- Source: Mironenko reflecting function
In applied mathematics, the reflecting function
F
(
t
,
x
)
{\displaystyle \,F(t,x)}
of a differential system
x
˙
=
X
(
t
,
x
)
{\displaystyle {\dot {x}}=X(t,x)}
connects the past state
x
(
−
t
)
{\displaystyle \,x(-t)}
of the system with the future state
x
(
t
)
{\displaystyle \,x(t)}
of the system by the formula
x
(
−
t
)
=
F
(
t
,
x
(
t
)
)
.
{\displaystyle \,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
)
{\displaystyle {\dot {x}}=X(t,x)}
with the general solution
φ
(
t
;
t
0
,
x
)
{\displaystyle \varphi (t;t_{0},x)}
in Cauchy form, the Reflecting Function of the system is defined by the formula
F
(
t
,
x
)
=
φ
(
−
t
;
t
,
x
)
.
{\displaystyle F(t,x)=\varphi (-t;t,x).}
Application
If a vector-function
X
(
t
,
x
)
{\displaystyle X(t,x)}
is
2
ω
{\displaystyle \,2\omega }
-periodic with respect to
t
{\displaystyle \,t}
, then
F
(
−
ω
,
x
)
{\displaystyle \,F(-\omega ,x)}
is the in-period
[
−
ω
;
ω
]
{\displaystyle \,[-\omega ;\omega ]}
transformation (Poincaré map) of the differential system
x
˙
=
X
(
t
,
x
)
.
{\displaystyle {\dot {x}}=X(t,x).}
Therefore the knowledge of the Reflecting Function give us the opportunity to find out the initial dates
(
ω
,
x
0
)
{\displaystyle \,(\omega ,x_{0})}
of periodic solutions of the differential system
x
˙
=
X
(
t
,
x
)
{\displaystyle {\dot {x}}=X(t,x)}
and investigate the stability of those solutions.
For the Reflecting Function
F
(
t
,
x
)
{\displaystyle \,F(t,x)}
of the system
x
˙
=
X
(
t
,
x
)
{\displaystyle {\dot {x}}=X(t,x)}
the basic relation
F
t
+
F
x
X
+
X
(
−
t
,
F
)
=
0
,
F
(
0
,
x
)
=
x
.
{\displaystyle \,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.
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
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