Dynamics of a nonstandard finite-difference scheme for delay differential equations with unimodal feedback

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• 作者：Yuanyuan Wang ; ¹ ; ^{wangyy@upc.edu.cn} ; ^{y-y-wang@163.com}
• 关键词：Unimodal feedback ; Nonstandard finite-difference ; Delay ; Hopf bifurcation
• 刊名：Communications in Nonlinear Science and Numerical Simulation
• 出版年：2012
• 出版时间：October, 2012
• 年：2012
• 卷：17
• 期：10
• 页码：3967-3978
• 全文大小：524 K

文摘

In this article, by a nonstandard finite-difference (NSFD) scheme we study the dynamics of the delay differential equation with unimodal feedback. First, under three cases local stability of the equilibria is discussed according to Schur polynomial and Hopf bifurcation theory of discrete system. Then, the explicit algorithms for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are derived, using the normal form method and center manifold theorem. In Section , numerical example using Nicholson¡¯s blowflies equation is provided to illustrate the theoretical results. Finally, it demonstrates significant superiority of nonstandard finite-difference scheme than Euler method under the means of describing approximately the dynamics of the original system.