Stability and periodic character of a rational third order difference equation M. Shojaei a , R. Saadati a,b, * , H. Adibi a a Department of Mathematics and Computer Science, Amirkabir University of Technology, 424 Hafez Avenue, Tehran 15914, Iran b Faculty of Sciences, University of Shomal, Amol 731, Iran Accepted 25 May 2007 Abstract The general solution, the local and global asymptotic stability of equilibrium points and period three cycles of the third order rational difference equation x nþ1 ¼ ax n2 b þ cx n2 x n1 x n ; n ¼ 0; 1; ... are studied in this paper. Ó 2007 Elsevier Ltd. All rights reserved. 1. Introduction Rational difference equations that are the ratio of two polynomials are one of the most important and practical classes of nonlinear difference equations. Fore more about the rational difference equations we refer the reader to [1,3–13,15,16]. Yang et al. [17] examined the rational second-order difference equation x n ¼ ax n1 þbx n2 cþdx n1 x n2 , these difference equations appear naturally as discrete analogues and as numerical solutions of differential and delay differential equa- tions having applications in biology, ecology, physics, etc. [2,15]. Also C ¸ inar [4] investigated the global behavior of all positive solutions of the rational second-order difference equation x nþ1 ¼ x n1 1 þ x n x n1 ; n ¼ 0; 1; ... Similarly, we investigate the stability and periodic character of the rational, third-order difference equation x nþ1 ¼ ax n2 b þ cx n2 x n1 x n ; n ¼ 0; 1; ... ð1Þ where the parameters a, b, c and the initial conditions x 2 , x 1 , x 0 are real numbers. To discuss the global behavior of Eq. (1) we use semiconjugacy and weak contraction arguments. 0960-0779/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2007.06.029 * Corresponding author. Present address: Institute for Studies in Applied Mathematics, 1, Fajr 4, Amol 46176–54553, Iran. E-mail address: rsaadati@eml.cc (R. Saadati). Chaos, Solitons and Fractals 39 (2009) 1203–1209 www.elsevier.com/locate/chaos