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

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  • 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.

    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 $\tau = 0.14 < \tau_0 = 0.15$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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