An explicit expression for the first return map in the center problem. (English) Zbl 1066.34026

After an appropriate change of variable, this paper is concerned with plane autonomous systems of the form \(dx/dt=-y+F(x,y)\), \(dy/dt =x+G(x,y)\), where \(F\) and \(G\) are real analytic functions whose Taylor expansions at the origin do not contain constant and linear terms. Generally, the origin can be either a center or a focus. The author investigates this classical Poincaré problem which characterizes \(F\) and \(G\) in order that the origin is a center. The author presents a method by which an explicit expression for the first return map is obtained for this center problem.


34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
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