Stability and Bifurcation Analysis in a Two-Dimensional Neutral Differential Equation
DOI:
https://doi.org/10.5890/DNC.2019.12.004Abstract
This paper discusses asymptotic stability and Hopf bifurcations occurs at the origin in certain two-dimensional neutral delay differential equations. We give necessary and sufficient conditions on the parameters to obtain the asymptotic stability and bifurcations. Global existence of periodic solutions is established using a global Hopf bifurcation result of Krawcewicz et al. Finally, some numerical simulations are carried out to support the analytic results. Our results are a generalization of M. Liu and X. Xu [1].References
[1] Liu, M. and Xu, X. (2013), Bifurcation Analysis in a Two-Dimensional Neutral Differential Equation, Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2013, Article ID 367589.
[2] Wei, J.J. and Ruan, S.G. (2002), Stability and global Hopf bifurcation for neutral differential equations, Acta Mathematica Sinica, 45(1), 93-104.
[3] Wang, C. and Wei, J. (2010), Hopf bifurcation for neutral functional differential equations, Nonlinear Analysis: Real World Applications, 11(3), 1269-1277.
[4] Qu, Y., Li, M.Y., and Wei, J. (2011), Bifurcation analysis in a neutral differential equation, Journal of Mathematical Analysis and Applications, 378(2), 387-402.
[5] Moussaid, A. and Talibi Alaoui, H. (2015), Study of numerical stability for neutral differential equations with delay by θ -method, J. British Journal of Applied Science & Technology, 13(1).
[6] Su, H., Li, W., and Ding, X. (2013), Preservation of Hopf bifurcation for neutral delay-differential equations by θ - methods, J. of Computational and Applied Mathematics, 248, 76-87.
[7] Chen, Y. and Wu. J. (2001), Slowly oscillating periodic solutions for a delayed frustrated network of two neurons, Journal of Mathematical Analysis and Applications, 259(1), 188-208.
[8] Wei, J. and Ruan, S. (1999), Stability and bifurcation in a neural network modelwith two delays, Physica D, 130(3-4), 255-272.
[9] Faria, T. (2000), On a planar systemmodelling a neuron networkwith memory, Journal of Differential Equations, 168(1), 129-149.
[10] Wei, J., Velarde,M.G., andMakarov,V.A. (2002), Oscillatory phenomena and stability of periodic solutions in a simple neural network with delay, Nonlinear Phenomena in Complex Systems, 5(4), 407-417.
[11] Wei, J.J., Zhang, C.R., and Li, X.L. (2005), Bifurcation in a twodimensional neural network model with delay, Applied Mathematics and Mechanics, 26(2), 193-200.
[12] Hale, J.K. and Lunel, M.V. (1993), Introduction to functional-differential equations, vol. 99 of AppliedMathematical Sciences, Springer, NewYork, NY, USA.
[13] Wang, C. and Wei, J. (2008), Normal forms forNFDEs with parameters and application to the lossless transmission line, Nonlinear Dynamics, 52(3), 199-206.
[14] Wiggins, S. (1990), Introduction to Applied Nonlinear Dynamical Systems and Chaos, vol. 2 of Texts in AppliedMathematics, Springer, New York, NY, USA.
[15] Hassard, B.D., Kazarinoff, N.D., and Wan, Y.H. (1981), Theory and Applications of Hopf Bifurcation, Cambridge University Press, Cambridge.
[16] Chen, Y. and Bi, Y.Q. (2014), Stability and Hopf Bifurcation Analysis of a Vector-Borne Disease with Time Delay, Hindawi Publishing Corporation Journal of Applied Mathematics Vol. 2014, Article ID 804204.
[17] Krawcewicz,W.,Wu, J., and Xia, H. (1993), GlobalHopf bifurcation theory for condensing fields and neutral equationswith applications to lossless transmission problems, The Canadian Applied Mathematics Quarterly, 1(2), 167-220.
[18] Wu, J. (1998), Symmetric functional differential equations and neural networks with memory, Trans. Amer. Math. Soc., 350, 4799-4838.
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