差分方程Xn+1=axn+bX2n/(cxn+dn-1)的解性质的研究
Studying of Property of the Solution of the Nonlinear Difference EquationXn+1=axn+bX2n/(cxn+dn-1)

作者: 颜丽敏 * , 许晓婕 , 刘静 :;

关键词: 有理性差分方程渐进稳定性全局吸引性Rational Difference Equation Asymptotical Stability Global Attractivity

摘要:

文献[1]中研究了差分方程Xn+1=axn+bX2n/(cxn+dn-1)的解性质的研究,(a,b,c,d是正实数)解的有界性、平衡点的局部及全局吸引性,并给出了其几类特殊情形下的解的表达式。本文首先指出文献[1]中的第23段的讨论方法是不正确的。然后本文中我们对平衡点的性质进行了讨论,分为两种情况进行研究,分别为 (1-a )(c+d)≠ b(1-a )(c+d)= b 。在第一种情况中可以得到平衡解在适当的条件下是全局吸引的,在第二种情况下得到任意实数都是平衡解且是不渐进稳定的。在每种情况中举了一些数值例子,并用相应的软件画出其图像,从而进一步证明了结论的正确性。

In paper [1], some properties are investigated about difference equation Xn+1=axn+bX2n/(cxn+dn-1)

Abstract: Studying of Property of the Solution of the Nonlinear Difference EquationXn+1=axn+bX2n/(cxn+dn-1)
where a>0,b>0,c>0,d>0, and the initial conditions x1,x 0∈0+∞such as the boundedness of the solution and the properties of the equilibrium point. In this paper we will prove that some methods used in paper [1] are wrong at first. Then we continue to study some properties of the difference equation under dif-ferent circumstances. We get some interesting conclusions. At last, we use some numerical examples to verify the conclusions that we get in this paper.

文章引用: 颜丽敏 , 许晓婕 , 刘静 (2011) 差分方程Xn+1=axn+bX2n/(cxn+dn-1)的解性质的研究。 理论数学, 1, 198-204. doi: 10.12677/pm.2011.13038

参考文献

[1] F. Arnault, T. P. Berger, C. Lauradoux, M. Minier and B. Pousse. A new approach for FCSRs. In: M. J. J. V. Rijmen Jr., R. Safavi-Naini (Eds.), Selected areas in cryptography. New York: Springer, 2009, 5867: 433-448.

[2] F. Arnault, T. P. Berger and B. Pousse. A matrix approach for FCSR automata. Cryptography and Communications, Springer Science + Business Media, 2011, 3(2): 109-139.

[3] T. P. Berger, M. Minier and B. Pousse. Software oriented stream ciphers based upon FCSRs in diversified mode. In: B. Roy, N. Sendrier (Eds.), INDOCRYPT 2009, Lecture notes in computer science. New York: Springer, 2009, 5922: 119-135.

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