Mironenko reflecting function
In applied mathematics, the reflecting function formula_1 of a differential system formula_2 connects the past state formula_3 of the system with the future state formula_4 of the system by the formula formula_5 The concept of the reflecting function was introduced by Uladzimir Ivanavich Mironenka. Definition. For the differential system formula_2 with the general solution formula_7 in Cauchy form, the Reflecting Function of the system is defined by the formula formula_8 Application. If a vector-function formula_9 is formula_10-periodic with respect to formula_11, then formula_12 is the in-period formula_13 transformation (Poincaré map) of the differential system  formula_14 Therefore the knowledge of the Reflecting Function give us the opportunity to find out the initial dates formula_15 of periodic solutions of the differential system formula_2 and investigate the stability of those solutions. For the Reflecting Function formula_1 of the system formula_2 the basic relation is holding. Therefore we have an opportunity sometimes to find Poincaré map of the non-integrable in quadrature systems even in elementary functions.