The structure of reflective function of polynomial differential systems (Q1021736)

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.