Mironenko reflecting function

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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 Uladzimir Ivanavich Mironenka.


For the differential system \dot x=X(t,x) with the general solution \varphi(t;t_0,x) in Cauchy form Reflecting Function is defined by formula F(t,x)=\varphi(-t;t,x).


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.


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