Mironenko reflecting function

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Short description: Mathematical function

In applied mathematics, the reflecting function

F(t,x)

of a differential system

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 x˙=X(t,x) with the general solution φ(t;t0,x) in Cauchy form, the Reflecting Function of the system is defined by the formula F(t,x)=φ(−t;t,x).

Application

If a vector-function X(t,x) is 2ω-periodic with respect to t, then F(−ω,x) is the in-period [−ω;ω] transformation (Poincaré map) of the differential system x˙=X(t,x). Therefore the knowledge of the Reflecting Function give us the opportunity to find out the initial dates (ω,x0) of periodic solutions of the differential system x˙=X(t,x) and investigate the stability of those solutions.

For the Reflecting Function F(t,x) of the system x˙=X(t,x) the basic relation

Ft+FxX+X(−t,F)=0,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



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