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05.06.2024

Effect of Leakage Delays on Bifurcation in Fractional-Order Bidirectional Associative Memory Neural Networks with Five Neurons and Discrete Delays

verfasst von: Yangling Wang, Jinde Cao, Chengdai Huang

Erschienen in: Cognitive Computation | Ausgabe 5/2024

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Abstract

As is well known that time delays are inevitable in practice due to the finite switching speed of amplifiers and information transmission between neurons. So the study on the Hopf bifurcation of delayed neural networks has aroused extensive attention in recent years. However, it’s worth mentioning that only the communication delays between neurons were generally considered in most existing relevant literatures. Actually, it has been proven that a kind of so-called leakage delays cannot be ignored because the self-decay process of a neuron’s action potential is not instantaneous in hardware implementation of neural networks. Though leakage delays have been taken into account in a few more recent works concerning the Hopf bifurcation of fractional-order bidirectional associative memory neural networks, the addressed neural networks were low-dimension or the involved time delays were single. In this paper, we propose a five-neuron fractional-order bidirectional associative memory neural network model, which includes leakage delays and discrete communication delays to meet the characteristics of real neural networks better. Then we use the stability theory of fractional differential equations and Hopf bifurcation theory to investigate its dynamic behavior of Hopf bifurcation. The Hopf bifurcation of the proposed model are studied by taking the involved two different leakage delays as the bifurcation parameter respectively, and two kinds of sufficient conditions for Hopf bifurcation are obtained. A numerical example as well as its simulation plots and phase portraits are given at last. Our results indicate that a Hopf bifurcation rises near the zero equilibrium point when the leakage delay reaches its critical value which is given by an explicit formula. Particularly, the results of numerical simulations show that the leakage delay would narrow the stability region of the proposed system and make the Hopf bifurcation occur earlier.

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Literatur
1.
Zurück zum Zitat Kosko B. Adaptive bi-directional associative memories. Appl Opt. 1987;26:4947–60.CrossRef Kosko B. Adaptive bi-directional associative memories. Appl Opt. 1987;26:4947–60.CrossRef
2.
Zurück zum Zitat Cao JD, Zhou DM. Stability analysis of delayed cellular neural networks. Neural Netw. 1998;11:1601–5.CrossRef Cao JD, Zhou DM. Stability analysis of delayed cellular neural networks. Neural Netw. 1998;11:1601–5.CrossRef
3.
Zurück zum Zitat Cao JD, Wang L. Exponential stability and periodic oscillatory solution in BAM networks with delays. IEEE Trans Neural Netw. 2002;13(2):457–63.CrossRef Cao JD, Wang L. Exponential stability and periodic oscillatory solution in BAM networks with delays. IEEE Trans Neural Netw. 2002;13(2):457–63.CrossRef
4.
Zurück zum Zitat Cao JD, Liang JL, James J. Exponential stability of high-order bidirectional associative memory neural networks with time delays. Physica D. 2004;199(3–4):425–36.MathSciNetCrossRef Cao JD, Liang JL, James J. Exponential stability of high-order bidirectional associative memory neural networks with time delays. Physica D. 2004;199(3–4):425–36.MathSciNetCrossRef
5.
Zurück zum Zitat Cao JD, Song QK. Stability in Cohen-Grossberg-type bidirectional associative memory neural networks with time-varying delays. Nonlinearity. 2006;19:1601–17.MathSciNetCrossRef Cao JD, Song QK. Stability in Cohen-Grossberg-type bidirectional associative memory neural networks with time-varying delays. Nonlinearity. 2006;19:1601–17.MathSciNetCrossRef
6.
Zurück zum Zitat Cao JD, Xiao M. Stability and Hopf bifurcation in a simplified BAM neural network with two time delays. IEEE Trans Neural Netw. 2007;18(2):416–30.CrossRef Cao JD, Xiao M. Stability and Hopf bifurcation in a simplified BAM neural network with two time delays. IEEE Trans Neural Netw. 2007;18(2):416–30.CrossRef
7.
Zurück zum Zitat Zhou FY, Ma CR. Global exponential stability of high-order BAM neural networks with reaction-diffusion terms. Int J Bifurcat Chaos. 2010;20(10):3209–23.MathSciNetCrossRef Zhou FY, Ma CR. Global exponential stability of high-order BAM neural networks with reaction-diffusion terms. Int J Bifurcat Chaos. 2010;20(10):3209–23.MathSciNetCrossRef
8.
Zurück zum Zitat Wang BX, Jian JG. Stability and Hopf bifurcation analysis on a four-neuron BAM neural network with distributed delays. Commun Nonlinear Sci Numer Simulat. 2010;15:189–204.MathSciNetCrossRef Wang BX, Jian JG. Stability and Hopf bifurcation analysis on a four-neuron BAM neural network with distributed delays. Commun Nonlinear Sci Numer Simulat. 2010;15:189–204.MathSciNetCrossRef
9.
Zurück zum Zitat Wu HX, Liao XF, Feng W, Guo ST. Mean square stability of uncertain stochastic BAM neural networks with interval time-varying delays. Cogn Neurodyn. 2021;6:443–58.CrossRef Wu HX, Liao XF, Feng W, Guo ST. Mean square stability of uncertain stochastic BAM neural networks with interval time-varying delays. Cogn Neurodyn. 2021;6:443–58.CrossRef
10.
Zurück zum Zitat Wang TY, Zhu QX. Stability analysis of stochastic BAM neural networks with reaction-diffusion, multi-proportional and distributed delays. Physica A. 2019;533:121935.MathSciNetCrossRef Wang TY, Zhu QX. Stability analysis of stochastic BAM neural networks with reaction-diffusion, multi-proportional and distributed delays. Physica A. 2019;533:121935.MathSciNetCrossRef
11.
Zurück zum Zitat Tao BB, Xiao M, Zheng WX, Cao JD. Dynamics analysis and design for a bidirectional super-ring-shaped neural network with \(n\) neurons and multiple delays. IEEE Trans Neural Netw Learn Syst. 2021;32:2978–92.MathSciNetCrossRef Tao BB, Xiao M, Zheng WX, Cao JD. Dynamics analysis and design for a bidirectional super-ring-shaped neural network with \(n\) neurons and multiple delays. IEEE Trans Neural Netw Learn Syst. 2021;32:2978–92.MathSciNetCrossRef
12.
Zurück zum Zitat Nicolis JS. Chaos and Information Processing. Singapore: World Scientific; 1991.CrossRef Nicolis JS. Chaos and Information Processing. Singapore: World Scientific; 1991.CrossRef
13.
Zurück zum Zitat Shilnikov AL, Cymbalyuk GS. Transition between Tonio-spiking and bursting in a neuron model via the blue-sky catastrophe. Phys Rev Lett. 2005;94(4):048101. Shilnikov AL, Cymbalyuk GS. Transition between Tonio-spiking and bursting in a neuron model via the blue-sky catastrophe. Phys Rev Lett. 2005;94(4):048101.
14.
Zurück zum Zitat Yu WW, Cao JD. Stability and Hopf bifurcation analysis on a four-neuron BAM neural network with time delays. Phys Lett A. 2006;351:64–78.CrossRef Yu WW, Cao JD. Stability and Hopf bifurcation analysis on a four-neuron BAM neural network with time delays. Phys Lett A. 2006;351:64–78.CrossRef
15.
Zurück zum Zitat Xu CJ, Zhang QM, Wu YS. Bifurcation analysis for two-neuron networks with discrete and distributed delays. Cogn Comput. 2016;8(6):1103–18.CrossRef Xu CJ, Zhang QM, Wu YS. Bifurcation analysis for two-neuron networks with discrete and distributed delays. Cogn Comput. 2016;8(6):1103–18.CrossRef
16.
Zurück zum Zitat Javidmanesh E, Dadi Z, Afsharnezhad Z. Existence and Stability Analysis of Bifurcating Periodic Solutions in a Delayed Five-Neuron BAM Neural Network Model. Nonlinear Dyn. 2013;72(1):149–64.MathSciNetCrossRef Javidmanesh E, Dadi Z, Afsharnezhad Z. Existence and Stability Analysis of Bifurcating Periodic Solutions in a Delayed Five-Neuron BAM Neural Network Model. Nonlinear Dyn. 2013;72(1):149–64.MathSciNetCrossRef
17.
Zurück zum Zitat Javidmanesh E, Dadi Z, Afsharnezhad Z, Effati S. Global stability analysis and existence of periodic solutions in an eight-neuron BAM neural network model with delays. J Intell Fuzzy Syst. 2014;27:391–406.MathSciNetCrossRef Javidmanesh E, Dadi Z, Afsharnezhad Z, Effati S. Global stability analysis and existence of periodic solutions in an eight-neuron BAM neural network model with delays. J Intell Fuzzy Syst. 2014;27:391–406.MathSciNetCrossRef
18.
Zurück zum Zitat Liu YW, Li SS, Liu ZR, Wang RQ. High codimensional bifurcation analysis to a six-neuron BAM neural network. Cogn Neurodyn. 2016;10:149–64.CrossRef Liu YW, Li SS, Liu ZR, Wang RQ. High codimensional bifurcation analysis to a six-neuron BAM neural network. Cogn Neurodyn. 2016;10:149–64.CrossRef
19.
Zurück zum Zitat Hilfer R. Applications of fractional calculus in physics. Singapore: World Scientific York; 2000. Hilfer R. Applications of fractional calculus in physics. Singapore: World Scientific York; 2000.
20.
Zurück zum Zitat Magin R. Fractional calculus in bioengineering. Crit Rev Biomed Eng. 2004;32(1):1–104.CrossRef Magin R. Fractional calculus in bioengineering. Crit Rev Biomed Eng. 2004;32(1):1–104.CrossRef
21.
Zurück zum Zitat Kibas AA, Srivastava HM, Trujillo JJ. Theory and application of fractional differential equations. New York: Elsevier; 2006. Kibas AA, Srivastava HM, Trujillo JJ. Theory and application of fractional differential equations. New York: Elsevier; 2006.
22.
Zurück zum Zitat Zhang JM, Wu JW, Bao HB, Cao JD. Synchronization analysis of fractional-order three-neuron BAM neural networks with multiple time delays. Appl Math Comput. 2018;339:441–50.MathSciNet Zhang JM, Wu JW, Bao HB, Cao JD. Synchronization analysis of fractional-order three-neuron BAM neural networks with multiple time delays. Appl Math Comput. 2018;339:441–50.MathSciNet
23.
Zurück zum Zitat Shafiya M, Nagamani G, Dafik D. Global synchronization of uncertain fractional-order BAM neural networks with time delay via improved fractional-order integral inequality. Math Comput Simulat. 2022;191:168–86.MathSciNetCrossRef Shafiya M, Nagamani G, Dafik D. Global synchronization of uncertain fractional-order BAM neural networks with time delay via improved fractional-order integral inequality. Math Comput Simulat. 2022;191:168–86.MathSciNetCrossRef
24.
Zurück zum Zitat Bao HB, Ju HP, Cao JD. Non-fragile state estimation for fractional-order delayed memristive BAM neural networks. Neural Netw. 2019;119:190–9.CrossRef Bao HB, Ju HP, Cao JD. Non-fragile state estimation for fractional-order delayed memristive BAM neural networks. Neural Netw. 2019;119:190–9.CrossRef
25.
Zurück zum Zitat Nagamani G, Shafiya M, Soundararajan G, Prakash M. Robust state estimation for fractional-order delayed BAM neural networks via LMI approach. J Franklin Inst. 2020;357:4964–82.MathSciNetCrossRef Nagamani G, Shafiya M, Soundararajan G, Prakash M. Robust state estimation for fractional-order delayed BAM neural networks via LMI approach. J Franklin Inst. 2020;357:4964–82.MathSciNetCrossRef
26.
Zurück zum Zitat Wu AL, Zeng ZG, Song XG. Global Mittag-Leffler stabilization of fractional-order bidirectional associative memory neural networks. Neurocomputing. 2016;177:489–96. Wu AL, Zeng ZG, Song XG. Global Mittag-Leffler stabilization of fractional-order bidirectional associative memory neural networks. Neurocomputing. 2016;177:489–96.
27.
Zurück zum Zitat Xu CJ, Aouiti C, Liu ZX. A further study on bifurcation for fractional order BAM neural networks with multiple delays. Neurocomputing. 2020;417:501–15.CrossRef Xu CJ, Aouiti C, Liu ZX. A further study on bifurcation for fractional order BAM neural networks with multiple delays. Neurocomputing. 2020;417:501–15.CrossRef
28.
Zurück zum Zitat Xu CJ, Liao MX, Li PL, Guo Y, Liu ZX. Bifurcation Properties for fractional order delayed BAM neural networks. Cogn Comput. 2021;13:322–56.CrossRef Xu CJ, Liao MX, Li PL, Guo Y, Liu ZX. Bifurcation Properties for fractional order delayed BAM neural networks. Cogn Comput. 2021;13:322–56.CrossRef
29.
Zurück zum Zitat Huang CD, Meng YJ, Cao JD, Alsaedi A, Alsaadi FE. New bifurcation results for fractional BAM neural network with leakage delay. Chaos, Solitons Fractals. 2017;100:31–44.MathSciNetCrossRef Huang CD, Meng YJ, Cao JD, Alsaedi A, Alsaadi FE. New bifurcation results for fractional BAM neural network with leakage delay. Chaos, Solitons Fractals. 2017;100:31–44.MathSciNetCrossRef
30.
Zurück zum Zitat Huang CD, Liu H, Chen YF, Chen XP, Song F. Dynamics of a fractional-order BAM neural network with leakage delay and communication delay. Fractals. 2021;29(3):2150073.CrossRef Huang CD, Liu H, Chen YF, Chen XP, Song F. Dynamics of a fractional-order BAM neural network with leakage delay and communication delay. Fractals. 2021;29(3):2150073.CrossRef
32.
Zurück zum Zitat Tehrani NF, Razvan MR. Bifurcation structure of two coupled FHN neurons with delay. Math Biosci. 2015;270:41–56.MathSciNetCrossRef Tehrani NF, Razvan MR. Bifurcation structure of two coupled FHN neurons with delay. Math Biosci. 2015;270:41–56.MathSciNetCrossRef
33.
Zurück zum Zitat Podlubny I. Fractional differential equations. New York: Academic Press; 1999. Podlubny I. Fractional differential equations. New York: Academic Press; 1999.
34.
Zurück zum Zitat Bandyopadhyay B, Kamal S. Stabilization and control of fractional order systems: a sliding mode approach. Lecture Notes Electr Eng. 2015;317:115–27.MathSciNetCrossRef Bandyopadhyay B, Kamal S. Stabilization and control of fractional order systems: a sliding mode approach. Lecture Notes Electr Eng. 2015;317:115–27.MathSciNetCrossRef
35.
Zurück zum Zitat Matignon D. Stability results for fractional differential equations with applications to control processing. Computat Eng Syst Appl. 1996;2:963–8. Matignon D. Stability results for fractional differential equations with applications to control processing. Computat Eng Syst Appl. 1996;2:963–8.
36.
Zurück zum Zitat Fan DJ, Wei JJ. Hopf bifurcation analysis in a tri-neuron network with time delay. Nonlinear Anal RWA. 2008;9:9–25.MathSciNetCrossRef Fan DJ, Wei JJ. Hopf bifurcation analysis in a tri-neuron network with time delay. Nonlinear Anal RWA. 2008;9:9–25.MathSciNetCrossRef
Metadaten
Titel
Effect of Leakage Delays on Bifurcation in Fractional-Order Bidirectional Associative Memory Neural Networks with Five Neurons and Discrete Delays
verfasst von
Yangling Wang
Jinde Cao
Chengdai Huang
Publikationsdatum
05.06.2024
Verlag
Springer US
Erschienen in
Cognitive Computation / Ausgabe 5/2024
Print ISSN: 1866-9956
Elektronische ISSN: 1866-9964
DOI
https://doi.org/10.1007/s12559-024-10305-0

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