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.
For the differential system x ˙ = X ( t , x ) with the general solution φ ( t ; t 0 , x ) in Cauchy form Reflecting Function is defined by formula F ( t , x ) = φ ( − t ; t , x ) .
If a vector-function X ( t , x ) is 2 ω -periodic with respect to t , then F ( − ω , x ) is the in-period [ − ω ; ω ] transformation (Poincaré map) of the differential system x ˙ = X ( t , x ) . Therefore the knowledge of the Reflecting Function give us the opportunity to find out the initial dates ( ω , x 0 ) of periodic solutions of the differential system x ˙ = X ( t , x ) and investigate the stability of those solutions.
For the Reflecting Function F ( t , x ) of the system x ˙ = X ( t , x ) the basic relation
F t + F x X + X ( − t , F ) = 0 , 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.
Мироненко В. И. Отражающая функция и периодические решения дифференциальных уравнений. — Минск, Университетское, 1986. — 76 с.Мироненко В. И. Отражающая функция и исследование многомерных дифференциальных систем. — Гомель: Мин. образов. РБ, ГГУ им. Ф. Скорины, 2004. — 196 с.