Short memory fractional differential equations for new memristor and neural network design

Abstract

Fractional derivatives hold memory effects, and they are extensively used in various real-world applications. However, they also need large storage space and cause poor efficiency. In this paper, some standard definitions are revisited. Then, short memory fractional derivatives and a short memory fractional modeling approach are introduced. Numerical solutions are given by the use of the predictor–corrector method. The short memory is adopted for fractional modeling of memristor, neural networks and materials’ relaxation property. Global stability conditions of variable-order neural networks are derived. The new features of short memory fractional differential equations are used to improve the performance of networks. The results are illustrated in comparison with standard ones. Finally, discussions are made about potential applications.

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Acknowledgements

This study is warmly invited by the associate editor Jun Ma. The authors also feel grateful to all the referees’ and editor’s valuable suggestions.

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Correspondence to Maokang Luo.

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This study was financially supported by Sichuan Science and Technology Support Program (Grant Nos. 2018JY0120, 2019JDTD0015).

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Appendix

Appendix

Firstly, the system (3.13) can be rewritten on the first sub-interval

$$\begin{aligned} \left\{ \begin{array}{l} x_{n+1} =x_0 +\frac{h^\nu }{\Gamma (\nu )}\sum \limits _{j=0}^n {\frac{\Gamma (n-j+\nu )}{\Gamma (n-j+1)}}\\ (\alpha (y_j -x_j +\zeta x_j -W(w_j )x_j )),\\ y_{n+1} =y_0 +\frac{h^\nu }{\Gamma (\nu )}\sum \limits _{j=0}^n {\frac{\Gamma (n-j+\nu )}{\Gamma (n-j+1)}} (x_j -y_j +z_j ),\\ z_{n+1} =z_0 +\frac{h^\nu }{\Gamma (\nu )}\sum \limits _{j=0}^n {\frac{\Gamma (n-j+\nu )}{\Gamma (n-j+1)}} (-\beta y_j -\gamma w_j ),\\ w_{n+1} =w_0 +\frac{h^\nu }{\Gamma (\nu )}\sum \limits _{j=0}^n {\frac{\Gamma (n-j+\nu )}{\Gamma (n-j+1)}} x_j .\\ \end{array}\right. \end{aligned}$$

The Jacobian matrix is used

$$\begin{aligned} J(n)=\begin{pmatrix} a_{11} (n)&{} a_{12} (n)&{} a_{13} (n)&{} a_{14} (n)\\ b_{21} (n)&{} b_{22} (n)&{} b_{23} (n)&{} b_{24} (n)\\ c_{31} (n)&{} c_{32} (n)&{} c_{33} (n)&{} c_{34} (n)\\ d_{41} (n)&{} d_{42} (n)&{} d_{43} (n)&{} d_{44} (n)\\ \end{pmatrix}, \end{aligned}$$

where \(a_{11}(0)=b_{22}(0)=c_{33}(0)=d_{44}(0)=1\) and \(J(0)=I\) is the identity matrix.

Each element can be calculated, and the system of tangent maps can be determined as

$$\begin{aligned} a_{11} (n+1)&=a_{11} (0)+\frac{h^\nu }{\Gamma (\nu )}\sum \limits _{j=0}^n {\frac{\Gamma (n-j+\nu )}{\Gamma (n-j+1)}} \\&\qquad [\alpha (b_{21} (j)-a_{11} (j)+\zeta a_{11} (j)\\&\quad -W(w_j )a_{11} (j))], \\ a_{12} (n+1)&=a_{12} (0)+\frac{h^\nu }{\Gamma (\nu )}\sum \limits _{j=0}^n {\frac{\Gamma (n-j+\nu )}{\Gamma (n-j+1)}} \\&\qquad [\alpha (b_{22} (j)-a_{12} (j)+\zeta a_{12} (j)\\&\quad -W(w_j )a_{12} (j))],\\ a_{13} (n+1)&=a_{13} (0)+\frac{h^\nu }{\Gamma (\nu )}\sum \limits _{j=0}^n {\frac{\Gamma (n-j+\nu )}{\Gamma (n-j+1)}} \\&\qquad [\alpha (b_{23} (j)-a_{13} (j)+\zeta a_{13} (j)\\&\quad -W(w_j )a_{13} (j))],\\ a_{14} (n+1)&=a_{14} (0)+\frac{h^\nu }{\Gamma (\nu )}\sum \limits _{j=0}^n {\frac{\Gamma (n-j+\nu )}{\Gamma (n-j+1)}} \\&\qquad [\alpha (b_{24} (j)-a_{14} (j)+\zeta a_{14} (j)\\&\quad -W(w_j )a_{14} (j))],\\ b_{21} (n+1)&=b_{21} (0)+\frac{h^\nu }{\Gamma (\nu )}\sum \limits _{j=0}^n {\frac{\Gamma (n-j+\nu )}{\Gamma (n-j+1)}} \\&\qquad [a_{11} (j)-b_{21} (j)+c_{31} (j)],\\ b_{22} (n+1)&=b_{22} (0)+\frac{h^\nu }{\Gamma (\nu )}\sum \limits _{j=0}^n {\frac{\Gamma (n-j+\nu )}{\Gamma (n-j+1)}} \\&\qquad [a_{12} (j)-b_{22} (j)+c_{32} (j)],\\ b_{23} (n+1)&=b_{23} (0)+\frac{h^\nu }{\Gamma (\nu )}\sum \limits _{j=0}^n {\frac{\Gamma (n-j+\nu )}{\Gamma (n-j+1)}} \\&\qquad [a_{13} (j)-b_{23} (j)+c_{33} (j)],\\ b_{24} (n+1)&=b_{24} (0)+\frac{h^\nu }{\Gamma (\nu )}\sum \limits _{j=0}^n {\frac{\Gamma (n-j+\nu )}{\Gamma (n-j+1)}} \\&\qquad [a_{14} (j)-b_{24} (j)+c_{34} (j)], \\ c_{31} (n+1)&=c_{31} (0)+\frac{h^\nu }{\Gamma (\nu )}\sum \limits _{j=0}^n {\frac{\Gamma (n-j+\nu )}{\Gamma (n-j+1)}} \\&\qquad [-\beta b_{21} (j)-\gamma d_{41} (j)],\\ c_{32} (n+1)&=c_{32} (0)+\frac{h^\nu }{\Gamma (\nu )}\sum \limits _{j=0}^n {\frac{\Gamma (n-j+\nu )}{\Gamma (n-j+1)}} \\&\qquad [-\beta b_{22} (j)-\gamma d_{42} (j)],\\ c_{33} (n+1)&=c_{33} (0)+\frac{h^\nu }{\Gamma (\nu )}\sum \limits _{j=0}^n {\frac{\Gamma (n-j+\nu )}{\Gamma (n-j+1)}} \\&\qquad [-\beta b_{23} (j)-\gamma d_{43} (j)],\\ c_{34} (n+1)&=c_{34} (0)+\frac{h^\nu }{\Gamma (\nu )}\sum \limits _{j=0}^n {\frac{\Gamma (n-j+\nu )}{\Gamma (n-j+1)}} \\&\qquad [-\beta b_{24} (j)-\gamma d_{44} (j)],\\ d_{41} (n+1)&=d_{41} (0)+\frac{h^\nu }{\Gamma (\nu )}\sum \limits _{j=0}^n {\frac{\Gamma (n-j+\nu )}{\Gamma (n-j+1)}} a_{11} (j),\\ d_{42} (n+1)&=d_{42} (0)+\frac{h^\nu }{\Gamma (\nu )}\sum \limits _{j=0}^n {\frac{\Gamma (n-j+\nu )}{\Gamma (n-j+1)}} a_{12} (j),\\ d_{43} (n+1)&=d_{43} (0)+\frac{h^\nu }{\Gamma (\nu )}\sum \limits _{j=0}^n {\frac{\Gamma (n-j+\nu )}{\Gamma (n-j+1)}} a_{13} (j),\\ d_{44} (n+1)&=d_{44} (0)+\frac{h^\nu }{\Gamma (\nu )}\sum \limits _{j=0}^n {\frac{\Gamma (n-j+\nu )}{\Gamma (n-j+1)}} a_{14} (j). \end{aligned}$$

Secondly, the singular value decomposition method is adopted to calculate all the eigenvalues \(\lambda _{i}(n)\) of J(n). As a result, the Lyapunov exponents spectrum is approximately obtained as \(\frac{\ln \vert \lambda _i (n)\vert }{n},i=1,2,3,4.\)

Finally, for the second and other sub-intervals, we can continue this method successfully and identify the chaotic states if there is one positive Lyapunov exponent.

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Wu, G., Luo, M., Huang, L. et al. Short memory fractional differential equations for new memristor and neural network design. Nonlinear Dyn 100, 3611–3623 (2020). https://doi.org/10.1007/s11071-020-05572-z

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Keywords

  • Fractional differential equations
  • Short memory
  • Variable-order modeling
  • Neural network
  • Memristor
  • Global stability and existences