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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 \(y\).

34A26 Geometric methods in ordinary differential equations
34C25 Periodic solutions to ordinary differential equations
Full Text: DOI
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