Abstract
In this article, we discuss the Poincaré center-focus problem of some cubic differential systems by using a new method (Mironenko’s method) and obtain some new sufficient conditions for a critical point to be a center. By using this method we not only solve a center-focus problem, but also at the same time, we open a class of differential systems, which do not have to be polynomial differential systems, with the same qualitative behavior at the critical point.
Highlights
Introduction and preliminariesConsider the cubic system x = p (x, y) + p (x, y) + p (x, y), ( . )y = q (x, y) + q (x, y) + q (x, y), where pi(x, y) and qi(x, y) are homogeneous polynomials in x and y of degree i (i =, )
We will apply the Mironenko’s method to study the Poincaré centerproblem of the certain cubic differential systems which are in the general form
We introduce the concept of the reflecting function, which will be used throughout the rest of this article
Summary
We will apply the Mironenko’s (reflecting function method) method to study the Poincaré centerproblem of the certain cubic differential systems which are in the general form. There are many papers which are devoted to investigations of qualitative behavior of solutions of differential systems by help of reflecting functions [ – ].
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