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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.

External links

The Reflecting Function Site
How to construct equivalent differential systems

Undergraduate Texts in Mathematics

Graduate Texts in Mathematics

Graduate Studies in Mathematics

Mathematics Encyclopedia

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