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

Udhayakumar Kandasamy; Rakkiyappan Rajan

doi:10.3934/dcdss.2020137

Article Contents

2020, Volume 13, Issue 9: 2537-2559. Doi: 10.3934/dcdss.2020137

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

Udhayakumar Kandasamy and
Rakkiyappan Rajan^,

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

^* Corresponding author: rakkigru@gmail.com
^* Corresponding author: rakkigru@gmail.com

Received: October 31, 2018

Revised: February 28, 2019

Early access: November 2019

Published: June 2020

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Abstract

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.

Keywords:

Mathematics Subject Classification: 92B20, 34D23, 34C23.

Citation:

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Figure 1. The waveform and the phase digram of real and imaginary parts of $v_0(t)$ when $\tau = 0.14 < \tau_0 = 0.15$

Download: Full-size image PowerPoint slide

Figure 2. The waveform and the phase digram of real and imaginary parts of $v_0(t)$ when $\tau = 0.39 > \tau_0 = 0.15$

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Figure 3. The waveform and the phase digram of real and imaginary parts of $w_0(t)$ when $\tau = 0.14 < \tau_0 = 0.15$

Download: Full-size image PowerPoint slide

Figure 4. The waveform and the phase digram of real and imaginary parts of $w_0(t)$ when $\tau = 0.39 > \tau_0 = 0.15$

Download: Full-size image PowerPoint slide

Figure 5. The waveform and the phase digram of real and imaginary parts of $v_1(t)$ when $\tau = 0.14 < \tau_0 = 0.15$

Download: Full-size image PowerPoint slide

Figure 6. The waveform and the phase digram of real and imaginary parts of $v_1(t)$ when $\tau = 0.39 > \tau_0 = 0.15$

Download: Full-size image PowerPoint slide

Figure 7. The waveform and the phase digram of real and imaginary parts of $w_1(t)$ when $\tau = 0.14 < \tau_0 = 0.15$

Download: Full-size image PowerPoint slide

Figure 8. The waveform and the phase digram of real and imaginary parts of $w_1(t)$ when $\tau = 0.39 > \tau_0 = 0.15$

Download: Full-size image PowerPoint slide

Figure 9. The waveform and the phase digram of real and imaginary parts of $v_2(t)$ when $\tau = 0.14 < \tau_0 = 0.15$

Download: Full-size image PowerPoint slide

Figure 10. The waveform and the phase digram of real and imaginary parts of $v_2(t)$ when $\tau = 0.39 > \tau_0 = 0.15$

Download: Full-size image PowerPoint slide

Figure 11. The waveform and the phase digram of real and imaginary parts of $w_2(t)$ when $\tau = 0.14 < \tau_0 = 0.15$

Download: Full-size image PowerPoint slide

Figure 12. The waveform and the phase digram of real and imaginary parts of $w_2(t)$ when $\tau = 0.39 > \tau_0 = 0.15$

Download: Full-size image PowerPoint slide

Figure 13. The waveform and the phase digram of real and imaginary parts of $v_3(t)$ when $\tau = 0.14 < \tau_0 = 0.15$

Download: Full-size image PowerPoint slide

Figure 14. The waveform and the phase digram of real and imaginary parts of $v_3(t)$ when $\tau = 0.39 > \tau_0 = 0.15$

Download: Full-size image PowerPoint slide

Figure 15. The waveform and the phase digram of real and imaginary parts of $w_3(t)$ when $\tau = 0.14 < \tau_0 = 0.15$

Download: Full-size image PowerPoint slide

Figure 16. The waveform and the phase digram of real and imaginary parts of $w_3(t)$ when $\tau = 0.39 > \tau_0 = 0.15$

Download: Full-size image PowerPoint slide

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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.
[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.
[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.
[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.
[5]	T. Dray and C. Manogue, The Geomerty of the Octonions, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2015. doi: 10.1142/8456.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[22]	C.-A. Popa, Global exponential stability of octonion-valued neural networks with leakage delay and mixed delays, Neural Netw., 105 (2018), 277-293.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.