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In applied mathematics, the reflecting function $${\displaystyle \,F(t,x)}$$ of a differential system $${\displaystyle {\dot {x}}=X(t,x)}$$ connects the past state $${\displaystyle \,x(-t)}$$ of the system with the future state $${\displaystyle \,x(t)}$$ of the system by the formula $${\displaystyle \,x(-t)=F(t,x(t)).}$$ The concept of the reflecting function was introduced 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, the Reflecting Function of the system is defined by the 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 $$\,t,$$ 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.
Literature

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.

The Reflecting Function Site
How to construct equivalent differential systems