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On the reflective function of polynomial differential system. (English) Zbl 1034.34008
The author investigates the planar differential system \[ \begin{aligned} \dot{x} &=P(t,x,y):=a(t,x)+b(t,x)y+c(t,x)y^2,\\ \dot{y} &=Q(t,x,y):=e(t,x)+f(t,x)y+g(t,x)y^2, \end{aligned} \tag{1} \] with continuously differentiable coefficients by the method of reflective functions (RF). The theory of RF was established in 1986 by V. I. Mironenko [Reflecting function and periodic solutions of differential equations. (Russian). Minsk: Izdatel’stvo “Universitetskoe” (1986; Zbl 0607.34038)] in order to overcome some difficulties in the construction of a PoincarĂ© mapping for a differential system \[ \dot{x}=X(t,x). \tag{2} \] A continuously differential vector function \(F(t,x):\mathbb{R}\times \mathbb{R}^n\to \mathbb{R}^n\) is called RF, if it is a solution to the Cauchy problem \[ F_{t}(t,x)+F_{x}(t,x)X(t,x)+X(-t,F(t,x))=0, \quad F(0,x)=x. \] The main results concern necessary and sufficient conditions for periodicity and stability of solutions to (1) when the first component of RF does not depend on \(y\).

MSC:
34A26 Geometric methods in ordinary differential equations
34C25 Periodic solutions to ordinary differential equations
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