In applied mathematics, the reflecting function F ( t , x ) {\displaystyle \,F(t,x)} of a differential system x ˙ = X ( t , x ) {\displaystyle {\dot {x}}=X(t,x)} connects the past state x ( − t ) {\displaystyle \,x(-t)} of the system with the future state x ( t ) {\displaystyle \,x(t)} of the system by the formula x ( − t ) = F ( t , x ( t ) ) . {\displaystyle \,x(-t)=F(t,x(t)).} The concept of the reflecting function was introduced by Uladzimir Ivanavich Mironenka.
Definition
For the differential system x ˙ = X ( t , x ) {\displaystyle {\dot {x}}=X(t,x)} with the general solution φ ( t ; t 0 , x ) {\displaystyle \varphi (t;t_{0},x)} in Cauchy form, the Reflecting Function of the system is defined by the formula F ( t , x ) = φ ( − t ; t , x ) . {\displaystyle F(t,x)=\varphi (-t;t,x).}
Application
If a vector-function X ( t , x ) {\displaystyle X(t,x)} is 2 ω {\displaystyle \,2\omega } -periodic with respect to t {\displaystyle \,t} , then F ( − ω , x ) {\displaystyle \,F(-\omega ,x)} is the in-period [ − ω ; ω ] {\displaystyle \,[-\omega ;\omega ]} transformation (Poincaré map) of the differential system x ˙ = X ( t , x ) . {\displaystyle {\dot {x}}=X(t,x).} Therefore the knowledge of the Reflecting Function give us the opportunity to find out the initial dates ( ω , x 0 ) {\displaystyle \,(\omega ,x_{0})} of periodic solutions of the differential system x ˙ = X ( t , x ) {\displaystyle {\dot {x}}=X(t,x)} and investigate the stability of those solutions.
For the Reflecting Function F ( t , x ) {\displaystyle \,F(t,x)} of the system x ˙ = X ( t , x ) {\displaystyle {\dot {x}}=X(t,x)} the basic relation
F t + F x X + X ( − t , F ) = 0 , F ( 0 , x ) = x . {\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.