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
x
˙
=
X
(
t
,
x
)
with the general solution
φ
(
t
;
t
0
,
x
)
in Cauchy form Reflecting Function is defined by formula
F
(
t
,
x
)
=
φ
(
−
t
;
t
,
x
)
.
If a vector-function
X
(
t
,
x
)
is
2
ω
-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
,
x
0
)
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
F
t
+
F
x
X
+
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.
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.
