# Mironenko reflecting function

The reflecting function $\,F(t,x)$ of a dynamical system connects the past state $\,x(-t)$ of it with the future state $\,x(t)$ of it by the formula $\,x(-t)=F(t,x(t)).$ The concept of the reflecting function was introduсed by Uladzimir Ivanavich Mironenka.

## Definition

For the differential system $\dot x=X(t,x)$ with the general solution $\varphi(t;t_0,x)$ in Cauchy form Reflecting Function is defined by formula $F(t,x)=\varphi(-t;t,x).$

## Application

If a vector-function $\,x(-t)$ is periodic in $\,2\omega$ with respect to $\,t$, then $\,F(-\omega,x)$ is a transformation (Poincaré map) periodic in $\,[-\omega;\omega]$ of the differential system $\dot x=X(t,x).$ Therefore the knowledge of the Reflecting Function give us the opportunity to find out the initial date $\,(\omega,x_0)$ of periodic solutions of the differential system and investigate the stability of those solutions.

For the Reflecting Function $\,F(t,x)$ of the system $\dot x=X(t,x)$ the basic relation

$\,F_t+F_x X+X(-t,F)=0,\qquad F(0,x)=x.$

is holding.

Therefore we have an opportunity sometimes to find Poincaré map of the non-integrable in quadrature systems even in elementary functions.