PERIODIC OSCILLATORY SOLUTIONS FOR A THREE-LAYER NETWORK MODEL WITH DELAYS

Main Article Content

CHUNHUA FENG

Abstract

This paper investigates a network model incorporating multiple discrete delays. The existence of periodic oscillatory solutions for such a three-layer neural network has been derived. Due to the multiple time delays, bifurcating method is hard to detect the existence of periodic oscillatory solutions for this model. Some criteria by means of the mathematical analysis method are provided to guarantee the existence of periodic oscillatory solutions. Our criterion is very easy to check.
Some simulations are presented to demonstrate the correctness of the results. Computer simulation indicates that the present theorems are only sufficient conditions.

Keywords:
Three-layer network model, instability, periodic oscillatory solution

Article Details

How to Cite
FENG, C. (2021). PERIODIC OSCILLATORY SOLUTIONS FOR A THREE-LAYER NETWORK MODEL WITH DELAYS. Asian Journal of Mathematics and Computer Research, 28(3), 22-32. Retrieved from https://www.ikprress.org/index.php/AJOMCOR/article/view/6960
Section
Original Research Article

References

Wang TS, Cheng ZS, Bu R, Ma RS. Stability and Hopf bifurcation analysis of a simplified six neuron tridiagonal two-layer neural network model with delay. Neurocomputing. 2019;112:203- 214.

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

Cheng ZS, Li DH, Cao JD. Stability and Hopf bifurcation of a three-layer neural network model with delays. Neurocomputing. 2016;175:355-370.

Bazighifan O, El-Nabulsi RA. Different techniques for studying oscillatory behavior of solution of differential equations. Rocky Mountain J. Math. 2021;51:77-86.

Feng; AJOMCOR, 28(3): 22-32, 2021

Moaaz O, El-Nabulsi RA, Bazighifan O. Behavior of non-oscillatory solutions of fourth-order neutral differential equations. Symmetry. 2020;12:477.

Available:https://doi.org/10.3390/sym12030477

Moaaz O, El-Nabulsi RA, Bazighifan O. Oscillatory behavior of fourth-order differential equations with neutral delay. Symmetry. 2020;12:371.

Available:https://doi.org/10.3390/sym12030371.

Hassan TS, El-Nabulsi RA, Menaem AA. Amended criteria of oscillation for nonlinear functional dynamic equations of second-order. Mathematics. 2021;9:1191.

Available:https://doi.org/10.3390/math9111191.

Santra SS, El-Nabulsi RA, Khedher KM. Oscillation of second-order differential equations with multiple and mixed delays under a canonical operator. Mathematics. 2021;9:1323. Available:https://doi.org/10.3390/math9121323

El-Nabulsi RA. Nonlocal-in-time kinetic energy in nonconservative fractional systems, disordered dynamics, jerk and snap and oscillatory motions in the rotating fluid tube. Int. J. Non-Linear Mechanics. 2017;93:65-81.

El-Nabulsi RA. Fractional differential operators and generalized oscillatory dynamics. Thai J. Mathematics. 2020;18:715-732.

El-Nabulsi RA. A generalized nonlinear oscillator from non-standard degenerate Lagrangians and its consequent Hamiltonian formalism, Proceedings of the National Academy of Sciences, India Section A: Physical Sciences. 2014;84:563-569.

Alzabut J, Selvam AG, El-Nabulsi RA, Dhakshinamoorthy V, Samei ME. Asymptotic stability of nonlinear discrete fractional pantograph equations with non-local initial conditions. Symmetry. 2021;13:473.

Available:https://doi.org/10.3390/sym1303047

Weg BP, Greve L, Andres M, Eller TK, Rosic B. Neural network-based surrogate model for a bifurcating structural fracture response. Engineering Fracture Mechanics. 2021;241:107424.

Selimefendigila F, Oztop HF. Thermoelectric generation in bifurcating channels and efficientmodeling by using hybrid CFD and artificial neural networks. Renewable Energy. 2021;172:582-598.

Tyagi S, Jain SK, Abbas S, Meherrem S, Ray RK. Time delay induced instabilities and Hopf bifurcation analysis in 2-neuron network model with reaction- diffusion term. Neurocomputing. 2018;313:306-315.

Zeng XC, Xiong ZL, Wang CJ. Hopf bifurcation for neutral-type neural network model with two delays. Appl. Math. Comput. 2016;282:17-31.

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

Gupta PD, Majee NC, Roy AB. Stability, bifurcation and global existence of a Hopf-bifurcating periodic solution for a class of three-neuron delayed network models. Nonlinear Analysis: TMA. 2007;62:2934-2954.

Yan XP. Bifurcation analysis in a simplified tri-neuron BAM network model with multiple delays. Nonlinear Analysis: TMA. 2008;9:963-976.

Wang X, Wang Z, Lu JW, Meng B. Stability, bifurcation and chaos of a discrete-time pair approximation epidemic model on adaptive networks. Math. Comput. Simul. 2021;182:182- 194.

Ferguson T. A new approach to bifurcations in the Kuramoto model. J, Math. Anal. Appl. 2021;502:125205.

Feng; AJOMCOR, 28(3): 22-32, 2021

Ramfrez OJ, Dominguez EJ, Juarez MA, Aragon JL, Medina RV. Experimental detection of Hopf bifurcation in two-dimensional dynamical systems. Chaos, Solitons and Fractals. 2021;X 6:100058.

Xu CJ, Liu ZX, Yao LY, Aouiti C. Further exploration on bifurcation of fractional-order six-neuron bi-directional associative memory neural networks with multi-delays. Appl. Math. Comput. 2021;410:126458.

Huang CD, Meng YJ, Cao JD, Alsaedi A, Alsaadi FE. New bifurcation results for fractional BAM neural network with leakage delay. Chaos, Solitons and Fractals. 2017;100:31-44.

Gopalsamy K. Stability and oscillations in delay different equations of population dynamics. Kluwer Academic, Dordrecht, Norwell, MA; 1992.

Hale JK, Verduyn SM. Introduction to functional differential equations. New York, Springer; 1993.

Chafee N. A bifurcation problem for a functional differential equation of finitely retarded type. J. Math Anal.Appl. 1971;35:312-348.

Feng C, Plamondon R. An oscillatory criterion for a time delayed neural ring network model. Neural Networks. 2012;29-30:70-79.