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September  2020, 13(9): 2537-2559. doi: 10.3934/dcdss.2020137

Hopf bifurcation of a fractional-order octonion-valued neural networks with time delays

 Department of Mathematics, Bharathiar University, Coimbatore-641 046, Tamil Nadu, India

* Corresponding author: rakkigru@gmail.com

Received  November 2018 Revised  March 2019 Published  September 2020 Early access  November 2019

In this paper, the hopf bifurcation of a fractional-order octonion-valued neural networks with time delays is investigated. With this constructed model all the parameters would belong to the normed division algebra of octonians. Because of the non-commutativity of the octonians, the fractional-order octonian-valued neural networks can be decomposed into four-dimensional real-valued neural networks. Furthermore, the conditions for the occurrence of Hopf bifurcation for the considered model are firstly given by taking time delay as a bifurcation parameter. Also we investigate their bifurcation when the system loses its stability. Finally, we give one numerical simulation to verify the effectiveness of the our proposed method.

Citation: Udhayakumar Kandasamy, Rakkiyappan Rajan. Hopf bifurcation of a fractional-order octonion-valued neural networks with time delays. Discrete & Continuous Dynamical Systems - S, 2020, 13 (9) : 2537-2559. doi: 10.3934/dcdss.2020137
References:
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C. Ho and J. Cao, Finite-time stability and settling-time estimation of nonlinear impulsive systems, Automatica J. IFAC, 99 (2019), 361-368.  doi: 10.1016/j.automatica.2018.10.024.  Google Scholar [16] X. Li and S. Song, Stabilization of delay systems: Delay-dependent impulsive control, IEEE Trans. Automat. Control, 62 (2017), 406-411.  doi: 10.1109/TAC.2016.2530041.  Google Scholar [17] X. Li and J. Cao, An impulsive delay inequality involving unbounded time-varying delay and applications, IEEE Trans. Automat. Control, 62 (2017), 3618-3625.  doi: 10.1109/TAC.2017.2669580.  Google Scholar [18] X. Li and J. Wu, Stability of nonlinear differential systems with state-dependent delayed impulses, Automatica J. IFAC, 64 (2016), 63-69.  doi: 10.1016/j.automatica.2015.10.002.  Google Scholar [19] X. Li and S. Song, Impulsive control for existence, uniqueness, and global stability of periodic solutions of recurrent neural networks with discrete and continuously distributed delays, IEEE Trans. Neural Netw. Learn. Syst., 24 (2013), 868-877.   Google Scholar [20] L. Li, Z. Wang, Y. Li, H. Shen and J. Lu, Hopf bifurcation analysis of a complex-valued neural network model with discrete and distributed delays, Appl. Math. Comput., 330 (2018), 152-169.  doi: 10.1016/j.amc.2018.02.029.  Google Scholar [21] C.-A. Popa, Global exponential stability of neutral-type octonion-valued neural networks with time-varying delays, Neurocomputing, 309 (2018), 117-133.  doi: 10.1016/j.neucom.2018.05.004.  Google Scholar [22] C.-A. Popa, Global exponential stability of octonion-valued neural networks with leakage delay and mixed delays, Neural Netw., 105 (2018), 277-293.   Google Scholar [23] R. Rakkiyappan, G. Velmurugan and J. Cao, Stability analysis of fractional-order complex-valued neural networks with time delays, Chaos, Solitons Fract., 78 (2015), 297-316.  doi: 10.1016/j.chaos.2015.08.003.  Google Scholar [24] R. Rakkiyappan, J. Cao and G. Velmurugan, Existence and uniform stability analysis of fractional-order complex-valued neural networks with time delays, IEEE Trans. Neural Netw. Learn. Syst., 26 (2015), 84-97.  doi: 10.1109/TNNLS.2014.2311099.  Google Scholar [25] R. Rakkiyappan, G. Velmurugan, F. A. Rihan and S. Lakshmanan, Stability analysis of memristor-based complex-valued recurrent neural networks with time delays, Complexity, 21 (2015), 14-39.  doi: 10.1002/cplx.21618.  Google Scholar [26] F. A. Rihan, S. Lakshmanan, A. H. Hashish, R. Rakkiyappan and E. Ahmed, Fractional-order delayed predator-prey systems with Holling type-Ⅱ functional response, Nonlinear Dyn., 80 (2015), 777-789.  doi: 10.1007/s11071-015-1905-8.  Google Scholar [27] L. Wang and X. Li, µ-stability of impulsive differential systems with unbounded time-varying delays and nonlinear perturbations, Math. Methods Appl. Sci., 36 (2013), 1440-1446.  doi: 10.1002/mma.2696.  Google Scholar [28] C. Wangn and T.-Z. Xu, Stability of the nonlinear fractional differential equations with the right-sided riemann-liouville fractional derivative, Discrete Contin. Dyn. Syst. Ser. S, 10 (2017), 505-521.  doi: 10.3934/dcdss.2017025.  Google Scholar [29] F. X. Wu, Stability and bifurcation of ring-structured genetic regulatory networks with time delays, IEEE Trans. Circuits Syst. I. Regul. Pap., 59 (2012), 1312-1320.  doi: 10.1109/TCSI.2011.2173385.  Google Scholar [30] M. Xiao, W. X. Zheng, G. Jiang and J. Cao, Undamped oscillations generated by Hopf bifurcations in fractional-order recurrent neural networks with Caputo derivative, IEEE Trans. Neural Netw. Learn. Syst., 26 (2015), 3201-3214.  doi: 10.1109/TNNLS.2015.2425734.  Google Scholar [31] W. Xu, J. Cao, M. Xiao, D. W. Ho and G. Wen, A new framework for analysis on stability and bifurcation in a class of neural networks with discrete and distributed delays, IEEE Trans. Cybern., 45 (2015), 2224-2236.  doi: 10.1109/TCYB.2014.2367591.  Google Scholar

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References:
 [1] P. Arena, R. Caponetto, L. Fortuna and D. Porto, Bifurcation and chaos in non-integer order cellular neural networks, Int. J. Bifurc. Chaos Appl. Sci. Eng., 8 (1998), 1527-1539.   Google Scholar [2] J. Cao and M. Xiao, Stability and Hopf bifurcation in a simplified BAM neural network with two time delays, IEEE Trans. Neural Netw., 18 (2007), 416-430.   Google Scholar [3] L. Chen, Y. Chai, R. Wu, T. Ma and H. Zhai, Dynamic analysis of a class of fractional-order neural networks with delay, Neurocomputing, 111 (2013), 190-194.  doi: 10.1016/j.neucom.2012.11.034.  Google Scholar [4] Z. Cheng, K. Xie, T. Wang and J. Cao, Stability and Hopf bifurcation of three-triangle neural networks with delays, Neurocomputing, 322 (2018), 206-215.  doi: 10.1016/j.neucom.2018.09.063.  Google Scholar [5] T. Dray and C. Manogue, The Geomerty of the Octonions, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2015. doi: 10.1142/8456.  Google Scholar [6] H. Hu and L. Huang, Stability and Hopf bifurcation analysis on a ring of four neurons with delays, Appl. Math. Comput., 213 (2009), 587-599.  doi: 10.1016/j.amc.2009.03.052.  Google Scholar [7] C. Huang, J. Cao, M. Xiao, A. Alsaedi and T. Hayat, Effects of time delays on stability and Hopf bifurcation in a fractional ring-structured network with arbitrary neurons, Commun. Nonlinear Sci. Numer. Simul., 57 (2018), 1-13.  doi: 10.1016/j.cnsns.2017.09.005.  Google Scholar [8] C. Huang, J. Cao, M. Xiao, A. Alsaedi and T. Hayat, Bifurcations in a delayed fractional complex-valued neural network, Appl. Math. Comput., 292 (2017), 210-227.  doi: 10.1016/j.amc.2016.07.029.  Google Scholar [9] C. Huang, J. Cao and Z. Ma, Delay-induced bifurcation in a tri-neuron fractional neural network, Internat. J. Systems Sci., 47 (2016), 3668-3677.  doi: 10.1080/00207721.2015.1110641.  Google Scholar [10] H. Huang and J. Cao, Impact of leakage delay on bifurcation in high-order fractional BAM neural networks, Neural Netw., 98 (2018), 223-235.  doi: 10.1016/j.neunet.2017.11.020.  Google Scholar [11] B. Karaagac, New exact solutions for some fractional order differential equations via improved sub-equation method, Discrete Contin. Dyn. Syst. Ser. S, 12 (2019), 447-454.   Google Scholar [12] E. Kaslik and I. R. Rădulescu, Dynamics of complex-valued fractional-order neural networks, Neural Netw., 89 (2017), 39-49.  doi: 10.1016/j.neunet.2017.02.011.  Google Scholar [13] A. Kilbas, H. Srivastava and J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204, Elsevier Science B.V., Amsterdam, 2006.  Google Scholar [14] I. Koca, Numerical analysis of coupled fractional differential equations with atangana-baleanu fractional derivative, Discrete Contin. Dyn. Syst. Ser. S, 12 (2019), 475-486.  doi: 10.3934/dcdss.2019031.  Google Scholar [15] X. Li, D. W. C. Ho and J. Cao, Finite-time stability and settling-time estimation of nonlinear impulsive systems, Automatica J. IFAC, 99 (2019), 361-368.  doi: 10.1016/j.automatica.2018.10.024.  Google Scholar [16] X. Li and S. Song, Stabilization of delay systems: Delay-dependent impulsive control, IEEE Trans. Automat. Control, 62 (2017), 406-411.  doi: 10.1109/TAC.2016.2530041.  Google Scholar [17] X. Li and J. Cao, An impulsive delay inequality involving unbounded time-varying delay and applications, IEEE Trans. Automat. Control, 62 (2017), 3618-3625.  doi: 10.1109/TAC.2017.2669580.  Google Scholar [18] X. Li and J. Wu, Stability of nonlinear differential systems with state-dependent delayed impulses, Automatica J. IFAC, 64 (2016), 63-69.  doi: 10.1016/j.automatica.2015.10.002.  Google Scholar [19] X. Li and S. Song, Impulsive control for existence, uniqueness, and global stability of periodic solutions of recurrent neural networks with discrete and continuously distributed delays, IEEE Trans. Neural Netw. Learn. Syst., 24 (2013), 868-877.   Google Scholar [20] L. Li, Z. Wang, Y. Li, H. Shen and J. Lu, Hopf bifurcation analysis of a complex-valued neural network model with discrete and distributed delays, Appl. Math. Comput., 330 (2018), 152-169.  doi: 10.1016/j.amc.2018.02.029.  Google Scholar [21] C.-A. Popa, Global exponential stability of neutral-type octonion-valued neural networks with time-varying delays, Neurocomputing, 309 (2018), 117-133.  doi: 10.1016/j.neucom.2018.05.004.  Google Scholar [22] C.-A. Popa, Global exponential stability of octonion-valued neural networks with leakage delay and mixed delays, Neural Netw., 105 (2018), 277-293.   Google Scholar [23] R. Rakkiyappan, G. Velmurugan and J. Cao, Stability analysis of fractional-order complex-valued neural networks with time delays, Chaos, Solitons Fract., 78 (2015), 297-316.  doi: 10.1016/j.chaos.2015.08.003.  Google Scholar [24] R. Rakkiyappan, J. Cao and G. Velmurugan, Existence and uniform stability analysis of fractional-order complex-valued neural networks with time delays, IEEE Trans. Neural Netw. Learn. Syst., 26 (2015), 84-97.  doi: 10.1109/TNNLS.2014.2311099.  Google Scholar [25] R. Rakkiyappan, G. Velmurugan, F. A. Rihan and S. Lakshmanan, Stability analysis of memristor-based complex-valued recurrent neural networks with time delays, Complexity, 21 (2015), 14-39.  doi: 10.1002/cplx.21618.  Google Scholar [26] F. A. Rihan, S. Lakshmanan, A. H. Hashish, R. Rakkiyappan and E. Ahmed, Fractional-order delayed predator-prey systems with Holling type-Ⅱ functional response, Nonlinear Dyn., 80 (2015), 777-789.  doi: 10.1007/s11071-015-1905-8.  Google Scholar [27] L. Wang and X. Li, µ-stability of impulsive differential systems with unbounded time-varying delays and nonlinear perturbations, Math. Methods Appl. Sci., 36 (2013), 1440-1446.  doi: 10.1002/mma.2696.  Google Scholar [28] C. Wangn and T.-Z. Xu, Stability of the nonlinear fractional differential equations with the right-sided riemann-liouville fractional derivative, Discrete Contin. Dyn. Syst. Ser. S, 10 (2017), 505-521.  doi: 10.3934/dcdss.2017025.  Google Scholar [29] F. X. Wu, Stability and bifurcation of ring-structured genetic regulatory networks with time delays, IEEE Trans. Circuits Syst. I. Regul. Pap., 59 (2012), 1312-1320.  doi: 10.1109/TCSI.2011.2173385.  Google Scholar [30] M. Xiao, W. X. Zheng, G. Jiang and J. Cao, Undamped oscillations generated by Hopf bifurcations in fractional-order recurrent neural networks with Caputo derivative, IEEE Trans. Neural Netw. Learn. Syst., 26 (2015), 3201-3214.  doi: 10.1109/TNNLS.2015.2425734.  Google Scholar [31] W. Xu, J. Cao, M. Xiao, D. W. Ho and G. Wen, A new framework for analysis on stability and bifurcation in a class of neural networks with discrete and distributed delays, IEEE Trans. Cybern., 45 (2015), 2224-2236.  doi: 10.1109/TCYB.2014.2367591.  Google Scholar
The waveform and the phase digram of real and imaginary parts of $v_0(t)$ when $\tau = 0.14 < \tau_0 = 0.15$
The waveform and the phase digram of real and imaginary parts of $v_0(t)$ when $\tau = 0.39 > \tau_0 = 0.15$
The waveform and the phase digram of real and imaginary parts of $w_0(t)$ when $\tau = 0.14 < \tau_0 = 0.15$
The waveform and the phase digram of real and imaginary parts of $w_0(t)$ when $\tau = 0.39 > \tau_0 = 0.15$
The waveform and the phase digram of real and imaginary parts of $v_1(t)$ when $\tau = 0.14 < \tau_0 = 0.15$
The waveform and the phase digram of real and imaginary parts of $v_1(t)$ when $\tau = 0.39 > \tau_0 = 0.15$
The waveform and the phase digram of real and imaginary parts of $w_1(t)$ when $\tau = 0.14 < \tau_0 = 0.15$
The waveform and the phase digram of real and imaginary parts of $w_1(t)$ when $\tau = 0.39 > \tau_0 = 0.15$
The waveform and the phase digram of real and imaginary parts of $v_2(t)$ when $\tau = 0.14 < \tau_0 = 0.15$
The waveform and the phase digram of real and imaginary parts of $v_2(t)$ when $\tau = 0.39 > \tau_0 = 0.15$
The waveform and the phase digram of real and imaginary parts of $w_2(t)$ when $\tau = 0.14 < \tau_0 = 0.15$
The waveform and the phase digram of real and imaginary parts of $w_2(t)$ when $\tau = 0.39 > \tau_0 = 0.15$
The waveform and the phase digram of real and imaginary parts of $v_3(t)$ when $\tau = 0.14 < \tau_0 = 0.15$
The waveform and the phase digram of real and imaginary parts of $v_3(t)$ when $\tau = 0.39 > \tau_0 = 0.15$
The waveform and the phase digram of real and imaginary parts of $w_3(t)$ when $\tau = 0.14 < \tau_0 = 0.15$
The waveform and the phase digram of real and imaginary parts of $w_3(t)$ when $\tau = 0.39 > \tau_0 = 0.15$
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