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Local analytic integrability for nilpotent centers. (English) Zbl 1037.34025

Authors’ abstract: Let \(X(x,y)\) and \(Y(x,y)\) be real analytic functions without constant and linear terms defined in a neighborhood of the origin. Assume that the analytic differential system \(\dot{x}=y+ X(x,y), \dot{y}=Y(x,y)\), has a nilpotent center at the origin. The first integrals, formal or analytic, will be real except if we say explicitly the converse. We prove the following:
(1) If \(X= y f(x,y^2)\) and \(Y= g(x,y^2)\), then the system has a local analytic first integral of the form \(H=y^2+F(x,y)\), where \(F\) starts with terms of order higher than two.
(2) If the system has a formal first integral, then it has a formal first integral of the form \(H=y^2+F(x,y)\), where \(F\) starts with terms of order higher than two. In particular, if the system has a local analytic first integral defined at the origin, then it has a local analytic first integral of the form \(H=y^2+F(x,y),\) where \(F\) starts with terms of order higher than two.
As an application, we characterize the nilpotent centers for the differential systems (3) \(\dot{x}=y+P_3(x,y)\), \(\dot{y}=Q_3(x,y)\), which have a local analytic first integral, where \(P_3\) and \(Q_3\) are homogeneous polynomials of degree three.

MSC:

34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34C20 Transformation and reduction of ordinary differential equations and systems, normal forms
34C25 Periodic solutions to ordinary differential equations
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