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Influence of high-dimension terms for qualitative structure of solutions of a second order linear difference system with ordinary coefficient in the neighborhood of a singular point. (Chinese. English summary) Zbl 0673.39002

This paper is devoted to the study of the following two-dimensional linear difference system \((1)\quad X(k+1)=\left( \begin{matrix} a\quad b\\ c\quad d\end{matrix} \right)X(k),\) a, b, c and d are constant, where X(k): \(J_+\to R^ 2\), and \(J_+\) being the set of all nonnegative integers. Assuming that the origin is the only isolated singular point of (1), the author proves five theorems which conclude that if the origin is a node, a focus, or a saddle point then the type, stability properties, and oscillatory properties will not change when higher order terms are added to the right member of (1). However, it is proved that if the origin is a center point of the system then it may change into a focus point by adding suitable higher order terms to the system (1).
The results obtained are discrete analogues of some known results in the qualitative theory of ordinary differntial equations. The author applied some definitions and results given by J. Hurt [SIAM 4, 582-596 (1967; Zbl 0264.65076)] and J. A. Heinen [Intern. J. Systems Sci. 10, No.6, 711-718 (1979; Zbl 0419.93062)].
Reviewer: En Hao Yang

MSC:

39A10 Additive difference equations
39A12 Discrete version of topics in analysis
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