On the Global Behavior of Solutions to a Planar System of Difference Equations

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dc.creatorBasu, Sukanya
dc.creatorMerino, Orlando
dc.date2008-12-17
dc.date.accessioned2026-07-07T12:16:16Z
dc.date.available2026-07-07T12:16:16Z
dc.descriptionWe establish the relation between local stability of equilibria and slopes of critical curves for a specific class of difference equations. We then use this result to give global behavior results for nonnegative solutions of the system of difference equations \begin{equation*} %\tag{LGIN} \begin{array}{rcl} x_{n+1} & = & \displaystyle \frac{b_1 x_n}{1+x_n+c_1 y_{n}} +h_1 y_{n+1} & = & \displaystyle \frac{b_2 y_n}{1+y_n+c_2 x_{n}} +h_2 \end{array} \quad n=0,1,..., \quad (x_0,y_0) \in [0,\infty)\times [0,\infty) \end{equation*} with positive parameters. In particular, we show that the system has between one and three equilibria, and that the number of equilibria determines global behavior as follows: if there is only one equilibrium, then it is globally asymptotically stable. If there are two equilibria, then one is a local attractor and the other one is nonhyperbolic. If there are three equilibria, then they are linearly ordered in the south-east ordering of the plane, and consist of a local attractor, a saddle point, and another local attractor. Finally, we give sufficient conditions for having a unique equilibrium.
dc.description13 pages
dc.identifierhttps://arxiv.org/abs/0812.3318
dc.identifierhttp://arxiv.org/abs/0812.3318
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/211750
dc.subjectDynamical Systems
dc.subject39A05; 39A11
dc.titleOn the Global Behavior of Solutions to a Planar System of Difference Equations
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