The reflective function of Mironenko

A reflecting function is a function that connects the past state of a system with its future state at a symmetrical moment in time. The concept of reflective function was introduced by Vladimir Ivanovich Mironenko .

Content

Definition

Let be $\varphi (t;t_{0},x)$ ${\ displaystyle \ varphi (t; t_ {0}, x)}$ there is a general solution in the Cauchy form of a system of differential equations ${\dot {x}}=X(t,x),$ ${\ displaystyle {\ dot {x}} = X (t, x),}$ whose decisions are uniquely determined by their initial data. The reflective function of this system is determined by the formula $F(t,x)=\varphi (-t;t,x).$ ${\ displaystyle F (t, x) = \ varphi (-t; t, x).}$

Application

For $2\omega$ ${\ displaystyle 2 \ omega}$ -periodic variable $t$ ${\ displaystyle t}$ systems of differential equations with reflective function $F(t,x)$ ${\ displaystyle F (t, x)}$ display $\Pi (x)$ ${\ displaystyle \ Pi (x)}$ during the period $[-\omega ;\omega ]$ ${\ displaystyle [- \ omega; \ omega]}$ ( Poincare map ) is found by the formula $\Pi (x)=F(-\omega ,x).$ ${\ displaystyle \ Pi (x) = F (- \ omega, x).}$ Therefore, knowledge of the reflective function allows you to find the initial data $(\omega ,x_{0})$ ${\ displaystyle (\ omega, x_ {0})}$ for $2\omega$ ${\ displaystyle 2 \ omega}$ -periodic solutions $\varphi (t;-\omega ,x_{0})$ ${\ displaystyle \ varphi (t; - \ omega, x_ {0})}$ system under consideration and investigate these solutions for Lyapunov stability . Reflective function $F(t,x)$ ${\ displaystyle F (t, x)}$ systems ${\dot {x}}=X(t,x)$ ${\ displaystyle {\ dot {x}} = X (t, x)}$ satisfies the so-called basic relation

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

Using this relation, it is established that for many systems of differential equations that are not integrable in quadratures, the map $\Pi (x)$ ${\ displaystyle \ Pi (x)}$ during the period $[-\omega ;\omega ]$ ${\ displaystyle [- \ omega; \ omega]}$ can be found even through elementary functions . In this, the reflective function can be compared with an integrating factor .

The reflective function is used to study the existence and stability of periodic solutions of boundary value problems for systems of differential equations.

Literature

Mironenko V.I. Reflecting function and periodic solutions of differential equations . - Minsk, Universitetskoye, 1986. - 76 p.
Mironenko V.I. Reflecting function and investigation of multidimensional differential systems. - Gomel: Min. images RB, GSU them. F. Skorins, 2004 .-- 196 p.

The reflective function of Mironenko

Content

Definition

Application

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Literature

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