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 $\phi (t;t_0,x)$ in Cauchy form, the Reflecting Function of the system is defined by the formula $F(t,x)=\phi (-t;t,x).$

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

$F_t+F_{x}X+X(-t,F)=0,F(0,x)=x.$

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