arXiv 0812.33182008-12-17math.DS

一类平面差分方程组解的整体性态

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

Sukanya Basu, Orlando Merino

备注：: 13 pages

摘要：

建立了一类特殊的差分方程平衡点的局部稳定性与临界曲线斜率之间的关系。然后利用这个结果给出了含正参数的差分方程系统\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*}非负解的整体性态结果。特别是，我们证明了系统有一个和三个之间的平衡点，平衡点的数量决定全局行为如下：如果只有一个平衡点，那么它是全局渐近稳定的。如果有两个平衡点，则一个是局部吸引子，另一个是非双曲的。如果有三个平衡点，则它们在平面的东南方向线性排列，并且由一个局部吸引子、一个鞍点和另一个局部吸引子组成。最后，我们给出了存在唯一平衡点的充分条件。

英文摘要：

We 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.

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摘要：

英文摘要：

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