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The structure of reflective function of polynomial differential systems. (English) Zbl 1179.34039
Recall that a reflective function $$F$$ for the initial value problem $x^{\prime}=X(t,x) ,\qquad x(t_{0}) =x_{0}\tag{1}$ is introduced by $$F(t,x) =\psi(-t;t,x),$$ where $$\psi(t;t_{0},x_{0})$$ is a solution of (1). It is known that $$x(t)$$ is a $$2\omega$$-periodic solution of (1) if and only if $$x(-\omega)$$ is a fixed point of the Poincaré map $$T(x) =F(-\omega,x)$$. The author of this very technical paper is concerned with the structure of the reflective function for polynomial systems of differential equations in the plane \begin{aligned}\dot{x} & =p_{0}(t,x) +p_{1}(t,x) y+p_{2}(t,x)y^{2},\\ \dot{y} & =q_{0}(t,x) +q_{1}(t,x) y+\cdots +q_{n}(t,x) y^{n}, \end{aligned} where $$p_{i}$$ and $$q_{i}$$ are continuously differentiable functions in $$x$$ and $$y$$. Two illustrative examples are considered.

##### MSC:
 34C25 Periodic solutions to ordinary differential equations
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##### References:
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