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Coexisting multi-stability of Hopfield neural network based on coupled fractional-order locally active memristor and its application in image encryption

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Abstract

Locally active memristor has become a research hotspot, and plays an important role in the research of neural networks. In order to study the dynamic behavior of synaptic crosstalk, coupled fractional-order locally active memristor is proposed to simulate the phenomenon of synaptic crosstalk in Hopfield neural network (HNN). It is found that the HNN model with coupled locally active memristor has multi-stability under different fractional order and coupling coefficient by phase diagram, bifurcation diagram, Lyapunov exponents, and attraction basin. Moreover, special phenomenon such as transient chaos is also found. Furthermore, the proposed memristive HNN model is implemented based on ARM platform by microcomputer, and the experimental results are in good agreement with the numerical simulation. Finally, based on the memristive HNN model, a color image encryption scheme based on DNA encoding and chaotic sequence is proposed, which has great keyspace and good encryption effect, and we implement encryption scheme in ARM platform by microcomputer.

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Appendix

Appendix

ADM for HNN based on a coupled locally active memristor.

The system (7) is rewritten as follows for facilitating the calculation.

$$ \left\{ \begin{gathered} D_{t}^{q} x = - x + w_{11} \tanh (x) + w_{12} \tanh (y) - k_{2} (a_{2} - b_{2} \tanh (w) + c_{2} \tanh (u))\tanh (z) \hfill \\ D_{t}^{q} y = - y + w_{21} \tanh (x) + w_{22} \tanh (y) + w_{23} \tanh (z) \hfill \\ D_{t}^{q} z = - z + k_{1} (a_{1} - b_{1} \tan (u) + c_{1} \tanh (w))\tanh (x) + w_{32} \tanh (y) + w_{33} \tanh (z) \hfill \\ D_{t}^{q} u = - u^{3} + u + \tanh (x) \hfill \\ D_{t}^{q} w = - w^{3} + w + \tanh (z) \hfill \\ \end{gathered} \right. $$
(28)

Let \(c_{10} = x(t_{0} ),c_{20} = y(t_{0} ),c_{30} = z(t_{0} ),c_{40} = u(t_{0} ),c_{50} = w(t_{0} )\), where \((x(t_{0} ),y(t_{0} ),z(t_{0} ),u(t_{0} ),w(t_{0} ))\) are the initial values of fractional-order HNN model.

$$ \left\{ \begin{gathered} c_{11} = - c_{10} + w_{11} \tanh (c_{10} ) + w_{12} \tanh (c_{20} ) - k_{2} (a_{2} - b_{2} \tanh (c_{50} ) + c_{2} \tanh (c_{40} ))\tanh (c_{30} ) \hfill \\ c_{21} = - c_{20} + w_{21} \tanh (c_{10} ) + w_{22} \tanh (c_{20} ) + w_{23} \tanh (c_{30} ) \hfill \\ c_{31} = - c_{30} + k_{1} (a_{1} - b_{1} \tanh (c_{40} ) + c_{1} \tanh (c_{50} ))\tanh (c_{10} ) + w_{32} \tanh (c_{20} ) + w_{33} \tanh (c_{30} ) \hfill \\ c_{41} = - c_{40}^{3} + c_{40} + \tanh (c_{10} ) \hfill \\ c_{51} = - c_{50}^{3} + c_{50} + \tanh (c_{30} ) \hfill \\ \end{gathered} \right. $$
(29)
$$ \left\{ \begin{gathered} c_{12} = - c_{11} + w_{11} ( - c_{11} (\tanh (c_{10} )^{2} - 1)) + w_{12} ( - w_{21} (\tanh (c_{20} )^{2} - 1)) - k_{2} a_{2} ( - c_{31} (\tanh (c_{30} )^{2} - 1)) \hfill \\ + k_{2} b_{2} ( - c_{31} \tanh (c_{50} )(\tanh (c_{30} )^{2} - 1) - c_{51} \tanh (c_{30} )(\tanh (c_{50} )^{2} - 1)) - k_{2} c_{2} ( - c_{31} \tanh (c_{40} )(\tanh (c_{30} )^{2} - 1) \hfill \\ - c_{41} \tanh (c_{30} )(\tanh (c_{40} )^{2} - 1)) \hfill \\ c_{22} = - c_{21} + w_{21} ( - c_{11} \tanh (c_{10} )^{2} - 1) + w_{22} ( - c_{21} (\tanh (c_{20} )^{2} - 1)) + w_{23} ( - c_{31} (\tanh (c_{30} )^{2} - 1)) \hfill \\ c_{32} = - c_{31} + k_{1} a_{1} ( - c_{11} (\tanh (c_{10} )^{2} - 1)) - k_{1} b_{1} ( - c_{11} \tanh (c_{40} )(\tanh (c_{10} )^{2} - 1) - c_{41} \tanh (c_{10} )(\tanh (c_{30} )^{2} - 1)) \hfill \\ + k_{1} c_{1} ( - c_{11} \tanh (c_{50} )(\tanh (c_{10} )^{2} - 1) - c_{51} \tanh (c_{10} )(\tanh (c_{50} )^{2} - 1)) + w_{32} ( - c_{21} (\tanh (c_{20} )^{2} - 1)) \hfill \\ + w_{33} ( - c_{31} (\tanh (c_{30} )^{2} - 1)) \hfill \\ c_{42} = - 3c_{40}^{2} c_{41} + c_{41} - c_{11} (\tanh (c_{10} )^{2} - 1) \hfill \\ c_{52} = - 3c_{50}^{2} c_{51} + c_{51} - c_{31} (\tanh (c_{30} )^{2} - 1) \hfill \\ \end{gathered} \right. $$
(30)
$$ \left\{ \begin{gathered} c_{13} = - c_{12} + w_{11} (c_{11}^{2} \tanh (c_{10} )(\tanh (c_{10} )^{2} - 1)\frac{\Gamma (2q + 1)}{{\Gamma^{2} (q + 1)}} - c_{12} (\tanh (c_{10} )^{2} - 1)) + w_{12} (c_{21}^{2} \tanh (c_{20} )(\tanh (c_{20} )^{2} - 1)\frac{\Gamma (2q + 1)}{{\Gamma^{2} (q + 1)}} \hfill \\ - c_{22} (\tanh (c_{20} )^{2} - 1)) - k_{2} a_{2} (c_{31}^{2} \tanh (c_{30} )(\tanh (c_{30} )^{2} - 1)\frac{\Gamma (2q + 1)}{{\Gamma^{2} (q + 1)}} - c_{32} (\tanh (c_{30} )^{2} - 1)) + \hfill \\ k_{2} b_{2} (c_{31} c_{51} (\tanh (c_{30} )^{2} - 1)(\tanh (c_{50} )^{2} - 1)\frac{\Gamma (2q + 1)}{{\Gamma^{2} (q + 1)}} - c_{52} \tanh (c_{30} )(\tanh (c_{50} )^{2} - 1) - c_{32} \tanh (c_{50} )(\tanh (c_{30} )^{2} - 1) + \hfill \\ c_{31}^{2} \tanh (c_{30} )\tanh (c_{50} )(\tanh (c_{30} )^{2} - 1)\frac{\Gamma (2q + 1)}{{\Gamma^{2} (q + 1)}} + c_{51} \tanh (c_{30} )\tanh (c_{50} )(\tanh (c_{50} )^{2} - 1)\frac{\Gamma (2q + 1)}{{\Gamma^{2} (q + 1)}}) - \hfill \\ k_{2} c_{2} (c_{31} c_{41} (\tanh (c_{30} )^{2} - 1)(\tanh (c_{40} )^{2} - 1)\frac{\Gamma (2q + 1)}{{\Gamma^{2} (q + 1)}} - c_{42} \tanh (_{{}} c_{30} )(\tanh (c_{40} )^{2} - 1) - c_{32} \tanh (c_{40} )(\tanh (c_{30} )^{2} - 1) \hfill \\ + c_{31}^{2} \tanh (c_{30} )\tanh (c_{40} )(\tanh (c_{30} )^{2} - 1)\frac{\Gamma (2q + 1)}{{\Gamma^{2} (q + 1)}} + c_{41}^{2} \tanh (c_{30} )\tanh (c_{40} )(\tanh (c_{30} )^{2} - 1)\frac{\Gamma (2q + 1)}{{\Gamma^{2} (q + 1)}}) \hfill \\ c_{23} = - c_{22} + w_{21} (c_{11}^{2} \tanh (c_{10} )(\tanh (c_{10} )^{2} - 1)\frac{\Gamma (2q + 1)}{{\Gamma^{2} (q + 1)}} - c_{12} (\tanh (c_{10} )^{2} - 1)) + w_{22} (c_{21}^{2} \tanh (c_{20} )(\tanh (c_{20} )^{2} - 1)\frac{\Gamma (2q + 1)}{{\Gamma^{2} (q + 1)}} \hfill \\ - c_{22} (\tanh (c_{20} )^{2} - 1)) + w_{23} (c_{31}^{2} \tanh (c_{30} )(\tanh (c_{30} )^{2} - 1)\frac{\Gamma (2q + 1)}{{\Gamma^{2} (q + 1)}} - c_{32} (\tanh (c_{30} )^{2} - 1)) \hfill \\ c_{33} = - c_{32} + k_{1} a_{1} (c_{11}^{2} \tanh (c_{10} )(\tanh (c_{10} )^{2} - 1)\frac{\Gamma (2q + 1)}{{\Gamma^{2} (q + 1)}} - c_{12} (\tanh (c_{10} )^{2} - 1)) - k_{1} b_{1} (c_{11} c_{41} (\tanh (c_{10} )^{2} - 1)(\tanh (c_{40} )^{2} - 1)\frac{\Gamma (2q + 1)}{{\Gamma^{2} (q + 1)}} \hfill \\ - c_{42} \tanh (c_{10} )(\tanh (c_{40} )^{2} - 1) - c_{12} \tanh (c_{40} )(\tanh (c_{10} )^{2} - 1) + c_{11}^{2} \tanh (c_{10} )\tanh (c_{40} )(\tanh (c_{10} )^{2} - 1)\frac{\Gamma (2q + 1)}{{\Gamma^{2} (q + 1)}} \hfill \\ + c_{41}^{2} \tanh (c_{10} )\tanh (c_{40} )(\tanh (c_{40} )^{2} - 1)\frac{\Gamma (2q + 1)}{{\Gamma^{2} (q + 1)}}) + k_{1} c_{1} (c_{11} c_{51} (\tanh (c_{10} )^{2} - 1)(\tanh (c_{50} )^{2} - 1)\frac{\Gamma (2q + 1)}{{\Gamma^{2} (q + 1)}} - \hfill \\ c_{52} \tanh (c_{10} )(\tanh (c_{50} )^{2} - 1) - c_{12} \tanh (c_{50} )(\tanh (c_{10} )^{2} - 1) + c_{11}^{2} \tanh (c_{10} )\tanh (c_{50} )(\tanh (c_{10} )^{2} - 1)\frac{\Gamma (2q + 1)}{{\Gamma^{2} (q + 1)}} + \hfill \\ c_{51}^{2} \tanh (c_{10} )\tanh (c_{50} )(\tanh (c_{50} )^{2} - 1)\frac{\Gamma (2q + 1)}{{\Gamma^{2} (q + 1)}}) + w_{32} (c_{12}^{2} \tanh (c_{20} )(\tanh (c_{20} )^{2} - 1)\frac{\Gamma (2q + 1)}{{\Gamma^{2} (q + 1)}} - c_{22} (\tanh (c_{20} )^{2} - 1)) + \hfill \\ w_{33} (c_{31}^{2} \tanh (c_{30} )(\tanh (c_{30} )^{2} - 1)\frac{\Gamma (2q + 1)}{{\Gamma^{2} (q + 1)}} - c_{32} (\tanh (c_{30} )^{2} - 1)) \hfill \\ c_{43} = - 3c_{42} c_{40}^{2} - 3c_{40} c_{41}^{2} + c_{42} + c_{11}^{2} \tanh (c_{10} )(\tanh (c_{10} )^{2} - 1)\frac{\Gamma (2q + 1)}{{\Gamma^{2} (q + 1)}} - c_{12} (\tanh (c_{10} )^{2} - 1) \hfill \\ c_{53} = - 3c_{52} c_{50}^{2} - 3c_{50} c_{51}^{2} + c_{52} + c_{31}^{2} \tanh (c_{30} )(\tanh (c_{30} )^{2} - 1)\frac{\Gamma (2q + 1)}{{\Gamma^{2} (q + 1)}} - c_{32} (\tanh (c_{30} )^{2} - 1) \hfill \\ \end{gathered} \right. $$
(31)
$$ \left\{ \begin{array}{l} \tilde x(t) = {c_{10}} + {c_{11}}\frac{{{{(t - {t_0})}^q}}}{{\Gamma (q + 1)}} + {c_{12}}\frac{{{{(t - {t_0})}^{2q}}}}{{\Gamma (2q + 1)}} + {c_{13}}\frac{{{{(t - {t_0})}^{3q}}}}{{\Gamma (3q + 1)}}\\ \tilde y(t) = {c_{20}} + {c_{21}}\frac{{{{(t - {t_0})}^q}}}{{\Gamma (q + 1)}} + {c_{22}}\frac{{{{(t - {t_0})}^{2q}}}}{{\Gamma (2q + 1)}} + {c_{23}}\frac{{{{(t - {t_0})}^{3q}}}}{{\Gamma (3q + 1)}}\\ \tilde z(t) = {c_{30}} + {c_{31}}\frac{{{{(t - {t_0})}^q}}}{{\Gamma (q + 1)}} + {c_{32}}\frac{{{{(t - {t_0})}^{2q}}}}{{\Gamma (2q + 1)}} + {c_{33}}\frac{{{{(t - {t_0})}^{3q}}}}{{\Gamma (3q + 1)}}\\ \tilde u(t) = {c_{40}} + {c_{41}}\frac{{{{(t - {t_0})}^q}}}{{\Gamma (q + 1)}} + {c_{42}}\frac{{{{(t - {t_0})}^{2q}}}}{{\Gamma (2q + 1)}} + {c_{43}}\frac{{{{(t - {t_0})}^{3q}}}}{{\Gamma (3q + 1)}}\\ \tilde w(t) = {c_{50}} + {c_{51}}\frac{{{{(t - {t_0})}^q}}}{{\Gamma (q + 1)}} + {c_{52}}\frac{{{{(t - {t_0})}^{2q}}}}{{\Gamma (2q + 1)}} + {c_{53}}\frac{{{{(t - {t_0})}^{3q}}}}{{\Gamma (3q + 1)}} \end{array} \right. $$
(32)

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Ding, D., Xiao, H., Yang, Z. et al. Coexisting multi-stability of Hopfield neural network based on coupled fractional-order locally active memristor and its application in image encryption. Nonlinear Dyn 108, 4433–4458 (2022). https://doi.org/10.1007/s11071-022-07371-0

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