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)

Cauchy Baron Augustin-Louis Cauchy ( , , ; ; 21 August 1789 – 23 May 1857) was a French mathematician, engineer, and physicist. He was one of the first to rigorously state and prove the key theorems of calculus (thereby creating real a ...

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)

\,2\omega

-periodic with respect to

\,t

, then

\,F(-\omega,x)

is the in-period

\, \omega;\omega /math> transformation (

Poincaré map In mathematics, particularly in dynamical systems, a first recurrence map or Poincaré map, named after Henri Poincaré, is the intersection of a periodic orbit in the state space of a continuous dynamical system with a certain lower-dimensiona ...

) 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)

periodic solution A periodic function, also called a periodic waveform (or simply periodic wave), is a function that repeats its values at regular intervals or periods. The repeatable part of the function or waveform is called a ''cycle''. For example, the tri ...

s of the differential system

\dot x=X(t,x)

and investigate the

stability Stability may refer to: Mathematics *Stability theory, the study of the stability of solutions to differential equations and dynamical systems ** Asymptotic stability ** Exponential stability ** Linear stability **Lyapunov stability ** Marginal s ...

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 In mathematics, an elementary function is a function (mathematics), function of a single variable (mathematics), variable (typically Function of a real variable, real or Complex analysis#Complex functions, complex) that is defined as taking addit ...

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
Differential equations