Reflecting function and solutions of two-point boundary value problems for nonautonomous two-dimensional differential systems (Q414063)

The reflecting function \(F(t,x)\) of a dynamical system introduced by Mironenko [\textit{V. I. Mironenko}, Reflecting function and periodic solutions of differential equations. Minsk: Izdatel'stvo ``Universitetskoe''. (1986; Zbl 0607.34038)] connects the past state and the future state of the system by the formula \(x(-t)=F(t,x(t))\). It is useful in studying periodic solutions of dynamical systems. The present paper establishes necessary and sufficient conditions for the coincidence of the reflecting functions of a second-order nonlinear system \[ \dot x=by+X(t,x,y),\quad \dot y=cx+ Y(t,x,y) \] and its linear approximation (harmonic oscillator) \[ \dot x=by,\quad \dot y=cx \quad (bc<0). \]