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Isochronous centers of a linear center perturbed by fourth degree homogeneous polynomial. (English) Zbl 0921.34032

The authors study the isochronous centers of two-dimensional autonomous systems in the plane \[ \dot x= -y+X_s(x, y),\quad \dot y=x+Y_s(x,y), \] where \(X_s(x,y)\) and \(Y_s (x,y\)) are homogeneous polynomials of degree \(s=4.\) A center is isochronous if the period of all integral curves in a neighborhood of the origin is constant. Necessary conditions for such isochronous center are obtained. Reversible systems that have an isochronous center at the origin are studied.

MSC:

34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34A05 Explicit solutions, first integrals of ordinary differential equations
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