in 2019. She is currently pursuing a Master’s degree with the School of Mathematics and Physics, Anqing Normal University, China. Her current research interests include fractional differential equations and dynamics of neural networks.
Hai Zhang is currently a Professor with the School of Mathematics and Physics, Anqing Normal University, China. He received his M.Sc. and Ph.D. degrees from Anhui University, China, in 2007 and 2010, respectively. From December 2012 to November 2014, he was a Postdoctoral Research Fellow at the Department of Mathematics, Southeast University, China. He is the author and co-author over 30 publications in the peer-reviewed journals. His current research interests include fractional differential equations, nonlinear dynamics, neural networks, control theory, and stability theory.
Weiwei Zhang is currently an Assistant Professor with the School of Mathematics and Physics, Anqing Normal University, Anqing, China. He received his M.S. degree from Anhui University, China, in 2009. From September 2019 to July 2020, he was a Visiting Scholar with the Department of Mathematics, Southeast University, China. He is currently working toward a Ph.D. degree at Nanjing University of Aeronautics and Astronautics, Nanjing, China. His current research interests include nonlinear systems, neural networks, complex networks, control theory, and stability theory.
Hongmei Zhang is currently an Assistant Professor with the School of mathematics and Physics, Anqing Normal University, Anqing, China. She received the bachelor degree in information and computing science from Northwestern University, Shanxi, China, in 2003. She received a Master’s degree in applied mathematics from East China Normal University Shanghai, China, in 2010. Her current research interests include stability theory and its application to the delay system, the nonlinear control, and complex networks.
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Cheng, J., Zhang, H., Zhang, W. et al. Quasi-projective Synchronization for Caputo Type Fractional-order Complex-valued Neural Networks with Mixed Delays. Int. J. Control Autom. Syst. 20, 1723–1734 (2022). https://doi.org/10.1007/s12555-021-0392-6
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DOI: https://doi.org/10.1007/s12555-021-0392-6