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Research on the properties of some planar polynomial differential equations. (English) Zbl 1251.34048
Consider the differential equation \[ x' = X(t,x), \] where \(X\) is a differentiable function. Let \(\phi(t;t_0,x_0)\) denote the general solution. The reflecting function is defined by \(F(t,x)=\phi(-t;t,x)\). If the equation is \(2\omega\)-periodic in \(t\) then the Poincaré mapping is \(T(x)=F(-\omega,x)\). Hence, a solution \(\phi(t;-\omega,x_0)\) is \(2\omega\)-periodic if and only if \(x_0\) is a fixed point of \(T\). The author uses the method of reflecting functions to study the behavior of solutions and gives sufficient conditions for the equations to have linear or fractional reflecting functions. The results are used to derive sufficient conditions for a center of polynomial two-dimensional systems.

MSC:
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34C07 Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert’s 16th problem and ramifications) for ordinary differential equations
34C25 Periodic solutions to ordinary differential equations
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