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Dynamic Behavior of Three-Layer Fractional-Order Neural Networks with Multiple Delays

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Abstract

Most of the complex network in the real world are not single-layer networks, and networks will be connected with each other. Networks with multi-layer is important because it means cognitive and artificial intelligence. Most current studies of networks consider the case that with n-nodes including ring network, small-word network, scale-free network, etc. This type of network is not enough to describe the complex structure of actual neural networks. However, it is more actual to study the dynamic behavior of multi-layer networks than single-layer networks. In this paper, the stability and bifurcation of a class of three-layer fractional-order neural networks with multiple delays was studied for the first time. By selecting the appropriate bifurcation parameter, the internal dynamic behavior of the given model was discussed by using the theory of Hopf bifurcation, and the critical value and criterion for Hopf bifurcation are derived. The influence of delay, fractional order, and the number of hidden neurons on the bifurcation point were discussed in detail. And the critical value of Hopf bifurcation is accurately calculated. The results show that the stability of the system can be destroyed by increasing the fractional order and the number of hidden neurons. The correctness of the theoretical results is verified by numerical simulation.

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Data Availability

The data that support the findings of this study are available from the corresponding author, [author initials], upon reasonable request.

References

  1. Zhang L, Wang F, Sun T, Xu B. A constrained optimization method based on BP neural network. Neural Comput Appl. 2016;29:413–21.

    Article  MATH  Google Scholar 

  2. Wang X, Li C, Huang T, Duan S. A weakly connected memristive neural network for associative memory. Neural Process Lett. 2013;40:275–88.

    Article  MATH  Google Scholar 

  3. Xia Y, Wang J. A general projection neural network for solving monotone variational inequalities and related optimization problems. IEEE Trans Neural Netw. 2004;15:318–28.

    Article  MATH  Google Scholar 

  4. Zhang J, Shao K, Luo X. Small sample image recognition using improved convolutional neural network. J Vis Commun Image R. 2018;55:640–7.

    Article  MATH  Google Scholar 

  5. Liu J, Gong M, He H. Deep associative neural network for associative memory based on unsupervised representation learning. Neural Netw. 2019;113:41–53.

    Article  MATH  Google Scholar 

  6. Tang Z, Ju H, Feng J. Novel approaches to pin cluster synchronization on complex dynamical networks in Lur’e forms. Commun Nonlinear Sci Numer Simul. 2018;57:422–38.

  7. Shen H, Park J. Finite-time synchronization control for uncertain Markov jump neural networks with input constraints. Nonlinear Dyn. 2014;77:1709–20.

    Article  MathSciNet  MATH  Google Scholar 

  8. Dong T, Bai J, Yang L. Bifurcation analysis of delayed complex-valued neural network with diffusions. Neural Process Lett. 2018;8:1–15.

    MATH  Google Scholar 

  9. Kundu A, Das P. Stability, bifurcations and synchronization in a delayed neural network model of n-identical neurons. Math Comput Simul. 2016;121:12–33.

    Article  MathSciNet  MATH  Google Scholar 

  10. Zhang Y, Xiao M, Zheng W, Cao J. Large-scale neural networks with asymmetrical three-ring structure: stability, nonlinear oscillations, and Hopf Bifurcation. IEEE Trans Cybern. 2022;52:9893–904.

    Article  MATH  Google Scholar 

  11. Mao X. Bifurcations and synchronization of delay-coupled neuronal networks with autaptic connections. Int J Bifurcat Chaos. 2018;28:1850071.

    Article  MathSciNet  MATH  Google Scholar 

  12. Guo Q, Dai C, Yu H, Liu H, Sun X. Stability and bifurcation analysis of a nutrient-phytoplankton model with time delay. Math Methods Appl Sci. 2019;43:3018–303.

    Article  MathSciNet  MATH  Google Scholar 

  13. Wang X, Wang Z, Huang X. Dynamic analysis of a fractional-order delayed SIR model with saturated incidence and treatment functions. Int J Bifurcat Chaos. 2018;28:1850180.

    Article  MATH  Google Scholar 

  14. Wang Z, Huang X, Zhou J. A numerical method for delayed fractional-order differential equations: Based on g-l defifinition. Appl Math Inform Sci. 2013;7:525–9.

    Article  MATH  Google Scholar 

  15. Sohail A, Beg O, Li Z, Celik S. Physics of fractional imaging in biomedicine. Prog Biophys Mol Biol. 2018;140:13–20.

    Article  MATH  Google Scholar 

  16. Zhang Y, Yu X, Sun H, Tick G, Wei W. Applicability of time fractional derivative models for simulating the dynamics and mitigation scenarios of COVID-19. Chaos Solit Fract. 2020;138: 109959.

    Article  MathSciNet  MATH  Google Scholar 

  17. Chen L, Yin H, Huang T, Yuan L, Zheng S. Chaos in fractional-order discrete neural networks with application to image encryption. Neural Netw. 2020;125:174–84.

    Article  MATH  Google Scholar 

  18. Xu C, Liao M, Li P, Yuan S. Impact of leakage delay on bifurcation in fractional-order complex-valued neural networks. Chaos Solit Fract. 2021;142: 110535.

    Article  MathSciNet  MATH  Google Scholar 

  19. Zhang X, Niu P, Liu N, Li G. Global synchronization in finite-time of fractional-order complexvalued delayed Hopfield neural networks. Int J Control Autom Syst. 2019;17:521–35.

    Article  MATH  Google Scholar 

  20. Huang C, Li Z, Ding D, Cao J. Bifurcation analysis in a delayed fractional neural network involving self-connection. Neurocomputing. 2018;314:186–97.

    Article  MATH  Google Scholar 

  21. Huang C, Meng Y, Cao J. New bifurcation results for fractional BAM neural network with leakage delay. Chaos Solit Fract. 2017;100:31–44.

  22. Huang C, Nie X, Zhao Z, Song Q. Novel bifurcation results for a delayed fractional-order quaternion-valued neural network. Neural Netw. 2019;117:67–93.

  23. Chen J, Li C, Yang X. Asymptotic stability of delayed fractional-order fuzzy neural networks with impulse effects. J Frankl Inst. 2018;355:7595–608.

    Article  MathSciNet  MATH  Google Scholar 

  24. Huang C, Tang J, Niu Y, Cao J. Enhanced bifurcation results for a delayed fractional neural network with heterogeneous orders. Physica A. 2019;526: 121014.

    Article  MathSciNet  MATH  Google Scholar 

  25. Zhang H, Ye M, Ye R, Cao J. Synchronization stability of Riemann-Liouville fractional delay-coupled complex neural networks. Physica A. 2018;508:155–65.

    Article  MathSciNet  MATH  Google Scholar 

  26. Fan Y, Huang X, Wang Z, Li Y. Improved quasi-synchronization criteria for delayed fractional-order memristor-based neural networks via linear feedback control. Neurocomputing. 2018;306:68–79.

    Article  MATH  Google Scholar 

  27. Hanif H. A computational approach for boundary layer flow and heat transfer of fractional Maxwell fluid. Math Comput Simul. 2022;191:1–13.

    Article  MathSciNet  MATH  Google Scholar 

  28. Liu F, Dong T, Wang H. Stability analysis and bifurcation control of a delayed incommensurate fractional-order gene regulatory network. Int J Bifurcat Chaos. 2020;30:2050089.

    Article  MathSciNet  MATH  Google Scholar 

  29. Xu C, Liu Z, Liao M, Li P. Fractional-order bidirectional associate memory neural networks with multiple delays: The case of Hopf bifurcation. Math Comput Simul. 2021;182:471–94.

    Article  MathSciNet  MATH  Google Scholar 

  30. Huang C, Liu H, Shi X, Chen X, Xiao M. Bifurcations in a fractional-order neural network with multiple leakage delays. Neural Netw. 2020;131:115–26.

    Article  MATH  Google Scholar 

  31. Ding D, Yao X, Zhang H. Complex projection synchronization of fractional-order complex-valued memristive neural networks with multiple delays. Neural Process Lett. 2020;51:325–45.

    Article  MATH  Google Scholar 

  32. Huang C, Zhao X, Wang X, Wang Z. Disparate delays-induced bifurcations in a fractional-order neural network. J Frankl Inst. 2019;356:2825–46.

    Article  MathSciNet  MATH  Google Scholar 

  33. Huang C, Cao J, Xiao M. Effects of time delays on stability and Hopf bifurcation in a fractional ring-structured network with arbitrary neurons. Commun Nonlinear Sci Numer Simulat. 2018;57:1–13.

    Article  MathSciNet  MATH  Google Scholar 

  34. Huang C, Wang J, Chen X, Cao J. Bifurcations in a fractional-order BAM neural network with four different delays. Neural Netw. 2021;141:344–54.

  35. Cao J, Chong K. Generalized dissipativity state estimation for genetic regulatory networks with interval time-delay signals and leakage delays. Commun Nonlinear Sci Numer Simulat. 2020;89: 105326.

    Article  MathSciNet  MATH  Google Scholar 

  36. Cheng Z, Li D, Cao J. Stability and Hopf bifurcation of a three-layer neural network model with delays. Neurocomputing. 2016;175:355–70.

    Article  MATH  Google Scholar 

Download references

Funding

This work was supported by National Natural Science Foundation of China (Grant Nos. 61374011, 62103215, 62103217), Mathematical Tianyuan Fund of National Natural Science Foundation of China (Grant No.12126345), and Natural Science Foundation of Shandong Province of China (Grant Nos. ZR2020MF080, ZR2020MF065, ZR202102270426).

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Authors and Affiliations

  1. School of Mathematics and Physics, Qingdao University of Science and Technology, Qingdao, 266061, China

    Xinyu Li, Zunshui Cheng, Youming Xin & Yun Shang

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  1. Xinyu Li
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  2. Zunshui Cheng
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Correspondence to Zunshui Cheng.

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Appendices

Appendix A

\(n_{1}=7k\),

\(n_{2}=-c_{13}c_{31}-c_{14}c_{41}-c_{23}c_{32}-c_{15}c_{51}-c_{24}c_{42}-c_{25}c_{52}-c_{36}c_{63}-c_{37}c_{73}-c_{46}c_{64}- c_{47}c_{74}-c_{56}c_{65}-c_{57}c_{75}\),

\(n_{3}=21k^{2}\),

\(n_{4}=5k(-c_{13}c_{31}-c_{14}c_{41}-c_{23}c_{32}-c_{15}c_{51}-c_{24}c_{42}-c_{25}c_{52}-c_{36}c_{63}-c_{37}c_{73}-c_{46}c_{64}- c_{47}c_{74}-c_{56}c_{65}-c_{57}c_{75})\),

\(n_{5}=35k^{3}\), \(n_{6}=c_{13}c_{24}c_{31}c_{42}-c_{13}c_{24}c_{32}c_{41}-c_{14}c_{23}c_{31}c_{42}+c_{14}c_{23}c_{32}c_{41}+c_{13}c_{25}c_{31}c_{52} -c_{13}c_{25}c_{32}c_{51}-c_{15}c_{23}c_{31}c_{52}+c_{15}c_{23}c_{32}c_{51}+c_{14}c_{25}c_{41}c_{52}-c_{14}c_{25}c_{42}c_{51}- c_{15}c_{24}c_{41}c_{52} + c_{15}c_{24}c_{42}c_{51} +c_{13}c_{31}c_{46}c_{64}-c_{13}c_{36}c_{41}c_{64}-c_{14}c_{31}c_{46}c_{63}+c_{14}c_{36}c_{41}c_{63}+c_{13}c_{31}c_{47}c_{74}+c_{13}c_{31}c_{56}c_{65}-c_{13}c_{36}c_{51}c_{65 }-c_{13}c_{37}c_{41}c_{74}-c_{14}c_{31}c_{47}c_{73}+c_{14}c_{37}c_{41}c_{73}-c_{15}c_{31}c_{56}c_{63}+c_{15}c_{36}c_{51}c_{63}+c_{23}c_{32}c_{46}c_{64} -c_{23}c_{36}c_{42}c_{64}-c_{24}c_{32}c_{46}c_{63}+c_{24}c_{36}c_{42}c_{63}+c_{13}c_{31}c_{57}c_{75}-c_{13}c_{37}c_{51}c_{75}+ c_{14}c_{41}c_{56}c_{65}-c_{14}c_{46}c_{51}c_{65}-c_{15}c_{31}c_{57}c_{73}+ c_{15}c_{37}c_{51}c_{73} -c_{15}c_{41}c_{56}c_{64}+ c_{15}c_{46}c_{51}c_{64}+c_{23}c_{32}c_{47}c_{74}+c_{23}c_{32}c_{56}c_{65}-c_{23}c_{36}c_{52}c_{65}-c_{23}c_{37}c_{42}c_{74}- c_{24}c_{32}c_{47}c_{73}+c_{24}c_{37}c_{42}c_{73}-c_{25}c_{32}c_{56}c_{63}+c_{25}c_{36}c_{52}c_{63}+c_{14}c_{41}c_{57}c_{75}- c_{14}c_{47}c_{51}c_{75}-c_{15}c_{41}c_{57}c_{74}+c_{15}c_{47}c_{51}c_{74}+c_{23}c_{32}c_{57}c_{75}-c_{23}c_{37}c_{52}c_{75}+ c_{24}c_{42}c_{56}c_{65}-c_{24}c_{46}c_{52}c_{65}-c_{25}c_{32}c_{57}c_{73}+c_{25}c_{37}c_{52}c_{73}-c_{25}c_{42}c_{56}c_{64}+ c_{25}c_{46}c_{52}c_{64}+c_{24}c_{42}c_{57}c_{75}-c_{24}c_{47}c_{52}c_{75}-c_{25}c_{42}c_{57}c_{74}+c_{25}c_{47}c_{52}c_{74} + c_{36}c_{47}c_{63}c_{74}-c_{36}c_{47}c_{64}c_{73}-c_{37}c_{46}c_{63} + c_{37}c_{46}c_{64}c_{73} + c_{36}c_{57}c_{63}c_{75}-c_{36}c_{57}c_{65}c_{73}-c_{37}c_{56}c_{63}c_{75}+c_{37}c_{56}c_{65}c_{73}+c_{46}c_{57}c_{64}c_{75}- c_{46}c_{57}c_{65}c_{74}- c_{47}c_{56}c_{64}c_{75} +c_{47}c_{56}c_{65}c_{74}\),

\(n_{7}=-10k^{2}(c_{13}c_{31}+c_{14}c_{41}+c_{23}c_{32}+c_{15}c_{51}+c_{24}c_{42}+c_{25}c_{52}+c_{36}c_{63}+c_{37}c_{73}+c_{47}c_{74}+c_{56}c_{65}+c_{57}c_{75})\),

\(n_{8}=35k^{4}\), \(n_{9}=3k(c_{13}c_{24}c_{31}c_{42}-c_{13}c_{24}c_{32}c_{41}-c_{14}c_{23}c_{31}c_{42}+c_{14}c_{23}c_{32}c_{41}+c_{13}c_{25}c_{31}c_{52} -c_{13}c_{25}c_{32}c_{51}-c_{15}c_{23}c_{31}c_{52}+c_{15}c_{23}c_{32}c_{51}+c_{14}c_{25}c_{41}c_{52}-c_{14}c_{25}c_{42}c_{51}- c_{15}c_{24}c_{41}c_{52} + c_{15}c_{24}c_{42}c_{51}+c_{13}c_{31}c_{46}c_{64}-c_{13}c_{36}c_{41}c_{64}-c_{14}c_{31}c_{46}c_{63}+c_{14}c_{36}c_{41}c_{63}+c_{13}c_{31}c_{47}c_{74}+c_{13}c_{31}c_{56}c_{65}-c_{13}c_{36}c_{51}c_{65 }-c_{13}c_{37}c_{41}c_{74}-c_{14}c_{31}c_{47}c_{73}+c_{14}c_{37}c_{41}c_{73}-c_{15}c_{31}c_{56}c_{63}+c_{15}c_{36}c_{51}c_{63}+c_{23}c_{32}c_{46}c_{64} -c_{23}c_{36}c_{42}c_{64}-c_{24}c_{32}c_{46}c_{63}+c_{24}c_{36}c_{42}c_{63}+c_{13}c_{31}c_{57}c_{75}-c_{13}c_{37}c_{51}c_{75}+ c_{14}c_{41}c_{56}c_{65}-c_{14}c_{46}c_{51}c_{65}-c_{15}c_{31}c_{57}c_{73}+ c_{15}c_{37}c_{51}c_{73} -c_{15}c_{41}c_{56}c_{64}+ c_{15}c_{46}c_{51}c_{64}+c_{23}c_{32}c_{47}c_{74}+c_{23}c_{32}c_{56}c_{65}-c_{23}c_{36}c_{52}c_{65}-c_{23}c_{37}c_{42}c_{74}- c_{24}c_{32}c_{47}c_{73}+c_{24}c_{37}c_{42}c_{73}-c_{25}c_{32}c_{56}c_{63}+c_{25}c_{36}c_{52}c_{63}+c_{14}c_{41}c_{57}c_{75}- c_{14}c_{47}c_{51}c_{75}-c_{15}c_{41}c_{57}c_{74}+c_{15}c_{47}c_{51}c_{74}+c_{23}c_{32}c_{57}c_{75}-c_{23}c_{37}c_{52}c_{75}+ c_{24}c_{42}c_{56}c_{65}-c_{24}c_{46}c_{52}c_{65}-c_{25}c_{32}c_{57}c_{73}+c_{25}c_{37}c_{52}c_{73}-c_{25}c_{42}c_{56}c_{64}+ c_{25}c_{46}c_{52}c_{64}+c_{24}c_{42}c_{57}c_{75}-c_{24}c_{47}c_{52}c_{75}-c_{25}c_{42}c_{57}c_{74}+c_{25}c_{47}c_{52}*c_{74} + c_{36}c_{47}c_{63}c_{74}-c_{36}c_{47}c_{64}c_{73}-c_{37}c_{46}c_{63} + c_{37}c_{46}c_{64}c_{73} + c_{36}c_{57}c_{63}c_{75}-c_{36}c_{57}c_{65}c_{73}-c_{37}c_{56}c_{63}c_{75}+c_{37}c_{56}c_{65}c_{73}+c_{46}c_{57}c_{64}c_{75}- c_{46}c_{57}c_{65}c_{74}- c_{47}c_{56}c_{64}c_{75} +c_{47}c_{56}c_{65}c_{74})\),

\(n_{10}=-10k^{3}(c_{13}c_{31}+c_{14}c_{41}+c_{23}c_{32}+c_{15}c_{51}+c_{24}c_{42}+c_{25}c_{52}+c_{36}c_{63}+c_{37}c_{73}+c_{47}c_{74}+c_{56}c_{65}+c_{57}c_{75}+c_{46}c_{64})\),

\(n_{11}=21k^{5}\), \(n_{12}=-c_{13} c_{24} c_{31} c_{42} c_{56} c_{65}+c_{13} c_{24} c_{31} c_{46}\) \( c_{52} c_{65}+c_{13} c_{24} c_{32}c_{41} c_{56} c_{65}-c_{13} c_{24} c_{32} c_{46} c_{51} c_{65}-c_{13} c_{24} c_{36}\) \( c_{41} c_{52} c_{65}+c_{13} c_{24} c_{36} c_{42} c_{51} c_{65}+c_{13} c_{25} c_{31} c_{42} c_{56} c_{64}-c_{13} c_{25}\) \( c_{31} c_{46} c_{52} c_{64}-c_{13} c_{25} c_{32} c_{41} c_{56} c_{64}+_{42} c_{51} c_{64}+c_{14} c_{23} c_{31} c_{42}\) \( c_{56} c_{65}-c_{14} c_{23} c_{31} c_{46} c_{52} c_{65}-c_{14} c_{23} c_{32} c_{41} c_{56} c_{65}+c_{14} c_{23} c_{32}\) \( c_{46} c_{51} c_{65}+c_{14} c_{23} c_{36} c_{41} c_{52} c_{65}-c_{14} c_{23} c_{36} c_{42} c_{51} c_{65}+c_{14} c_{25}\) \( c_{31} c_{42} c_{56} c_{63}+c_{14} c_{25} c_{31} c_{46} c_{52} c_{63}+ c_{14} c_{25} c_{32} c_{41} c_{56} c_{63}-c_{14} c_{25} c_{32} c_{46} c_{51} c_{63}-c_{14} c_{25} c_{36} c_{41} c_{52} c_{63}+c_{14} c_{25} c_{36} c_{42} c_{51}\) \( c_{63}-c_{15} c_{23} c_{31} c_{42} c_{56} c_{64}+c_{15} c_{23} c_{31} c_{46} c_{52} c_{64}c_{15} c_{23} c_{36} c_{41}\) \( c_{52} c_{64}+c_{15} c_{23} c_{36} c_{42} c_{51} c_{64}+c_{15} c_{24} c_{31} c_{42} c_{56} c_{63}-c_{15} c_{24} c_{31}\) \( c_{46} c_{52} c_{63}-c_{15} c_{24} c_{32} c_{41} c_{56} c_{63}+c_{15} c_{24} c_{32} c_{46} c_{51} c_{63}+c_{15} c_{24}\) \( c_{36} c_{41} c_{52} c_{63}-c_{15} c_{24} c_{36} c_{42} c_{51} c_{63}-c_{13} c_{24} c_{31} c_{42} c_{57} c_{75}+c_{13} c_{24} c_{31} c_{47} c_{52} c_{75}+c_{13} c_{24} c_{32} c_{41} c_{57} c_{75}-c_{13} c_{24} c_{32} c_{47} c_{51}\) \( c_{75}-c_{13} c_{24} c_{37} c_{41} c_{52} c_{75}+c_{13} c_{24} c_{37} c_{42} c_{51} c_{75}+c_{13} c_{25} c_{31} c_{42}\) \( c_{57} c_{74}-c_{13} c_{25} c_{31} c_{47} c_{52} c_{74}-c_{13} c_{25} c_{32} c_{41} c_{57} c_{74}+c_{13} c_{25} c_{32}\) \( c_{47} c_{51} c_{74}+c_{13} c_{25} c_{37} c_{41} c_{52} c_{74}-c_{13} c_{25} c_{37} c_{42} c_{51} c_{74}+c_{14} c_{23}\) \( c_{31} c_{42} c_{57} c_{75}-c_{14} c_{23} c_{31} c_{47} c_{52} c_{75}-c_{14} c_{23} c_{32} c_{41} c_{57} c_{75}+c_{14} c_{23} c_{32} c_{47} c_{51} c_{75}+c_{14} c_{23} c_{37} c_{41} c_{52} c_{75}-c_{14} c_{23} c_{37} c_{42} c_{51}\) \( c_{75}-c_{14} c_{25} c_{31} c_{42} c_{57} c_{73}+c_{14} c_{25} c_{31} c_{47} c_{52} c_{73}+c_{14} c_{25} c_{32} c_{41}\) \( c_{57} c_{73}-c_{14} c_{25} c_{32} c_{47} c_{51} c_{73}-c_{14} c_{25} c_{37} c_{41} c_{52} c_{73}+c_{14} c_{25} c_{37}\) \( c_{42} c_{51} c_{73}-c_{15} c_{23} c_{31} c_{42} c_{57} c_{74}+c_{15} c_{23} c_{31} c_{47} c_{52} c_{74}+c_{15} c_{23}\) \( c_{32} c_{41} c_{57} c_{74}-c_{15} c_{23} c_{32} c_{47} c_{51} c_{74}-c_{15} c_{23} c_{37} c_{41} c_{52} c_{74}+c_{15} c_{23} c_{37} c_{42} c_{51} c_{74}+c_{15} c_{24} c_{31} c_{42} c_{57} c_{73}-c_{15} c_{24} c_{31} c_{47} c_{52}\) \( c_{73}-c_{15} c_{24} c_{32} c_{41} c_{57} c_{73}+c_{15} c_{24} c_{32} c_{47} c_{51} c_{73}+c_{15} c_{24} c_{37} c_{41}\) \( c_{52} c_{73}-c_{15} c_{24} c_{37} c_{42} c_{51} c_{73}-c_{13} c_{31} c_{46} c_{57} c_{64} c_{75}+c_{13} c_{31} c_{46}\) \( c_{57} c_{65} c_{74}+c_{13} c_{31} c_{47} c_{56} c_{64} c_{75}-c_{13} c_{31} c_{47} c_{56} c_{65} c_{74}+c_{13} c_{36}\) \( c_{41} c_{57} c_{64} c_{75}-c_{13} c_{36} c_{41} c_{57} c_{65} c_{74}-c_{13} c_{36} c_{47} c_{51} c_{64} c_{75}+c_{13} c_{36} c_{47} c_{51} c_{65} c_{74}-c_{13} c_{37} c_{41} c_{56} c_{64} c_{75}+c_{13} c_{37} c_{41} c_{56} c_{65}\) \( c_{74}+c_{13} c_{37} c_{46} c_{51} c_{64} c_{75}-c_{13} c_{37} c_{46} c_{51} c_{65} c_{74}+c_{14} c_{31} c_{46} c_{57}\) \( c_{63} c_{75}-c_{14} c_{31} c_{46} c_{57} c_{65} c_{73}-c_{14} c_{31} c_{47} c_{56} c_{63} c_{75}+c_{14} c_{31} c_{47}\) \( c_{56} c_{65} c_{73}-c_{14} c_{36} c_{41} c_{57} c_{63} c_{75}+c_{14} c_{36} c_{41} c_{57} c_{65} c_{73}+c_{14}\) \( c_{36} c_{47} c_{51} c_{63} c_{75}-c_{14} c_{36} c_{47} c_{51} c_{65} c_{73}+c_{14} c_{37} c_{41} c_{56} c_{63} c_{75}-c_{14} c_{37} c_{41} c_{56} c_{65} c_{73}-c_{14} c_{37} c_{46} c_{51} c_{63} c_{75}+c_{14} c_{37} c_{46} c_{51} c_{65}\) \( c_{73}-c_{15} c_{31} c_{46} c_{57} c_{63} c_{74}+c_{15} c_{31} c_{46} c_{57} c_{64} c_{73}+c_{15} c_{31} c_{47} c_{56}\) \( c_{63} c_{74}-c_{15} c_{31} c_{47} c_{56} c_{64} c_{73}+c_{15} c_{36} c_{41} c_{57} c_{63} c_{74}-c_{15} c_{36} c_{41}\) \( c_{57} c_{64} c_{73}-c_{15} c_{36} c_{47} c_{51} c_{63} c_{74}+c_{15} c_{36} c_{47} c_{51} c_{64} c_{73}-c_{15} c_{37}\) \( c_{41} c_{56} c_{63} c_{74}+c_{15} c_{37} c_{41} c_{56} c_{64} c_{73}+c_{15} c_{37} c_{46} c_{51} c_{63} c_{74}-c_{15} c_{37} c_{46} c_{51} c_{64} c_{73}-c_{23} c_{32} c_{46} c_{57} c_{64} c_{75}+c_{23} c_{32} c_{46} c_{57} c_{65}\) \( c_{74}+c_{23} c_{32} c_{47} c_{56} c_{64} c_{75}-c_{23} c_{32} c_{47} c_{56} c_{65} c_{74}+c_{23} c_{36} c_{42} c_{57}\) \( c_{64} c_{75}-c_{23} c_{36} c_{42} c_{57} c_{65} c_{74}-c_{23} c_{36} c_{47} c_{52} c_{64} c_{75}+c_{23} c_{36} c_{47}\) \( c_{52} c_{65} c_{74}-c_{23} c_{37} c_{42} c_{56} c_{64} c_{75}+c_{23} c_{37} c_{42} c_{56} c_{65} c_{74}+c_{23}\) \( c_{37} c_{46} c_{52} c_{64} c_{75}-c_{23} c_{37} c_{46} c_{52} c_{65} c_{74}+c_{24} c_{32} c_{46} c_{57} c_{63} c_{75}-c_{24} c_{32} c_{46} c_{57} c_{65} c_{73}-c_{24} c_{32} c_{47} c_{56} c_{63} c_{75}+c_{24} c_{32} c_{47} c_{56} c_{65}\) \( c_{73}-c_{24} c_{36} c_{42} c_{57} c_{63} c_{75}+c_{24} c_{36} c_{42} c_{57} c_{65} c_{73}+c_{24} c_{36} c_{47} c_{52}\) \( c_{63} c_{75}-c_{24} c_{36} c_{47} c_{52} c_{65} c_{73}+c_{24} c_{37} c_{42} c_{56} c_{63} c_{75}-c_{24} c_{37} c_{42}\) \( c_{56} c_{65} c_{73}-c_{24} c_{37} c_{46} c_{52} c_{63} c_{75}+c_{24} c_{37} c_{46} c_{52} c_{65} c_{73}-c_{25} c_{32}\) \( c_{46} c_{57} c_{63} c_{74}+c_{25} c_{32} c_{46} c_{57} c_{64} c_{73}+c_{25} c_{32} c_{47} c_{56} c_{63} c_{74}-c_{25} c_{32} c_{47} c_{56} c_{64} c_{73}+c_{25} c_{36} c_{42} c_{57} c_{63} c_{74}-c_{25} c_{36} c_{42} c_{57} c_{64}\) \( c_{73}-c_{25} c_{36} c_{47} c_{52} c_{63} c_{74}+c_{25} c_{36} c_{47} c_{52} c_{64} c_{73}-c_{25} c_{37} c_{42} c_{56}\) \( c_{63} c_{74}+c_{25} c_{37} c_{42} c_{56} c_{64} c_{73}+c_{25} c_{37} c_{46} c_{52} c_{63} c_{74}- c_{25} c_{37} c_{46}\) \( c_{52} c_{64} c_{73}\),

\(n_{13}=3k^{2}(c_{13}c_{24}c_{31}c_{42}-c_{13}c_{24}c_{32}c_{41}-c_{14}c_{23}c_{31}c_{42}+c_{14}c_{23}c_{32}c_{41}+c_{13}c_{25}c_{31}c_{52} -c_{13}c_{25}c_{32}c_{51}-c_{15}c_{23}c_{31}c_{52}+c_{15}c_{23}c_{32}c_{51}+c_{14}c_{25}c_{41}c_{52}-c_{14}c_{25}c_{42}c_{51}- c_{15}c_{24}c_{41}c_{52} + c_{15}c_{24}c_{42}c_{51} +c_{13}c_{31}c_{46}c_{64}-c_{13}c_{36}c_{41}c_{64}-c_{14}c_{31}c_{46}c_{63}+c_{14}c_{36}c_{41}c_{63}+c_{13}c_{31}c_{47}c_{74}+c_{13}c_{31}c_{56}c_{65}-c_{13}c_{36}c_{51}c_{65 }-c_{13}c_{37}c_{41}c_{74}-c_{14}c_{31}c_{47}c_{73}+c_{14}c_{37}c_{41}c_{73}-c_{15}c_{31}c_{56}c_{63}+c_{15}c_{36}c_{51}c_{63}+c_{23}c_{32}c_{46}c_{64} -c_{23}c_{36}c_{42}c_{64}-c_{24}c_{32}c_{46}c_{63}+c_{24}c_{36}c_{42}c_{63}+c_{13}c_{31}c_{57}c_{75}-c_{13}c_{37}c_{51}c_{75}+ c_{14}c_{41}c_{56}c_{65}-c_{14}c_{46}c_{51}c_{65}-c_{15}c_{31}c_{57}c_{73}+ c_{15}c_{37}c_{51}c_{73} -c_{15}c_{41}c_{56}c_{64}+ c_{15}c_{46}c_{51}c_{64}+c_{23}c_{32}c_{47}c_{74}+c_{23}c_{32}c_{56}c_{65}-c_{23}c_{36}c_{52}c_{65}-c_{23}c_{37}c_{42}c_{74}- c_{24}c_{32}c_{47}c_{73}+c_{24}c_{37}c_{42}c_{73}-c_{25}c_{32}c_{56}c_{63}+c_{25}c_{36}c_{52}c_{63}+c_{14}c_{41}c_{57}c_{75}- c_{14}c_{47}c_{51}c_{75}-c_{15}c_{41}c_{57}c_{74}+c_{15}c_{47}c_{51}c_{74}+c_{23}c_{32}c_{57}c_{75}-c_{23}c_{37}c_{52}c_{75}+ c_{24}c_{42}c_{56}c_{65}-c_{24}c_{46}c_{52}c_{65}-c_{25}c_{32}c_{57}c_{73}+c_{25}c_{37}c_{52}c_{73}-c_{25}c_{42}c_{56}c_{64}+ c_{25}c_{46}c_{52}c_{64}+c_{24}c_{42}c_{57}c_{75}-c_{24}c_{47}c_{52}c_{75}-c_{25}c_{42}c_{57}c_{74}+c_{25}c_{47}c_{52}c_{74} + c_{36}c_{47}c_{63}c_{74}-c_{36}c_{47}c_{64}c_{73}-c_{37}c_{46}c_{63} + c_{37}c_{46}c_{64}c_{73} + c_{36}c_{57}c_{63}c_{75}-c_{36}c_{57}c_{65}c_{73}-c_{37}c_{56}c_{63}c_{75}+c_{37}c_{56}c_{65}c_{73}+c_{46}c_{57}c_{64}c_{75}- c_{46}c_{57}c_{65}c_{74}- c_{47}c_{56}c_{64}c_{75} +c_{47}c_{56}c_{65}c_{74})\),

\(n_{14}=-5k^{4}(c_{13}c_{31}+c_{14}c_{41}+c_{23}c_{32}+c_{15}c_{51}+c_{24}c_{42}+c_{25}c_{52}+c_{36}c_{63}+c_{37}c_{73}+c_{47}c_{74}+c_{56}c_{65}+c_{57}c_{75}+c_{46}c_{64})\), \(n_{15}=7k^{6}\), \(n_{16}= -c_{13} c_{24} c_{31} c_{42} c_{56} c_{65} {k}+c_{13} c_{24} c_{31} c_{46} c_{52} c_{65} {k} + c_{13} c_{24} c_{32} c_{41} c_{56} c_{65} {k} -c_{13} c_{24} c_{32} c_{46}\) \( c_{51} c_{65} {k} - c_{13} c_{24} c_{36} c_{41} c_{52} c_{65} {k} + c_{13} c_{24} c_{36} c_{42} c_{51} c_{65} {k} + c_{13} c_{25}\) \( c_{31} c_{42} c_{56} c_{64} {k} - c_{13} c_{25} c_{31} c_{46} c_{52} c_{64} {k} - c_{13} c_{25} c_{32} c_{41} c_{56} c_{64} {k} + c_{13} c_{25} c_{32} c_{46} c_{51} c_{64} {k} + c_{13} c_{25} c_{36} c_{41} c_{52} c_{64} {k} - c_{13} c_{25} c_{36} c_{42}\) \( c_{51} c_{64} {k} + c_{14} c_{23} c_{31} c_{42} c_{56} c_{65} {k} - c_{14} c_{23} c_{31} c_{46} c_{52} c_{65} {k} - c_{14} c_{23}\) \( c_{32} c_{41} c_{56} c_{65} {k} + c_{14} c_{23} c_{32} c_{46} c_{51} c_{65} {k} + c_{14} c_{23} c_{36} c_{41} c_{52} c_{65} {k} - c_{14} c_{23} c_{36} c_{42} c_{51} c_{65} {k} - c_{14} c_{25} c_{31} c_{42} c_{56} c_{63} {k} + c_{14} c_{25} c_{31} c_{46} c_{52}\) \( c_{63} {k} + c_{14} c_{25} c_{32} c_{41} c_{56} c_{63} {k} -c_{14} c_{25} c_{32} c_{46} c_{51} c_{63} {k} - c_{14} c_{25} c_{36}\) \( c_{41} c_{52} c_{63} {k} + c_{14} c_{25} c_{36} c_{42} c_{51} c_{63} {k} - c_{15} c_{23} c_{31} c_{42} c_{56} c_{64} {k} + c_{15} c_{23} c_{31} c_{46} c_{52} c_{64} {k} + c_{15} c_{23} c_{36} c_{41} c_{52} c_{64} {k} + c_{15} c_{23} c_{36} c_{42} c_{51}\) \( c_{64} {k} + c_{15} c_{24} c_{31} c_{42} c_{56} c_{63} {k} - c_{15} c_{24} c_{31} c_{46} c_{52} c_{63} {k} - c_{15} c_{24} c_{32}\) \( c_{41} c_{56} c_{63} {k} + c_{15} c_{24} c_{32} c_{46} c_{51} c_{63} {k} + c_{15} c_{24} c_{36} c_{41} c_{52} c_{63} {k} - c_{15} c_{24} c_{36} c_{42} c_{51} c_{63} {k} - c_{13} c_{24} c_{31} c_{42} c_{57} c_{75} {k} + c_{13} c_{24} c_{31} c_{47}\) \( c_{52} c_{75} {k} + c_{13} c_{24} c_{32} c_{41} c_{57} c_{75} {k} - c_{13} c_{24} c_{32} c_{47} c_{51} c_{75} {k} - c_{13}\) \( c_{24} c_{37} c_{41} c_{52} c_{75} {k} + c_{13} c_{24} c_{37} c_{42} c_{51} c_{75} {k} + c_{13} c_{25} c_{31} c_{42} c_{57} c_{74}\) \( {k} - c_{13} c_{25} c_{31} c_{47} c_{52} c_{74} {k} - c_{13} c_{25} c_{32} c_{41} c_{57} c_{74} {k} + c_{13} c_{25} c_{32} c_{47}\) \( c_{51} c_{74} {k} + c_{13} c_{25} c_{37} c_{41} c_{52} c_{74} {k} - c_{13} c_{25} c_{37} c_{42} c_{51} c_{74} {k} + c_{14} c_{23}\) \( c_{31} c_{42} c_{57} c_{75} {k} - c_{14} c_{23} c_{31} c_{47} c_{52} c_{75} {k} - c_{14} c_{23} c_{32} c_{41} c_{57} c_{75} {k} + c_{14} c_{23} c_{32} c_{47} c_{51} c_{75} {k} + c_{14} c_{23} c_{37} c_{41} c_{52} c_{75} {k} - c_{14} c_{23} c_{37} c_{42} c_{51}\) \( c_{75} {k} - c_{14} c_{25} c_{31} c_{42} c_{57} c_{73} {k} + c_{14} c_{25} c_{31} c_{47} c_{52} c_{73} {k} + c_{14} c_{25} c_{32}\) \( c_{41} c_{57} c_{73} {k} - c_{14} c_{25} c_{32} c_{47} c_{51} c_{73} {k} - c_{14} c_{25} c_{37} c_{41} c_{52} c_{73} {k} + c_{14} c_{25} c_{37} c_{42} c_{51} c_{73} {k} - c_{15} c_{23} c_{31} c_{42} c_{57} c_{74} {k} + c_{15} c_{23} c_{31} c_{47}\) \( c_{52} c_{74} {k} + c_{15} c_{23} c_{32} c_{41} c_{57} c_{74} {k} - c_{15} c_{23} c_{32} c_{47} c_{51} c_{74} {k} - c_{15} c_{23}\) \( c_{37} c_{41} c_{52} c_{74} {k} + c_{15} c_{23} c_{37} c_{42} c_{51} c_{74} {k} + c_{15} c_{24} c_{31} c_{42} c_{57} c_{73}\) \( {k} - c_{15} c_{24} c_{31} c_{47} c_{52} c_{73} {k} - c_{15} c_{24} c_{32} c_{41} c_{57} c_{73} {k} + c_{15} c_{24} c_{32} c_{47}\) \( c_{51} c_{73} {k} + c_{15} c_{24} c_{37} c_{41} c_{52} c_{73} {k} - c_{15} c_{24} c_{37} c_{42} c_{51} c_{73} {k} - c_{13} c_{31}\) \( c_{46} c_{57} c_{64} c_{75} {k} + c_{13} c_{31} c_{46} c_{57} c_{65} c_{74} {k} + c_{13} c_{31} c_{47} c_{56} c_{64} c_{75} {k} - c_{13} c_{31} c_{47} c_{56} c_{65} c_{74} {k} + c_{13} c_{36} c_{41} c_{57} c_{64} c_{75} {k} - c_{13} c_{36} c_{41} c_{57} c_{65}\) \( c_{74} {k} - c_{13} c_{36} c_{47} c_{51} c_{64} c_{75} {k} + c_{13} c_{36} c_{47} c_{51} c_{65} c_{74} {k} - c_{13} c_{37}\) \( c_{41} c_{56} c_{64} c_{75} {k} + c_{13} c_{37} c_{41} c_{56} c_{65} c_{74} {k} + c_{13} c_{37} c_{46} c_{51} c_{64} c_{75} {k} - c_{13} c_{37} c_{46} c_{51} c_{65} c_{74} {k} + c_{14} c_{31} c_{46} c_{57} c_{63} c_{75} {k} - c_{14} c_{31} c_{46} c_{57} c_{65}\) \( c_{73} {k} - c_{14} c_{31} c_{47} c_{56} c_{63} c_{75} {k} + c_{14} c_{31} c_{47} c_{56} c_{65} c_{73} {k} - c_{14} c_{36} c_{41}\) \( c_{57} c_{63} c_{75} {k} + c_{14} c_{36} c_{41} c_{57} c_{65} c_{73} {k} + c_{14} c_{36} c_{47} c_{51} c_{63} c_{75} {k} - c_{14} c_{36} c_{47} c_{51} c_{65} c_{73} {k} + c_{14} c_{37} c_{41} c_{56} c_{63} c_{75} {k} - c_{14} c_{37} c_{41} c_{56} c_{65}\) \( c_{73} {k} - c_{14} c_{37} c_{46} c_{51} c_{63} c_{75} {k} + c_{14} c_{37} c_{46} c_{51} c_{65} c_{73} {k} - c_{15} c_{31}\) \( c_{46} c_{57} c_{63} c_{74} {k} + c_{15} c_{31} c_{46} c_{57} c_{64} c_{73} {k} + c_{15} c_{31} c_{47} c_{56} c_{63} c_{74} {k} - c_{15} c_{31} c_{47} c_{56} c_{64} c_{73} {k} + c_{15} c_{36} c_{41} c_{57} c_{63} c_{74} {k} - c_{15} c_{36} c_{41} c_{57}\) \( c_{64} c_{73} {k} - c_{15} c_{36} c_{47} c_{51} c_{63} c_{74} {k} + c_{15} c_{36} c_{47} c_{51} c_{64} c_{73} {k} - c_{15} c_{37}\) \( c_{41} c_{56} c_{63} c_{74} {k} + c_{15} c_{37} c_{41} c_{56} c_{64} c_{73} {k} + c_{15} c_{37} c_{46} c_{51} c_{63} c_{74} {k} - c_{15} c_{37} c_{46} c_{51} c_{64} c_{73} {k} - c_{23} c_{32} c_{46} c_{57} c_{64} c_{75} {k} + c_{23} c_{32} c_{46} c_{57}\) \( c_{65} c_{74} {k} + c_{23} c_{32} c_{47} c_{56} c_{64} c_{75} {k} - c_{23} c_{32} c_{47} c_{56} c_{65} c_{74} {k} + c_{23} c_{36}\) \( c_{42} c_{57} c_{64} c_{75} {k} - c_{23} c_{36} c_{42} c_{57} c_{65} c_{74} {k} - c_{23} c_{36} c_{47} c_{52} c_{64} c_{75} {k} + c_{23} c_{36} c_{47} c_{52} c_{65} c_{74} {k} - c_{23} c_{37} c_{42} c_{56} c_{64} c_{75} {k} +c_{23} c_{37} c_{42} c_{56} c_{65}\) \( c_{74} {k} + c_{23} c_{37} c_{46} c_{52} c_{64} c_{75} {k} - c_{23} c_{37} c_{46} c_{52} c_{65} c_{74} {k} + c_{24} c_{32} c_{46}\) \( c_{57} c_{63} c_{75} {k} - c_{24} c_{32} c_{46} c_{57} c_{65} c_{73} {k} - c_{24} c_{32} c_{47} c_{56} c_{63} c_{75} {k} + c_{24} c_{32} c_{47} c_{56} c_{65} c_{73} {k} - c_{24} c_{36} c_{42} c_{57} c_{63} c_{75} {k} + c_{24} c_{36} c_{42} c_{57}\) \( c_{65} c_{73} {k} + c_{24} c_{36} c_{47} c_{52} c_{63} c_{75} {k} - c_{24} c_{36} c_{47} c_{52} c_{65} c_{73} {k} + c_{24} c_{37}\) \( c_{42} c_{56} c_{63} c_{75} {k} - c_{24} c_{37} c_{42} c_{56} c_{65} c_{73} {k} - c_{24} c_{37} c_{46} c_{52} c_{63} c_{75} {k} + c_{24} c_{37} c_{46} c_{52} c_{65} c_{73} {k} - c_{25} c_{32} c_{46} c_{57} c_{63} c_{74} {k} + c_{25} c_{32} c_{46} c_{57}\) \( c_{64} c_{73} {k} + c_{25} c_{32} c_{47} c_{56} c_{63} c_{74} {k} - c_{25} c_{32} c_{47} c_{56} c_{64} c_{73} {k} + c_{25} c_{36}\) \( c_{42} c_{57} c_{63} c_{74} {k} - c_{25} c_{36} c_{42} c_{57} c_{64} c_{73} {k} - c_{25} c_{36} c_{47} c_{52} c_{63} c_{74}\) \( {k} + c_{25} c_{36} c_{47} c_{52} c_{64} c_{73} {k} - c_{25} c_{37} c_{42} c_{56} c_{63} c_{74} {k} + c_{25} c_{37} c_{42}\) \( c_{56} c_{64} c_{73} {k} + c_{25} c_{37} c_{46} c_{52} c_{63} c_{74} {k} - c_{25} c_{37} c_{46} c_{52} c_{64} c_{73} {k}\),

\(n_{17}={k}^{3} (c_{13} c_{24} c_{31} c_{42}-c_{13} c_{24} c_{32} c_{41}-c_{14} c_{23} c_{31} c_{42}+ c_{14} c_{23} c_{32} c_{41}+c_{13} c_{25} c_{31} c_{52}- c_{13} c_{25} c_{32} c_{51} - c_{15} c_{23} c_{31} c_{52} + c_{15} c_{23} c_{32} c_{51} + c_{14} c_{25} c_{41} c_{52} - c_{14} c_{25} c_{42} c_{51}- c_{15} c_{24} c_{41} c_{52}+ c_{15} c_{24} c_{42} c_{51} + c_{13} c_{31} c_{46} c_{64} - c_{13} c_{36} c_{41} c_{64} -c_{14} c_{31} c_{46} c_{63}+ c_{14} c_{36} c_{41} c_{63}+ c_{13} c_{31} c_{47} c_{74} +c_{13} c_{31} c_{56} c_{65}- c_{13} c_{36} c_{51} c_{65} - c_{13} c_{37} c_{41} c_{74}-c_{14} c_{31} c_{47} c_{73}+c_{14} c_{37} c_{41} c_{73}-c_{15} c_{31} c_{56} c_{63}+c_{15} c_{36} c_{51} c_{63} + c_{23} c_{32} c_{46} c_{64} - c_{23} c_{36} c_{42} c_{64} - c_{24} c_{32} c_{46} c_{63} + c_{24} c_{36} c_{42} c_{63}+c_{13} c_{31} c_{57} c_{75}-c_{13} c_{37} c_{51} c_{75}+c_{14} c_{41} c_{56} c_{65}-c_{14} c_{46} c_{51} c_{65}-c_{15} c_{31} c_{57} c_{73} + c_{15} c_{37} c_{51} c_{73} - c_{15} c_{41} c_{56} c_{64}+ c_{15} c_{46} c_{51} c_{64} + c_{23} c_{32} c_{47} c_{74}+ c_{23} c_{32} c_{56} c_{65}- c_{23} c_{36} c_{52} c_{65} - c_{23} c_{37} c_{42} c_{74}- c_{24} c_{32} c_{47} c_{73} +c_{24} c_{37} c_{42} c_{73}- c_{25} c_{32} c_{56} c_{63} {k}^{3}\) \( + c_{25} c_{36} c_{52} c_{63} +c_{14} c_{41} c_{57} c_{75}- c_{14} c_{47} c_{51} c_{75} - c_{15} c_{41} c_{57} c_{74}+c_{15} c_{47} c_{51} c_{74} + c_{23} c_{32} c_{57} c_{75} -c_{23} c_{37} c_{52} c_{75} +c_{24} c_{42} c_{56} c_{65} -c_{24} c_{46} c_{52} c_{65} -c_{25} c_{32} c_{57} c_{73}+c_{25} c_{37} c_{52} c_{73} -c_{25} c_{42} c_{56} c_{64}+c_{25} c_{46} c_{52} c_{64} + c_{24} c_{42} c_{57} c_{75}- c_{24} c_{47} c_{52} c_{75} -c_{25} c_{42} c_{57} c_{74}+c_{25} c_{47} c_{52} c_{74} +c_{36} c_{47} c_{63} c_{74} - c_{36} c_{47} c_{64} c_{73} - c_{37} c_{46} c_{63} c_{74}+ c_{37} c_{46} c_{64} c_{73} +c_{36} c_{57} c_{63} c_{75} -c_{36} c_{57} c_{65} c_{73} - c_{37} c_{56} c_{63} c_{75}+ c_{37} c_{56} c_{65} c_{73} + c_{46} c_{57} c_{64} c_{75} -c_{46} c_{57} c_{65} c_{74} - c_{47} c_{56} c_{64} c_{75}+c_{47} c_{56} c_{65} c_{74})\),

\(n_{18}=-{k}^{5}(c_{13} c_{31}+c_{14} c_{41}+c_{23} c_{32}+c_{15} c_{51}+c_{24} c_{42}+c_{25} c_{52}+c_{36} c_{63}+c_{37} c_{73}+c_{46} c_{64}+c_{47} c_{74}+c_{56} c_{65}+ c_{57} c_{75})\),

\(n_{19}={k}^{7}\).

Appendix B

Concrete expressions in system (30)

$$\begin{aligned} \left\{ \begin{aligned} D_{1}=&E_{1},\\ D_{2}=&E_{1}E_{2}-E_{3},\\ D_{3}=&-E^{2}_{1}E_{4}+E_{1}E_{2}E_{3}+E_{1}E_{5}-E^{2}_{3}E_{2},\\ D_{4}=&E^{2}_{1}E_{2}E_{6}-E^{2}_{1}E^{2}_{4}-E_{1}E^{2}_{2}E_{5}+E_{1}E_{2}E_{3}E_{4}-E_{1}E_{2}E_{7}\\&-E_{1}E_{3}E_{6}\!+\!2E_{1}E_{4}E_{5}\!+\!E_{2}E_{3}E_{5}-E^{2}_{3}E_{4}+E_{3}E_{7}\!-\!E^{2}_{5},\\ D_{5}=&E^{3}_{1}E^{2}_{6}-E^{2}_{1}E_{2}E_{4}E_{7}+2E^{2}_{1}E_{2}E_{5}E_{6}+E^{2}_{1}E_{3}E_{4}E_{6}\\&-E^{2}_{1}E^{2}_{4}E_{5}+2E^{2}_{1}E_{6}E_{7}+E_{1}E^{2}_{2}E_{3}E_{7}-E_{1}E^{2}_{2}E^{2}_{5}\\&-E_{1}E_{2}E^{2}_{3}E_{6}+E_{1}E_{2}E_{3}E_{4}E_{5}-E_{1}E_{2}E_{5}E_{7}\\&-3E_{1}E_{3}E_{5}E_{6}+2E_{1}E_{4}E^{2}_{5}-E_{1}E^{2}_{7}-E_{2}E^{2}_{3}E_{7}\\&+E_{2}E_{3}E^{2}_{5}+E^{3}_{3}E_{6}-E^{2}_{3}E_{4}E_{5}+2E_{3}E_{5}E_{7}-E^{3}_{5},\\ D_{6}=&-E^{3}_{1}E^{3}_{6}-3E^{2}_{1}E_{2}E_{4}E_{6}E_{7}+2E^{2}_{1}E_{2}E_{5}E^{2}_{6}+E^{2}_{1}E_{3}E_{4}E^{2}_{6}\\&+E^{2}_{1}E^{3}_{4}E_{7}-E^{2}_{1}E^{2}_{4}E_{5}E_{6}+3E^{2}_{1}E^{2}_{6}E_{7}-E_{1}E^{3}_{2}E^{2}_{7}\\&+2E_{1}E^{2}_{2}E_{3}E_{6}E_{7}+E_{1}E^{2}_{2}E_{4}E_{5}E_{7}-E_{1}E^{2}_{2}E^{2}_{5}E_{6}\\&-E_{1}E_{2}E^{2}_{3}E^{2}_{6}-E_{1}E_{2}E_{3}E^{2}_{4}E_{7}+E_{1}E_{2}E_{3}E_{4}E_{5}E_{6}\\&+3E_{1}E_{2}E_{4}E^{2}_{7}-E_{1}E_{2}E_{5}E_{6}E_{7}+E_{1}E_{3}E_{4}E_{6}E_{7}\\&-3E_{1}E_{3}E_{5}E^{2}_{6}\!-\!2E_{1}E^{2}_{4}E_{5}E_{7}\!+\!2E_{1}E_{4}E^{2}_{5}E_{6}\!-\!3E_{1}E_{6}E^{2}_{7}\\&+E^{2}_{2}E_{3}E^{2}_{7}-2E_{2}E^{2}_{3}E_{6}E_{7}\!-\!E_{2}E_{3}E_{4}E_{5}E_{7}\!+\!E_{2}E_{3}E^{2}_{5}E_{6}\\&-E_{2}E_{5}E^{2}_{7}+E^{3}_{3}E^{2}_{6}+E^{2}_{3}E^{2}_{4}E_{7}-E^{2}_{3}E_{4}E_{5}E_{6}\\&-2E_{3}E_{4}E^{2}_{7}+3E_{3}E_{5}E_{6}E_{7}+E_{4}E^{2}_{5}E_{7}-E^{3}_{5}E_{6}+E^{3}_{7},\\ D_{7}=&E_{7}D_{6}. \end{aligned} \right. \end{aligned}$$

Appendix C

\(P'_{1}(s)=7qs^{7q-1}+6qn_{1}s^{6q-1}+5qn_{3}s^{5q-1}+4qn_{5}s^{4q-1}+3qn_{8}s^{3q-1}+2qn_{11}s^{2q-1}+qn_{15}s^{q-1}\),

\(P'_{2}(s)\!=\!5qn_{2}s^{5q-1}+4qn_{4}s^{4q-1}+3qn_{7}s^{3q-1}+2qn_{10}s^{2q-1}\) \(+qn_{14}s^{q-1}\),

\(P'_{3}(s)=3qn_{6}s^{3q-1}+2qn_{9}s^{2q-1}+qn_{13}s^{q-1}\),

\(P'_{4}(s)=qn_{12}s^{q-1}\).

Appendix D

\(A'_{1}(\omega )=7q\omega ^{7q-1}cos((7q-1)\pi /2)+6qn_{1}\omega ^{6q-1}cos((6q-1)\pi /2)+5qn_{3}\omega ^{5q-1}cos((5q-1)\pi /2)+4qn_{5}\omega ^{4q-1}cos((4q-1)\pi /2)+3qn_{8}\omega ^{3q-1}cos((3q-1)\pi /2)+2qn_{11}\omega ^{2q-1}cos((2q-1)\pi /2)+qn_{15}\omega ^{q-1}cos((q-1)\pi /2),\)

\(B'_{1}(\omega )=7q\omega ^{7q-1}sin((7q-1)\pi /2)+6qn_{1}\omega ^{6q-1}sin((6q-1)\pi /2)+5qn_{3}\omega ^{5q-1}sin((5q-1)\pi /2)+4qn_{5}\omega ^{4q-1}sin((4q-1)\pi /2)+3qn_{8}\omega ^{3q-1}\) \(sin((3q-1)\pi /2)+2qn_{11}\omega ^{2q-1}sin((2q-1)\pi /2)+qn_{15}\omega ^{q-1}sin((q-1)\pi /2),\)

\(A'_{2}(\omega )=5qn_{2}\omega ^{5q-1}cos((5q-1)\pi /2)+4qn_{4}\omega ^{4q-1}cos((4q-1)\pi /2)+3qn_{7}\omega ^{3q-1}cos((3q-1)\pi /2)+2qn_{10}\omega ^{2q-1}cos((2q-1)\pi /2)+qn_{14}\omega ^{q-1}cos((q-1)\pi /2)\),

\(B'_{2}(\omega )=5qn_{2}\omega ^{5q-1}sin((5q-1)\pi /2)+4qn_{4}\omega ^{4q-1}sin((4q-1)\pi /2)+3qn_{7}\omega ^{3q-1}sin((3q-1)\pi /2)+2qn_{10}\omega ^{2q-1}sin((2q-1)\pi /2)+qn_{14}\omega ^{q-1}sin((q-1)\pi /2)\),

\(A'_{3}(\omega )=3qn_{6}\omega ^{3q-1}cos((3q-1)\pi /2)+2qn_{9}\omega ^{2q-1}cos((2q-1)\pi /2)+qn_{13}\omega ^{q-1}cos((q-1)\pi /2)\),

\(B'_{3}(\omega )=3qn_{6}\omega ^{3q-1}sin((3q-1)\pi /2)+2qn_{9}\omega ^{2q-1}sin((2q-1)\pi /2)+qn_{13}\omega ^{q-1}sin((q-1)\pi /2)\),

\(A'_{4}(\omega )=qn_{12}\omega ^{q-1}cos((q-1)\pi /2)\),

\(B'_{4}(\omega )=qn_{12}\omega ^{q-1}cos((q-1)\pi /2)\),

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Li, X., Cheng, Z., Xin, Y. et al. Dynamic Behavior of Three-Layer Fractional-Order Neural Networks with Multiple Delays. Cogn Comput 17, 48 (2025). https://doi.org/10.1007/s12559-025-10411-7

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