Mironenko reflecting function

The reflecting function $\,F(t,x)$ of a dynamical system connects the past state $\,x(-t)$ of it with the future state $\,x(t)$ of it by the formula $\,x(-t)=F(t,x(t)).$ The concept of the reflecting function was introduсed by Vladimir Ivanovich Mironenko.

Definition

For the differential system ${\dot {x}}=X(t,x)$ with the general solution $\varphi (t;t_{0},x)$ in Cauchy form Reflecting Function is defined by formula $F(t,x)=\varphi (-t;t,x).$

Application

If a vector-function $\,x(-t)$ is periodic in $\,2\omega$ with respect to $\,t$ , then $\,F(-\omega ,x)$ is a transformation (Poincaré map) periodic in $\,[-\omega ;\omega ]$ 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 date $\,(\omega ,x_{0})$ of periodic solutions of the differential system 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.

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