Linear recursion formulas of quantities of singular point and applications

In this paper, we study the algorithm of focus quantities and saddle quantities of a real planar autonomous differential system. Two linear recursion formulas for computation of quantities of singular point are obtained. Using the recursion formulas to compute focus quantities or saddle quantities of a real planar autonomous differential system, we need only make use of four fundamental operation of arithmetic in the coefficients of the system, then avoid complex integrating recursion operations, which is necessary in more usual approaches. Therefore, the calculation can be readily done with using computer symbol operation system such as Mathematica. They are far more practical than more usual approaches from the algorithmic point of view, and by which, the focus quantities and saddle quantities of a polynomial system can be computed and simplified in a much simpler way. As a result of straightforward application, we obtain with relative ease the first 8 focus quantities at origin of a cubic system and the sufficient and necessary conditions of integrability in the neighborhood of the origin of this system. (C) 2003 Published by Elsevier Inc.