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 φ(t;t₀,x) in Cauchy form Reflecting Function is defined by formula F(t,x) = φ( − 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.

Links

• The Reflecting Function Site
• How to construct equivalent differential systems