# American Institute of Mathematical Sciences

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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  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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##### References:
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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