Mironenko reflecting function

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

Definition

For the differential system ${\displaystyle {\dot {x}}=X(t,x)}$ with the general solution ${\displaystyle \varphi (t;t_{0},x)}$ in Cauchy form Reflecting Function is defined by formula ${\displaystyle F(t,x)=\varphi (-t;t,x).}$

Application

If a vector-function ${\displaystyle X(t,x)}$ is ${\displaystyle \,2\omega }$-periodic with respect to ${\displaystyle \,t}$, then ${\displaystyle \,F(-\omega ,x)}$ is the in-period ${\displaystyle \,[-\omega ;\omega ]}$ transformation (Poincaré map) of the differential system ${\displaystyle {\dot {x}}=X(t,x).}$ Therefore the knowledge of the Reflecting Function give us the opportunity to find out the initial dates ${\displaystyle \,(\omega ,x_{0})}$ of periodic solutions of the differential system ${\displaystyle {\dot {x}}=X(t,x)}$ and investigate the stability of those solutions.

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

${\displaystyle \,F_{t}+F_{x}X+X(-t,F)=0,\qquad F(0,x)=x.}$

is holding.

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