Mironenko reflecting function

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In applied mathematics, the reflecting function of a differential system connects the past state of the system with the future state of the system by the formula The concept of the reflecting function was introduced by Uladzimir Ivanavich Mironenka.

Definition

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For the differential system   with the general solution   in Cauchy form, the Reflecting Function of the system is defined by the formula  

Application

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If a vector-function   is  -periodic with respect to  , then   is the in-period   transformation (Poincaré map) of the differential system   Therefore the knowledge of the Reflecting Function give us the opportunity to find out the initial dates   of periodic solutions of the differential system   and investigate the stability of those solutions.

For the Reflecting Function   of the system   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.

Literature

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