Document Type : Research Article
Authors
1 School of Economics and Statistics, Guangzhou University, P.O. Box. 510006, Guangzhou, China
2 Electrical and Computer Engineering Department, University of Windsor, Windsor, ON, Canada
3 Department of Automation, Biomechanics and Mechatronics, Lods University of Technology, Poland
Abstract
The present study introduces a kind of fractional-order Hopfield neural network (FOHNN), and its complex dynamic behavior is investigated through chaos analyses. With the use of phase space analysis and bifurcation diagrams and maximal Lyapunov exponent (MLE) it is demonstrated that for the values of 0.87 < α < 1, as the fractional-order (FO), the dynamical behavior of the mentioned FOHNN is chaotic. Then, the bounded trait of chaotic systems is utilized to derive an adaptive model-free control technique to suppress of complex dynamics of the FOHNN. Furthermore, according to the matrix analysis theorem of non-integer-order systems and the adaptive model-free control methodology, analytical consequences of the designed controller are evidenced. Eventually, two examples are reported to illustrate the applicability of the mentioned model-free control method.
Keywords
[1] Aghababa M.P. (2012). “Comments on H∞synchronization of uncertain fractional-order chaotic systems: Adaptive fuzzy approach [ISA Trans 50 (2011) 548–556]”, ISA Transactions, 51, 11-12.
[2] Aghababa M.P. (2015). “Fractional modeling and control of a complex nonlinear energy supply-demand system”, Complexity, 20, 74-86.
[3] Aghababa M.P., Borjkhani M. (2014). “Chaotic fractional-order model for muscular blood vessel and its control via fractional control scheme”, Complexity, 20 (2), 37-46.
[4] Asl M.S., Javidi M. (2017). “An improved PC scheme for nonlinear fractional differential equations: Error and stability analysis”, Journal of Computational and Applied Mathematics, 324, 101-117.
[5] Aghababa M.P., Haghighi A.R., Roohi M. (2015). “Stabilization of unknown fractional-order chaotic systems: an adaptive switching control strategy with application to power systems”, IET Generation, Transmission & Distribution, 9, 1883-1893.
[6] Asakawa S., Kyoya I. (2013). “Hopfield neural network model for explaining double dissociation in semantic memory impairment”, BMC Neuroscience, 14, 233.
[7] Banerjee S., Majumdar D. (2000). “Shape matching in multimodal medical images using point landmarks with Hopfield net”, Neuro computing, 30, 103-116.
[8] Chen Y., Tang C., Roohi M. (2021). “Design of a model-free adaptive sliding mode control to synchronize chaotic fractional-order systems with input saturation: An application in secure communications”, Journal of the Franklin Institute, 8109-8137.
[9] Curran P.F., Chua L.O. (1997). “Absolute stability theory and the synchronization problem”, International Journal of Bifurcation and Chaos, 7, 1375-1382.
[10] Danca M.F., Garrappa R., Tang K.S., Chen G. (2013). “Sustaining stable dynamics of a fractional-order chaotic financial system by parameter switching”, Computers & Mathematics with Applications, 66, 702-716.
[11] Diks C., Wagener F.O.O. (2008). “A bifurcation theory for a class of discrete time Markovian stochastic systems”, Physica D: Nonlinear Phenomena, 237, 3297-3306.
[12] Ding Y., Wang Z., Ye H. (2012). “Optimal control of a fractional-order HIV-immune system with memory”, IEEE Transactions on Control Systems Technology, 20, 763-769.
[13] Du J., Sheng Z., Mei Q., Huang H. (2009). “Study on output dynamic competition model and its global bifurcation”, International Journal of Nonlinear Sciences and Numerical Simulation, 10, 129-136.
[14] Effati S., Moghaddas M. (2016). “A novel neural network based on NCP function for solving constrained nonconvex optimization problems”, Complexity, 21, 130-141.
[15] Esfahani Z., Roohi M., Gheisarnejad M., Dragicević T., Khooban M.H. (2019). “Optimal non-integer sliding mode control for frequency regulation in stand-alone modern power grids”, Applied Sciences, 9, 3411.
[16] Fei J., Lu C. (2018). “Adaptive fractional-order sliding mode controller with neural estimator”, Journal of the Franklin Institute, 355, 2369-2391.
[17] Fradkov A.L., Evans R. (2005). “Control of chaos: Methods and applications in engineering”, Annual Reviews in Control, 29, 33-56.
[18] Gomaa Haroun A., Yin-Ya L. (2019). “A novel optimized fractional-order hybrid fuzzy intelligent PID controller for interconnected realistic power systems”, Transactions of the Institute of Measurement and Control, 41, 3065-3080.
[19] Haghighi A., Aghababa M.P., Asghary N., Roohi M. (2020). “A nonlinear control scheme for stabilization of fractional-order dynamical chaotic systems”, Journal of Advanced Mathematical Modeling, 10, 19-38.
[20] Hamza M.F., Yap H.J., Choudhury I.A. (2017). “Recent advances on the use of meta-heuristic optimization algorithms to optimize the type-2 fuzzy logic systems in intelligent control”, Neural Computing and Applications, 28, 979-999.
[21] He S., Sun K., Wang H. (2016). “Solution and dynamics analysis of a fractional-order hyperchaotic system”, Mathematical Methods in the Applied Sciences, 39, 2965-2973.
[22] Hong C.H., Choi K.C., Kim B.S. (2009). “Applications of adaptive neural network control to an unmanned airship”, International Journal of Control, Automation and Systems, 7, 911.
[23] Hopfield J.J. (1984). “Neurons with graded response have collective computational properties like those of two-state neurons”, Proceedings of the national academy of sciences, 81, 3088-3092.
[24] Huang W.Z., Huang Y. (2008). “Chaos of a new class of Hopfield neural networks”, 206, 1-11.
[25] Jafari P., Teshnehlab M., Tavakoli-Kakhki M. (2018). “Adaptive type-2 fuzzy system for synchronization and stabilization of chaotic non-linear fractional order systems”, IET Control Theory & Applications, 12, 183-193.
[26] Javan D.S., Mashhadi H.R., Rouhani M. (2013). “A fast static security assessment method based on radial basis function neural networks using enhanced clustering”, International Journal of Electrical Power and Energy Systems, 44, 988-996.
[27] Li J., Liu F., Guan Z., Li T. (2013). “A new chaotic Hopfield neural network and its synthesis via parameter switchings”, Neurocomputing, 117, 33-39.
[28] Li C., Tong Y. (2013). “Adaptive control and synchronization of a fractional-order chaotic system”, Pramana, 80, 583-592.
[29] Lin T.C., Kuo C.H. (2011). “H∞ synchronization of uncertain fractional-order chaotic systems: Adaptive fuzzy approach”, ISA Transactions, 50, 548-556.
[30] Liu H., Pan Y., Li S., Chen Y. (2018). “Synchronization for fractional-order neural networks with full/under-actuation using fractional-order sliding mode control”, International Journal of Machine Learning and Cybernetics, 9, 1219-1232.
[31] Liu H., Wen Y., Luan F., Gao Y. (2009). “Application of experimental design and radial basis function neural network to the separation and determination of active components in traditional Chinese medicines by capillary electrophoresis”, Analytica Chimica Acta, 88-93.
[32] Mahmoud E.E., Jahanzib L.S., Trikha P., Almaghrabi O.A. (2021). “Analysis and control of the fractional chaotic Hopfield neural network. Advances in Difference Equations”, Advances in Difference Equations, 126.
[33] Mofid O., Mobayen S. (2018). “Adaptive synchronization of fractional-order quadratic chaotic flows with nonhyperbolic equilibrium”, Journal of Vibration and Control, 24, 4971-4987.
[34] Ouannas A., Azar A.T., Vaidyanathan S. (2017). “A robust method for new fractional hybrid chaos synchronization”, Mathematical Methods in the Applied Sciences, 40, 1804-1812.
[35] Peng X., Wu H. (2020). “Non-fragile robust finite-time stabilization and H∞ performance analysis for fractional-order delayed neural networks with discontinuous activations under the asynchronous switching”, Neural Computing and Applications, 32, 4045–4071.
[36] Petrás I. (2011). “An effective numerical method and its utilization to solution of fractional models used in bioengineering applications”, Advances in Difference Equations, 652789.
[37] Podlubny I. (1998). “Fractional differential equations: An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications”, Elsevier Science.
[38] Podlubny I. (2002). “Geometric and physical interpretation of fractional integration and fractional differentiation”, Fractional Calculus and Applied Analysis, 5, 367-386.
[39] Ramezani A., Safarinejadian B., Zarei J. (2019). “Fractional-order chaotic cryptography in colored noise environment by using fractional order interpolatory cubature Kalman filter”, Transactions of the Institute of Measurement and Control, 41, 3206-3222.
[40] Roohi M., Aghababa M.P., Haghighi A.R. (2015). “Switching adaptive controllers to control fractional-order complex systems with unknown structure and input nonlinearities”, Complexity, 21, 211-223.
[41] Roohi M., Hadian H., Aghababa M.P., Amiri S.M.M. (2017). “Design of an active control method for complete stabilization of unknown fractional-order non-autonomous systems”, International Workshop on Mathematics and Decision Science: Fuzzy Information and Engineering and Decision, 646, 131-142.
[42] Roohi M., Khooban M.H., Esfahani Z., Aghababa M.P., Dragicevic T. (2019). “A switching sliding mode control technique for chaos suppression of fractional-order complex systems”, Transactions of the Institute of Measurement and Control, 41, 2932-2946.
[43] Roohi M., Zhang C., Chen Y. (2020). “Adaptive model-free synchronization of different fractional-order neural networks with an application in cryptography”, Nonlinear Dynamics, 2020, 3979-4001.
[44] Rosenstein M.T., Collins J.J., De Luca C.J. (1993). “A practical method for calculating largest Lyapunov exponents from small data sets”, Physica D: Nonlinear Phenomena, 65, 117-134.
[45] Shrivastava Y., Singh B. (2019). “Stable cutting zone prediction in computer numerical control turning based on empirical mode decomposition and artificial neural network approach”, Transactions of the Institute of Measurement and Control, 41, 193-209.
[46] Sun H., Zhang Y., Baleanu D., Chen W., Chen Y.Q. (2018). “A new collection of real-world applications of fractional calculus in science and engineering”, Communications in Nonlinear Science and Numerical Simulation, 64, 213-231.
[47] Suykens J.A., Vandewalle J. (2005). “Cellular neural networks, multi-scroll chaos and synchronization”, World Scientific Series on Nonlinear Science Series A, 50, 248.
[48] Tank D., Hopfield J.J. (1986). “Simple neural optimization networks: An A/D converter, signal decision circuit, and a linear programming circuit”, IEEE Transactions on Circuits and Systems, 33, 533-541.
[49] Wang B., Liu Y. J., Yu F., Wang J., Yang J. (2017). “Robust finite-time control of fractional-order nonlinear systems via frequency distributed model”, Nonlinear Dynamics, 85, 2133-2142.
[50] Wang L., Li S., Tian F., Fu X. (2004). “A noisy chaotic neural network for solving combinatorial optimization problems: Stochastic chaotic simulated annealing”, IEEE Transactions on Systems, Man, and Cybernetics, Part B, 34, 2119-2125.
[51] Wang, R, Liu, Y.J, Yu, F.S, Wang, J.Y, Yang, J.L. (2017). “Adaptive variable universe of discourse fuzzy control for a class of nonlinear systems with unknown dead zones”, International Journal of Adaptive Control and Signal Processing, 31 (12), 1934-1951.
[52] Wang X.Y., Li Z.M. (2019). “A color image encryption algorithm based on Hopfield chaotic neural network. Optics and Lasers in Engineering”, Optics and Lasers in Engineering, 115, 107-118.
[53] Wei J.D., Cheng H.J., Chang C.W. (2019). “Hopfield network-based approach to detect seam-carved images and identify tampered regions”, Neural Computing and Applications, 31, 6479-6492.
[54] Xu G., Xu J., Liu F., Zang F. (2017). “Secure communication based on the synchronous control of hysteretic chaotic neuron”, Neurocomputing, 227, 108-112.
[55] Yin C., Dadras S., Zhong S.M. (2012). “Design an adaptive sliding mode controller for drive-response synchronization of two different uncertain fractional-order chaotic systems with fully unknown parameters”, Journal of the Franklin Institute, 349, 3078-3101.
[56] Yu J., Pan W.Z., Ng T.W. (2008). “Bifurcation and chaos in the generalized Korteweg-de Vries equation under a harmonic excitation”, International Journal of Nonlinear Sciences and Numerical Simulation, 9, 37-40.
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