Dynamics analysis of fractional-order Hopfield neural networks

References

• 1. S. Zhang, Y. Yu and H. Wang, Mittag–Leffler stability of fractional-order Hopfield neural networks, Nonlinear Analysis: Hybrid Syst. 16 (2014) 104–121. Web of Science, Google Scholar
• 2. S. Hayman, The mcculloch-pitts model, International Joint Conference on Neural Networks (IJCNN’99), Vol. 6 (IEEE, 1999), pp. 4438–4439. Google Scholar
• 3. J. Hopfield, Neural networks and physical systems with emergent collective computational abilities, Proc. Natl. Acad. Sci. 79(8) (1982) 2554–2558. Web of Science, Google Scholar
• 4. H. Huang, Q. Du and X. Kang, Global exponential stability of neutral high-order stochastic Hopfield neural networks with Markovian jump parameters and mixed time delays, ISA Trans. 52(6) (2013) 759–767. Web of Science, Google Scholar
• 5. H.-P. Hu, J.-K. Wang and F.-L. Xie, Dynamics analysis of a new fractional-order Hopfield neural network with delay and its generalized projective synchronization, Entropy 21(1) (2019) 1. Web of Science, Google Scholar
• 6. J. Hopfield, Neurons with graded response have collective computational properties like those of two-state neurons, Proc. Natl. Acad. Sci. USA 81 (1984) 3088–3092. Web of Science, Google Scholar
• 7. H. Wang, Y. Yu, G. Wen, S. Zhang and J. Yu, Global stability analysis of fractional-order Hopfield neural networks with time delay, Neurocomputing 154 (2015) 15–23. Web of Science, Google Scholar
• 8. E. Kaslik and S. Sivasundaram, Nonlinear dynamics and chaos in fractional-order neural networks, Neural Network 32 (2012) 245–256. Web of Science, Google Scholar
• 9. J. L. Mata-Machuca and R. Aguilar-Lopez, Adaptative synchronization in multi-output fractional-order complex dynamical networks and secure communications, Eur. Phys. J. Plus 133(14) (2018) 14. Web of Science, Google Scholar
• 10. E. Tlelo-Cuautle, D. L. F. L. Gerardo, V. T. Pham, C. Volos, S. Jafari and A. D. J. Quintas-Valles, Dynamics, FPGA realization and application of a chaotic system with an infinite number of equilibrium points, Nonlinear Dyn. 89 (2017) 1129–1139. Web of Science, Google Scholar
• 11. C. H. Yang, Z. M. Ge, C. M. Chang and S. Y. Li, Chaos synchronization and chaos control of quantum-CNN chaotic system by variable structure control and impulse control, Nonlinear Anal. Real World Appl. 11 (2010) 977–1985. Web of Science, Google Scholar
• 12. L. P. Chen, Y. Chai and R. C. Wu, Linear matrix inequality criteria for robust synchronization of uncertain fractional-order chaotic systems, Chaos 21 (2011) 043107. Web of Science, Google Scholar
• 13. J. P. Singh and B. K. Roy, Second order adaptive time varying sliding mode control for synchronization of hidden chaotic orbits in a new uncertain 4-D conservative chaotic system, Trans. Inst. Meas. Control 40 (2018) 3573–3586. Web of Science, Google Scholar
• 14. J. M. Munoz-Pacheco, E. Zambrano-Serrano, C. Volos, S. Jafari, J. Kengne and K. Rajagopal, A new fractional-order chaotic system with different families of hidden and self-excited attractors, Entropy 20 (2018) 564. Web of Science, Google Scholar
• 15. B. Kagstrom, A perturbation analysis of the generalized Sylvester equation, Siam J. Matrix Anal. Appl. 15(4) (1994) 1045–1060. Web of Science, Google Scholar
• 16. M. Caputo, Linear models of dissipation whose Q is almost frequency independent: part II, Geophys. J. Int. 13 (1967) 529–539. Web of Science, Google Scholar
• 17. R. B. Albadarneh, I. M. Batiha and M. Zurigat, Numerical solutions for linear fractional differential equations of order $1<\alpha <2$ using finite difference method (FFDM), J. Math. Computer Sci. 16(1) (2016) 103–111. Web of Science, Google Scholar
• 18. I. Podlubny, Fractional Differential Equations (Academic Press, San Diego, USA, 1999). Google Scholar
• 19. I. M. Batiha, R. El-Khazali, A. AlSaedi and S. Momani, The general solution of singular fractional-order linear time-invariant continuous systems with regular pencils, Entropy Dyn. Syst. 20(6) (2018) 400. Google Scholar
• 20. R. B. Albadarneh, M. Zerqat and I. M. Batiha, Numerical solutions for linear and non-linear fractional differential equations, Int. J. Pure Appl. Math. 106(3) (2016) 859–871. Google Scholar
• 21. K. Sun, Chaotic Secure Communication (Tsinghua University Press and Walter de Gruyter GmbH, Germany, 2016). Google Scholar
• 22. D. Kai, An algorithm for the numerical solution of differential equations of fractional order, Electron. Trans. Numer. Anal. 5 (1998) 1–6. Google Scholar
• 23. D. Kai, N. J. Ford and A. D. Freed, A predictor-corrector approach for the numerical solution of fractional differential equations, Nonlinear Dyn. 29 (2002) 3–22. Web of Science, Google Scholar
• 24. D. Kai and N. J. Ford, Analysis of fractional differential equations, J. Math. Anal. Appl. 265(2) (2002) 229–48. Web of Science, Google Scholar
• 25. M. Danca and N. Kuznetsov, Matlab code for Lyapunov exponents of fractional order systems, Int. J. Bifurc. Chaos 28(5) (2018) 1850067. Link, Web of Science, Google Scholar
• 26. M. Tavazoei and M. Haeri, A proof for non existence of periodic solutions in time invariant fractional order systems, Automatica 45 (2009) 1886–1890. Web of Science, Google Scholar
• 27. M.-F. Danca, M. Fečkan, N. Kuznetsov and G. Chen, Complex dynamics, hidden attractors and continuous approximation of a fractional-order hyperchaotic PWC system, Nonlinear Dyn. 91 (2018) 2523–2540. Web of Science, Google Scholar
• 28. K. Diethelm and N. Ford, Analysis of fractional differential equations, J. Math. Anal. Appl. 265 (2002) 229–248. Web of Science, Google Scholar
• 29. C. Li, Z. Gong, D. Qian and Y. Chen, On the bound of the Lyapunov exponents for the fractional differential systems, Chaos 20 (2010) 013127. Web of Science, Google Scholar
• 30. Z. Momani, M. Al Shridah, O. A. Arqub, M. Al-Momani and S. Momani, Modeling and analyzing neural networks using reproducing Kernel Hilbert space algorithm, Appl. Math. Inf. Sci. 12(1) (2018) 89–99. Google Scholar
• 31. C. F. Barenghi, Introduction to Chaos: Theoretical and Numerical Methods (Create Space Independent Publishing Platform, California, 2014). Google Scholar
• 32. R. Agarwal, D. O’Regan, S. Hristova and M. Cicek, Practical stability with respect to initial time difference for Caputo fractional differential equations, Commun. Nonlinear Sci. Numer. Simul. 42 (2017) 106–120. Web of Science, Google Scholar
• 33. Y. Li, Y. Chen and I. Podlubny, Mittag–Leffler stability of fractional order nonlinear dynamic systems, Automatica 45(8) (2009) 1965–1969. Web of Science, Google Scholar
• 34. R. Agarwal, S. Hristova and D. O’Regan, Applications of lyapunov functions to caputo fractional differential equations, Mathematics 6(11) (2018) 229. Web of Science, Google Scholar
• 35. B. Lundstrom, M. Higgs and W. Spain, Fractional differentiation by neocortical pyramidal neurons, Nature Neurosci. 11(11) (2008) 1335–1342. Web of Science, Google Scholar
• 36. R. Wu, X. Hei and L. Chen, Finite-time stability of fractional-order neural networks with delay, Commun. Theor. Phys. (Beijing) 60 (2013) 189–193. Web of Science, Google Scholar
• 37. J. Yu, C. Hu and H. Jiang, $\alpha$-stability and $\alpha$-synchronization for fractional-order neural networks, Neural Netw. 5 (2012) 82–87. Web of Science, Google Scholar
• 38. Y. Li, Y. Chen and I. Podlubny, Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag–Leffler stability, Comput. Math. Appl. 59(5) (2010) 1810–1821. Web of Science, Google Scholar
• 39. C. Liu, Y. He and B. Zhang, Design of BLDCM emulator for transmission control units, AIP Conf. Proc. 1955 (2018) 040067. Google Scholar
• 40. A. Coronel-Escamilla, J. F. Gómez-Aguilar, D. Baleanu, T. Córdova-Fraga, R. F. Escobar-Jiménez, V. H. Olivares-Peregrino and M. M. Al Qurashi, Bateman–Feshbach Tikochinsky and Caldirola–Kanai Oscillators with new fractional differentiation, Entropy 19 (2017) 55. Web of Science, Google Scholar