Mironenko reflecting function
In applied mathematics, the reflecting function [math]\displaystyle{ \,F(t,x) }[/math] of a differential system [math]\displaystyle{ \dot x=X(t,x) }[/math] connects the past state [math]\displaystyle{ \,x(-t) }[/math] of the system with the future state [math]\displaystyle{ \,x(t) }[/math] of the system by the formula [math]\displaystyle{ \,x(-t)=F(t,x(t)). }[/math] The concept of the reflecting function was introduced by Uladzimir Ivanavich Mironenka.
Definition
For the differential system [math]\displaystyle{ \dot x=X(t,x) }[/math] with the general solution [math]\displaystyle{ \varphi(t;t_0,x) }[/math] in Cauchy form, the Reflecting Function of the system is defined by the formula [math]\displaystyle{ F(t,x)=\varphi(-t;t,x). }[/math]
Application
If a vector-function [math]\displaystyle{ X(t,x) }[/math] is [math]\displaystyle{ \,2\omega }[/math]-periodic with respect to [math]\displaystyle{ \,t }[/math], then [math]\displaystyle{ \,F(-\omega,x) }[/math] is the in-period [math]\displaystyle{ \,[-\omega;\omega] }[/math] transformation (Poincaré map) of the differential system [math]\displaystyle{ \dot x=X(t,x). }[/math] Therefore the knowledge of the Reflecting Function give us the opportunity to find out the initial dates [math]\displaystyle{ \,(\omega,x_0) }[/math] of periodic solutions of the differential system [math]\displaystyle{ \dot x=X(t,x) }[/math] and investigate the stability of those solutions.
For the Reflecting Function [math]\displaystyle{ \,F(t,x) }[/math] of the system [math]\displaystyle{ \dot x=X(t,x) }[/math] the basic relation
- [math]\displaystyle{ \,F_t+F_x X+X(-t,F)=0,\qquad F(0,x)=x. }[/math]
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
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