Mironenko reflecting function

In applied mathematics, the reflecting function $$\,F(t,x)$$ of a differential system $$\dot x=X(t,x)$$ connects the past state $$\,x(-t)$$ of the system with the future state $$\,x(t)$$ of the system by the formula $$\,x(-t)=F(t,x(t)).$$ The concept of the reflecting function was introduced by Uladzimir Ivanavich Mironenka.

Definition
For the differential system $$\dot x=X(t,x)$$ with the general solution $$\varphi(t;t_0,x)$$ in Cauchy form, the Reflecting Function of the system is defined by the formula $$F(t,x)=\varphi(-t;t,x).$$

Application
If a vector-function $$X(t,x)$$ is $$\,2\omega$$-periodic with respect to $$\,t$$, then $$\,F(-\omega,x)$$ is the in-period $$\,[-\omega;\omega]$$ transformation (Poincaré map) of the differential system $$\dot x=X(t,x).$$ Therefore the knowledge of the Reflecting Function give us the opportunity to find out the initial dates $$\,(\omega,x_0)$$ of periodic solutions of the differential system $$\dot x=X(t,x)$$ and investigate the stability of those solutions.

For the Reflecting Function $$\,F(t,x)$$ of the system $$\dot x=X(t,x)$$ the basic relation


 * $$\,F_t+F_x X+X(-t,F)=0,\qquad F(0,x)=x.$$

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.