Discrete and Continuous Dynamical Systems - Series B (DCDS-B)

 Global bifurcation for discrete competitive systems in the plane Pages: 133 - 149, Volume 12, Issue 1, July 2009 M. R. S. Kulenović - University of Rhode Island, Kingston, RI 02881, United States (email) Orlando Merino - University of Rhode Island, Kingston, RI 02881, United States (email) Abstract: A global bifurcation result is obtained for families of competitive systems of difference equations $x_{n+1} = f_\alpha(x_n,y_n)$ $y_{n+1} = g_\alpha(x_n,y_n)$ where $\alpha$ is a parameter, $f_\alpha$ and $g_\alpha$ are continuous real valued functions on a rectangular domain $\mathcal{R}_\alpha \subset \mathbb{R}^2$ such that $f_\alpha(x,y)$ is non-decreasing in $x$ and non-increasing in $y$, and $g_\alpha(x, y)$ is non-increasing in $x$ and non-decreasing in $y$. A unique interior fixed point is assumed for all values of the parameter $\alpha$.     As an application of the main result for competitive systems a global period-doubling bifurcation result is obtained for families of second order difference equations of the type $x_{n+1} = F_\alpha(x_n, x_{n-1}), \quad n=0,1, \ldots$ where $\alpha$ is a parameter, $F_\alpha:\mathcal{I_\alpha}\times \mathcal{I_\alpha} \rightarrow \mathcal{I_\alpha}$ is a decreasing function in the first variable and increasing in the second variable, and $\mathcal{I_\alpha}$ is a interval in $\mathbb{R}$, and there is a unique interior equilibrium point. Examples of application of the main results are also given. Keywords:  bifurcation, competitive, map, global stable manifold, monotonicity, period-two solution. Mathematics Subject Classification:  Primary: 37G35 Secondary: 39A10, 39A11. Received: September 2008;      Revised: February 2009;      Published: May 2009. 2012 Impact Factor.88 Journal home Editorial board Readers Email alert Bookmark this page Recommend to friend RSS this journal   Authors Submit an article Track your article Guide for authors Tex file preparation Open access Editors Instruction Login Referees Instruction Librarians Order information Abstracted in More Call for special issues