Skip to main content
SpringerLink
Log in

Bifurcation Mechanism for Fractional-Order Three-Triangle Multi-delayed Neural Networks

  • Published:
Neural Processing Letters Aims and scope Submit manuscript

Abstract

This article is basically concerned with the stability and Hopf bifurcation problem of fractional-order three-triangle multi-delayed neural networks. Based on laplace transform, we obtain the characteristic equation of the considered fractional-order three-triangle multi-delayed neural networks. By discussing the distribution of the roots for the characteristic equation, the delay-independent stability condition and delay-induced bifurcation criterion are built. The research manifests that time delay is an important factor which affects the stability and the onset of Hopf bifurcation for fractional-order three-triangle multi-delayed neural networks. The computer simulation results and bifurcation figures are displayed to support the established main conclusions. The derived fruits of this article have great theoretical values in dominating neural networks.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

References

  1. Li YK, Shen SP (2020) Almost automorphic solutions for Clifford-valued neutral-type fuzzy cellular neural networks with leakage delays on time scales. Neurocomputing 417:23–35

    Google Scholar 

  2. Xiu CB, Zhou RX, Liu YX (2020) New chaotic memristive cellular neural network and its application in secure communication system. Chaos Solitons Fractals 141:110316

    MathSciNet  Google Scholar 

  3. Ji LP, Chang MZ, Shen YL, Zhang Q (2020) Recurrent convolutions of binary-constraint cellular neural network for texture recognition. Neurocomputing 387:161–171

    Google Scholar 

  4. Cui WX, Wang ZJ, Jin WB (2021) Fixed-time synchronization of Markovian jump fuzzy cellular neural networks with stochastic disturbance and time-varying delays. Fuzzy Sets Syst 411:68–84

    MathSciNet  MATH  Google Scholar 

  5. Huang CX, Su RL, Cao JD, Xiao SL (2020) Asymptotically stable high-order neutral cellular neural networks with proportional delays and D operators. Math Comput Simul 171:127–135

    MathSciNet  MATH  Google Scholar 

  6. Wang Z, Wang XH, Xia JW, Shen H, Meng B (2020) Adaptive sliding mode output tracking control based-FODOB for a class of uncertain fractional-order nonlinear time-delayed systems. Sci China Technol Sci 63(9):1854–1862

    Google Scholar 

  7. Meng B, Wang XH, Zhang ZY, Wang Z.: Necessary and sufficient conditions for normalization and sliding mode control of singular fractional-order systems with uncertainties. Sci China Inf Sci 63(5):152202:1-152202:10 (2020)

  8. Jia J, Huang X, Li YX, Cao CD, Ahmed A (2020) Global stabilization of fractional-order memristor-based neural networks with time delay. IEEE Trans Neural Netw Learning Sys 31(3):997–1009

    MathSciNet  Google Scholar 

  9. Xu CJ, Liao MX, Li PL, Liu ZX, Yuan S (2021) New results on pseudo almost periodic solutions of quaternion-valued fuzzy cellular neural networks with delays. Fuzzy Sets Syst 411:25–47

    MathSciNet  MATH  Google Scholar 

  10. Hsu CH, Lin JJ (2019) Stability of traveling wave solutions for nonlinear cellular neural networks with distributed delays. J Math Anal Appl 470(1):388–400

    MathSciNet  MATH  Google Scholar 

  11. Tang RQ, Yang XS, Wan XX (2019) Finite-time cluster synchronization for a class of fuzzy cellular neural networks via non-chattering quantized controllers. Neural Netw 113:79–90

    MATH  Google Scholar 

  12. Li YK, Qin JL (2018) Existence and global exponential stability of periodic solutions for quaternion-valued cellular neural networks with time-varying delays. Neurocomputing 292:91–103

    Google Scholar 

  13. Wang WT (2018) Finite-time synchronization for a class of fuzzy cellular neural networks with time-varying coefficients and proportional delays. Fuzzy Sets Syst 338:40–49

    MathSciNet  MATH  Google Scholar 

  14. Zheng MW, Li LX, Peng HP, Xiao JH, Yang YX, Zhang YP, Zhao H (2018) Fixed-time synchronization of memristor-based fuzzy cellular neural network with time-varying delay. J Franklin Inst 355(14):6780–6809

    MathSciNet  MATH  Google Scholar 

  15. Kumar R, Das S (2020) Exponential stability of inertial BAM neural network with time-varying impulses and mixed time-varying delays via matrix measure approach. Commun Nonlinear Sci Numer Simul 81:105016

    MathSciNet  MATH  Google Scholar 

  16. Kong FC, Zhu QX, Wang K, Nieto JJ (2019) Stability analysis of almost periodic solutions of discontinuous BAM neural networks with hybrid time-varying delays and D operator. J Franklin Inst 356(18):11605–11637

    MathSciNet  MATH  Google Scholar 

  17. Xu CJ, Tang XH, Liao MX (2011) Stability and bifurcation analysis of a six-neuron BAM neural network model with discrete delays. Neurocomputing 74(5):689–707

    Google Scholar 

  18. Gopalsamy K, He X (1994) Delay-independent stability in bidirectional associative memory networks. IEEE Trans Neural Netw 5(6):998–1002

    Google Scholar 

  19. Liao XF, Li SW, Wong KW (2003) Hopf bifurcation on a two-neuron system with distributed delays: a frequency domain approach. Nonlinear Dyn 31:299–326

    MathSciNet  MATH  Google Scholar 

  20. Mao XC (2012) Stability and Hopf bifurcation analysis of a pair of three-neuron loops with time delays. Nonlinear Dyn 68:151–159

    MathSciNet  MATH  Google Scholar 

  21. Ge JH, Xu J, Li ZQ (2017) Zero-Hopf bifurcation and multistability coexistence on a four-neuron network model with multiple delays. Nonlinear Dyn 87:2357–2366

    MathSciNet  MATH  Google Scholar 

  22. Xu CJ, Tang XH, Liao MX (2013) Stability and bifurcation analysis on a ring of five neurons with discrete delays. J Dyn Control Syst 19:237–275

    MathSciNet  MATH  Google Scholar 

  23. Liu YW, Li SS, Liu ZR, Wang RQ (2016) High codimensional bifurcation analysis to a six-neuron BAM neural network. Cogn Neurodyn 10:149–164

    Google Scholar 

  24. Javidmanesh E, Afsharnezhad Z, Effati S (2013) Existence and stability analysis of bifurcating periodic solutions in a delayed five-neuron BAM neural network model. Nonlinear Dyn 72:149–164

    MathSciNet  MATH  Google Scholar 

  25. Ge JH (2018) Effects of multiple delays on dynamics of a five-neuron network model. Nonlinear Dyn 94:87–98

    MATH  Google Scholar 

  26. Xu CJ, Liao MX, Li PL, Guo Y (2019) Bifurcation analysis for simplified five-neuron bidirectional associative memory neural networks with four delays. Neural Process Lett 50:2219–2245

    Google Scholar 

  27. Liu X (2014) Zero singularity of codimension two or three in a four-neuron BAM neural network model with multiple delays. Nonlinear Dyn 77:1783–1794

    MathSciNet  MATH  Google Scholar 

  28. Xu CJ, Liu ZX, Liao MX, Li PL, Xiao QM, Yuan S (2021) Fractional-order bidirectional associate memory (BAM) neural networks with multiple delays: the case of Hopf bifurcation. Math Comput Simul 182:471–494

    MathSciNet  MATH  Google Scholar 

  29. Yousef FB, Yousef A, Maji C (2021) Effects of fear in a fractional-order predator–prey system with predator density-dependent prey mortality. Chaos Solitons Fractals 145:110711

    MathSciNet  MATH  Google Scholar 

  30. Huang CD, Wang J, Chen XP, Cao JD (2021) Bifurcations in a fractional-order BAM neural network with four different delays. Neural Netw 141:344–354

    MATH  Google Scholar 

  31. Xu CJ, Aouiti C, Liu ZX (2020) A further study on bifurcation for fractional order BAM neural networks with multiple delays. Neurocomputing 417:501–515

    Google Scholar 

  32. Xu CJ, Liao MX, Li PL, Guo Y, Xiao QM, Yuan S (2019) Influence of multiple time delays on bifurcation of fractional-order neural networks. Appl Math Comput 361:565–582

    MathSciNet  MATH  Google Scholar 

  33. Xu CJ, Aouiti C (2020) Comparative analysis on Hopf bifurcation of integer order and fractional order two-neuron neural networks with delay. Int J Circuit Theory Appl 48(9):1459–1475

    Google Scholar 

  34. Xu CJ, Liao MX, Li PL, Guo Y, Liu ZX (2021) Bifurcation properties for fractional order delayed BAM neural networks. Cogn Comput 13(2):322–356

    Google Scholar 

  35. Huang CD, Liu H, Shi XY, Chen XP, Xiao M, Wang ZX, Cao JD (2020) Bifurcations in a fractional-order neural network with multiple leakage delays. Neural Netw 131:115–126

    MATH  Google Scholar 

  36. Huang CD, Cao JD, Xiao M, Alsaedi A, Hayat T (2017) Bifurcations in a delayed fractional complex-valued neural network. Appl Math Comput 292:210–227

    MathSciNet  MATH  Google Scholar 

  37. Xu CJ, Mu D, Liu ZX, Pang YC , Liao MX, L PL, Yao LY , Qin, QW (2022) Comparative exploration on bifurcation behavior for integer-order and fractional-order delayed BAM neural networks. Nonlinear Anal Model Control (2022) https://doi.org/10.15388/namc.2022.27.28491

  38. Xu CJ, Zhang W, Aouiti C, Liu ZX, Yao LY (2022) Further analysis on dynamical properties of fractional-order bi-directional associative memory neural networks involving double delays. Math Methods Appl Sci. https://doi.org/10.1002/mma.8477

    Article  MathSciNet  Google Scholar 

  39. Xu CJ, Zhang W, Liu ZX, Yao LY (2022) Delay-induced periodic oscillation for fractional-order neural networks with mixed delays. Neurocomputing 488:681–693

    Google Scholar 

  40. Xu CJ, Zhang W, Liu ZX, Li PL, Yao LY (2022) Bifurcation study for fractional-order three-layer neural networks involving four time delays. Cogn Comput 14:714–732

    Google Scholar 

  41. Cheng ZS, Xie KH, Wang TS, Cao JD (2018) Stability and Hopf bifurcation of three-triangle neural networks with delays. Neurocomputing 322:206–215

    Google Scholar 

  42. Podlubny I (1999) Fractional differential equations. Academic Press, New York

    MATH  Google Scholar 

  43. Matignon D (1996) Stability results for fractional differential equations with applications to control processing. Comput Eng Syst Appl 2:963–968

    Google Scholar 

  44. Wang XH, Wang Z, Xia JW (2019) Stability and bifurcation control of a delayed fractional-order eco-epidemiological model with incommensurate orders. J Frankl Inst 356(15):8278–8295

    MathSciNet  MATH  Google Scholar 

  45. Deng WH, Li CP, Lü JH (2007) Stability analysis of linear fractional differential system with multiple time delays. Nonlinear Dyn 48(4):409–416

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This work is supported by National Natural Science Foundation of China (No.12261015 and No.62062018) and Project of High-level Innovative Talents of Guizhou Province ([2016]5651), Basic Research Program of Guizhou Province (ZK[2022]025), Natural Science Project of the Education Department of Guizhou Province (KY[2021]031), University Science and Technology Top Talents Project of Guizhou Province (KY[2018]047), Foundation of Science and Technology of Guizhou Province ([2019]1051), Guizhou University of Finance and Economics(2018XZD01). The authors would like to thank the referees and the editor for helpful suggestions incorporated into this paper. Joint Fund Project of Guizhou University of Finance and Economics and Institute of International Trade and Economic Cooperation of Ministry of Commerce on Contiguous areas of extreme poverty Poor peasant psychological Poverty alleviation (2017SWBZD09).

Author information

Authors and Affiliations

  1. Guizhou Key Laboratory of Economics System Simulation, Guizhou University of Finance and Economics, Guiyang, 550004, People’s Republic of China

    Changjin Xu

  2. School of Mathematics and Statistics, Guizhou University of Finance and Economics, Guiyang, 550004, People’s Republic of China

    Zixin Liu

  3. School of Mathematics and Statistics, Henan University of Science and Technology, Luoyang, 471023, People’s Republic of China

    Peiluan Li & Jinling Yan

  4. Library, Guizhou University of Finance and Economics, Guiyang, 550004, People’s Republic of China

    Lingyun Yao

Authors
  1. Changjin Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zixin Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Peiluan Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Jinling Yan
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Lingyun Yao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Changjin Xu.

Ethics declarations

Competing interests

The authors declare that they have no competing interests.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Xu, C., Liu, Z., Li, P. et al. Bifurcation Mechanism for Fractional-Order Three-Triangle Multi-delayed Neural Networks. Neural Process Lett 55, 6125–6151 (2023). https://doi.org/10.1007/s11063-022-11130-y

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11063-022-11130-y

Keywords

Search

Navigation