Applied Mathematics, 2010, 1, 76-80
doi:10.4236/am.2010.11009 Published Online May 2010 (http://www.SciRP.org/journal/am)
The Structure of Reflective Function of Higher
Dimensional Differential System
Zhengxin Zhou
Department of Mathematics, Yangzhou University, Yangzhou, China
Received March 16, 2010; revised April 18, 2010; accepted April 29, 2010
Abstract
In this article, we discuss the structure of reflective function of the higher dimensional differential systems
and apply the results to study the existence of periodic solutions of these systems.
Keywords: Reflecting Function, Periodic Solution, Higher Dimensional System
1. Introduction
As we know, to study the property of the solutions of
differential system
'(, )x Xtx=
(1)
is very important not only for the theory of ordinary
differential equation but also for practical reasons. If
( 2,)(,)Xtx Xtx ω += (
ω
is a positive constant), to
study the solutions’ behavior of (1), we could use, as
introduced in [1], the Poincare mapping. But it is very
difficult to find the Poincar e mapping for many sys-
tems which cannot be integrated in quadratures. In the
1980’s the Russian mathematician Mironenko [2] first
established the theory of reflective functions (RF).
Since then a quite new method to study (1) has been
found.
In the present section, we introduce the concept of the
reflective function, which will be used throughout the
rest of this article.
Now consider the system (1) with a continuously
differentiable right-hand side and with a general solu-
tion
00
(; ,)tt x ψ
. For each such system, the reflective
function (RF) of (1) is defined as
(,())( ;,)Ftxt ttx ψ = −
.
Then for any solution
()xt
of (1), we have( ,( ))Ft xt=
()xt= −
. If system (1) is
2 ω −
periodic with respect to
t
,
and (, )Ftx is its RF, then (,)(;,)Fx x ωψω ω −= − is
the Poincare [1-2] mapping of (1) over the period
[ ,] ωω −
. So, for any solution
()xt
of (1) defined on
[ ,] ωω −
, it will be 2
ω
--periodic if and only if
()x ω −
is
a fixed point of the Poincare mapping
()( ,)Tx Fx ω = −
.
A function
(, )Ftx
is a reflective function of system (1)
if and only if it is a solution of the partial differential
equation (called a basic relation, BR)
''
(,)( ,)0
tx
FFXtxX tx++− =
(2)
with the initial condition (0, )
Fxx
=. It implies that for
non-integrable periodic systems we also can find out its
Poincare mapping. If, for example,
(,)( ,)0XtxXtx+− =
, then ()Tx x=.
If (, )Ftxis the RF of (1), then it is also the RF of the
system
1
'(, )(, )(,(, ))
x
xXtxF RtxRtFtx
−
= +−−
,
whe re
(, )Rt x
is an arbitrary vector function such that the
solutions of the above systems are uniquely determined
by their initial conditions. Therefore, all these
2
ω
−periodic systems have a common Poincare mapping
over the period [−
ω
,
ω
], and the behavior of the peri-
odic solutions of these systems are the same.
To find out the reflective function is very important
for studying the qualitative behavior of solutions of dif-
ferential systems. The literatures [5-8] have discussed the
structure of the reflective function of some second order
quadric systems and linear systems and obtained many
good results.
Now, we consider the higher dimensional polynomial
differential system
12 3
22
123 456
22
123 456
'(, ,, )
'(, ,, )
'(, ,, )
xppypzPt x y z
yqqyqz qyqyz qzQtxyz
zrryrz ryryz rzRtxyz
=++=

=++++ +=

=++++ +=

(3)
where
(, ),(,),(, )
(1, 2,3;1, 2,..., 6)
i ij jjj
pp txqqtxrr tx
ij
== =
= =
are continuously differentiable functions in R2, and
22
23
0pp+≠
（in some deleted neighborhood of
0t=

Z. X Zh o u
Copyright © 2010 SciR es. AM
77
and |
t
| being small enough,
22
23
0pp+≠
is different
from zero）, and there exists a unique solution for the
initial value problem of (3). And suppose that
123
(, ,, )((, ,, ),(,,,),(, ,, ))
T
FtxyzFtxyzFtxyz Ftxyz=
is
the RF of (3).
In this paper, we will discuss the structure of
(,,,)(2, 3)
i
Ftxyz i=
when
1
(, ,, )(, )Ftxyz ftx=
. At the
same time, we obtain the good results that
12 3
(, , , )(,)(,)(, )(2,3)
iii i
Ftxyzftxftxyftxzi=++ =
.
The obtained results are used for research of problems of
existence of periodic solution of the system (3) and es-
tablish the sufficient conditions under which the first
component of the solution of (3) is even function.
In the following, we will denote
(, );(, );(, );(, ,,),
iijjjji i
pptx qqtxrrtx FFtxyz=− =−==
1, 2,3,1, 2,...,6ij= =
. The notation
(, )0
i
p tx ≠
means
that, in some deleted neighborhood of
0t=
and |
t
| being
small enough,
(, )
i
p tx
is different from zero,
(, ,, )(, ,, )(,,, ).
AA AA
DAPtxyz QtxyzRtxyz
tx yz
∂∂ ∂ ∂
=+++
∂∂ ∂∂
2. Main Results
Without loss of generality, we suppose that
(, )ftx x=. Otherwise, we take the tra nsformation
( ,),,ftxyz ξ ηζ == =
.
Now, let’s consider the system (3)
Lemma 1. For the system (3), suppose that
1
Fx=
.
The n
(,)0,1, 2, 3.
i
p txi= =
(4)
Proof. Using the relation (2), we get
23
(,,,)(,,,)0PtxyzPtxF F+− =
,
i.e.
11232233 0pppypzpF pF+++++=
. (5)
Putting
0t=
, we get
12 3
(0, )(0,)(0, )0,,,pxpxypxzxyz++ ≡∀
.
It implies that the relation (4) is valid.
In the following discussion, we always assume (4)
holds without further mention.
Case 1.
3
0p≠
.
From (5), we get
31 22
FF λλ = +
, (6)
where
3 112
111 1213
3 33
2
2
3
,
.
p pp p
y zyz
p pp
p
p
λλλ λ
λ
+
=−− − =++
= −
Differentiating relation (6) respect to
t
implies
2
0 1222
0A AF AF++ =
, (7)
whe re
2
0121113231626
22
01 0203040506
( )()
;
A Dqrrqrq
aay azayayz az
λλλ λλλ = − ++−+−
=++++ +
1222223231 525
1 2626111213
( )( )
2( );
ADqrr qr q
rqaay az
λλλλλ λ
λλ λ
= − ++−+−
+− =++
;
2
222442 5252626
( ,)()(),AA txqrrqrq λ λλλλ ==− ++−+−
In which
''
0111111 121 131121
2
1132362 611
()() ;
tx
apqrrq
rqrq
λλλ λλ
λ λλλ
= ++++−+
+−+−
'' '
0212122121122 132
1232362611 12
() 2();
txx
appqr
rq rq
λλλλ λ
λλλ λλ
=+++ ++
+− +−
'' '
0313113133123 133
1332362611 13
() 2();
tx x
ap pqr
rq rq
λλλλ λ
λλλ λλ
=+++++
+− +−
'2
04122 124 13462612
( );
x
apqrrq λλ λλλ =+++−
''
0512313212513 56261213
2() ;
xx
appq rrq λλλλλλλ =++++−
'2
0613312 613662 613
( );
x
apqrrq λλ λλλ =+++−
''
1122122 2232 3
1152 56262 11
()
() 2();
tx
apr qr q
rq rq
λλλ λλ
λλλ λλ
= ++−+−+
+− +−
'
12221252 562 62 12
() 2();
x
aprq rq λλλλ λλ =+−+−
'
132 313525626 213
() 2();
x
aprqrq λλλλ λλ =+− +−
3 11 2
1112 13
3 33
, ,.
p pp p
p pp
λ λλ
+
=− =−=−
Lemma 2.Le t
12
,0F xA= ≠
and
0
02
lim(1, 2,...,6)
i
t
ai
A
→
=
exist. Then
0
0
2
lim0(1, 3, 6),
j
t
a
j
A
→
= =
02 11
02
lim 0,
t
aa
A
→
+=
04 1205 13
00
22
lim1, lim0.
tt
aa aa
AA
→→
++
=−=
Proof. Using the relation (7), we have
22
01 0203040506
02
lim
t
aay azayayz az
A
→
++++ ++
2 11 1213
22
00
2
limlim 0,
tt
aay az
FF
A
→→
++
+=
As
2
(0, ,, )Fxyz y=
, it follows that the results of
Lemma 2 are true.
Theorem 1. Let the conditions of Lemma 1 and
Lemma 2 satisfy and
12
02
lim2 0
t
a
A
→
+>
. Then
123
(, )(, )(, )(2,3).
ii ii
Ff txftxyftxzi=++ =

Z. X Zhou
Copyright © 2010 SciR es. AM
78
Proof. AS 20A≠. From (7), it follows
2
23
0121010 2
22 2
22
222
,
AAFAAAA A
FF F
AAA
+−
=−=+
(8)
Differentiating relation (7) respect to
t
implies
2
0122212 3
2 232
( ,,,)
2(, ,,)0.
DADAFDAFAQtxFF
AQtx FFF
+ +−−
−− =
Substituting (6) into the above, we get
23
0 12 3242
0,BBF BFBF++ +=
(9)
whe re
00 11111221
, 2,BDAABDA AA µ µµ =− =−−
22 1322423
2, 2,
BDA AABA
µµ µ =−−=−
in which
2
1131612232 51621
2
34 52 62
, 2,
.
qq qqqqq
qq q
µλλ µλλλλ
µ λλ
=++ =+++
=++
Substituting (8) into (9), we have
012 0C CF
+=
, (10)
whe re
0 10
0 023
2
22
2
1 02
1
1 1232
22
,
.
A AA
C BBB
AA
A AA A
C BBB
AA
=−+
−
=−+
*
1.
If
1
0C≡
, from (10) follows 00C≡. By simple
computation, we obtain
00 01 1
1 23
2
2222
2,
AA AA
A
DAAA A
µ µµ =−+
(11)
2
0 1 11
12 3
2
22 2 2
2( 2)
A A AA
DAA A
A
µµ µ =−+ −
. (12)
Let
20
1
22
() 4
A
A
AA
∆= −
. Using (11) (12) we get
1
32
2
2( )
A
DA
µµ
∆=−∆. (13)
Since
22 0 1
11 1213
2
22 2
22
2 010203040506
22
123 456
2
52
4
44
1
( )4(()
4 ())
( ),
22
A Aaay az
AA A
Aaay azayayz az
ddy dz dydyzdz
dd
dy zW
dd
∆=−=++−
++++ +
=++++ +
=++ +
whe re
22 2
4653425 142
4
1[(4)2(2) 4],
4
Wdddzdd ddzdd d
d
=−+−+−
2
1112 01211122 02
22
22
2
311132034122 04
22
22
2
513122 05613206
22
22
11
(4),(24 ),
11
(24),(4 ),
11
(24 ),(4 ).
daAadaaAa
AA
da aAadaAa
AA
da aAadaAa
AA
=−= −
= −=−
= −=−
By Lemma 2 we get
2
46 534 25
00
2
412
0
lim(4)0,lim(3)0,
lim(4) 0,
tt
t
dd ddddd
dd d
→→
→
−=− =
−=
thus, 0 lim(,,)0.
tWtxz →= In the identity (13) taking
52
44
22
dd
yz
dd
φ
==−−
. We obtain
'' '
1
32
2
(, , )(,, ,)
(, ,, )
2(( ,)( ,,,)).
(, )
tx z
W WPtxWQtxz
Atx z
Wtxtx z
A tx
φφ
φµ µφ
++ =
−
By the uniqueness of solution of initial problem of li-
near partial differential equation, we get
(, , )0Wtxz ≡
.
Therefore
2 52
4
44
( ).
22
dd
dy z
dd
∆= ++
Using the relation (7), we obtain
11 121352
24
2 44
21 2223
1()
2222
(, )(, )(, ).
aay azdd
Fdy z
A dd
f txftxyftxz
++
=−+ ++
=++
By the relation (6), we get
3122313233
(, )(, )(, ).FFf txftxyftxz λλ =+=++
*
2.
If 10C≠. From (10) follows 0
2
1
.
C
FC = − By the
express of
,(0,1, 2,0,1,2, 3)
ij
AB ij= =
, we know that
1
C is a quadratic polynomial r espect of
,yz
,
0
C
is a
cubic polynomial respect to
,.yz
Substituting 2 F=
01
CC
−into relation (7), we get 1 0110
()
C CACA
−=
2
20
AC
. It implies that 1 Cis divided by
0
C
or
2
A
,
and
3
22
0
(, ),
ij
ij
ij
Fftx yz
+=
=∑ substituti ng it into (7) and
equating the coefficients of like powers of
y
and
z
im-
plies
2
( ,)0,1
ij
ftxij= +>
. Thus,
123
(, )(, )(, ),2,3.
ii ii
Fftxftxyftxz i=++ =

Z. X Zh o u
Copyright © 2010 SciR es. AM
79
Summarizing the above, the proof is completed.
Obviously, from the relation (7) implies
Theorem 2. Let
1 21
,(0,)0(1,2, 3),0,0
i
F xpxiAA====≠.
The n
1
22
001 0203040506
2
111 1213
3 22
,
.
Aaay az ayayz az
FAaay az
FF
λλ
++++ +
=−=− ++
= +
Case 2.
32
( ,)0,0
p txp≡≠
.
Applying identity (5) yields
112
2 12
22
,
pp p
F yy
pp
ζζ
+
=− −=+
where
112
12
22
,.
pp p
pp
ζζ
+
=−=−
Differentiating this identity respect to
t
implies
2
0 1323
0MMF MF++ =
,
whe re
2
0121 2243
22
01 0203040506
()
,
MDyq qFqF
mmy mzmymyz mz
ζζ =+ ++ +=
++++ +
1352111226
,,MqqFmm y Mq=+=+ =
'' 2
01111211 21 41
;
tx
mpqqq q ζζζζ ζ =+ ++++
'''
02122122 222412
2;
x xt
mp pqq q ζζζζζζζ =+++++
'2
04222442; x
mp qq ζζ ζ = ++
032 305250626
113511252
,,,
,.
m qm qmq
m qqmq
ζζζ
ζζ
== =
=+=
Similarly, we obtain the following conc l usi on:
Lemma 3. Let
3 261
0,0, 0,
ppqFx= ≠≠=
and
0
06
lim(1, 2,..., 5)
i
t
mi
q
→
=
and 1
02
lim
t
p
p
→ exist. Then
011 2
0 00
6 22
lim0(1, 2, 4), lim0,lim1
j
t tt
mpp p
j
q pp
→ →→
+
==== −
03 1105126
00 0
6 66
lim0, lim0,lim1.
tt t
mm mmq
q qq
→→ →
++
=== −
Theorem 3. Let the conditions of Lemma 1 and
Lemma 3 satisfy. Then
21 2
3 313233
(, )(, ),
(, )(, )(, ).
F txtxy
Ff txftxyf txz
ζζ = +
=++
Theorem 4. Let
36
22
11 121
(0,) 0(1,2), (,)0,(,)0
0,
i
pxiqtxqtx
m mFx
== ≡≡
+≠= , Then
21 2
2
01 02030405
3
11 12
(, )(, ),
.
F txtxy
mmy mzmymyz
Fm my
ζζ = +
++++
= −+
Theorem 5. For the system (3), if the following con-
ditions satisfy
112 213 31222223 32
0, 0,p ppfpfppfpf+++= ++=
32233 332131
0,(0, )0,(0, )0,p pfpffxfx++== =
''
222321 31 21 21
1 11
''
23332131
31 31
( ,,,)
( ,,,)
0,
tx
tx
ffQ txff
ff
pqr
ffRtxff
ff
−
  
++ ++
  
−
 
 
=
''' '''
2223222321 221 3
1
''' '''
3233323331 232 3
''
22232 3
''
32332 3( ,,0,0)
0,
tt xxx x
tt xxx x
yz
yz
tx
fffffpfp
p
fffffp fp
ff qqQQ
f f rrRR
 
++ +
 
 


+=



''
22 2346 22223 3
''
32 334 6 32233 3
22
4622 23 523 33 5
22 23
22
462223 52333 5 32 33
0,
xx
xx
ff qq fp fp
f f rr
fp fp
qqffqffq ff
rrffrffr ff

 
++

 
 


 
+=
  
 

''
22 235 3
22 23
''
32 33 5 2 32 33
4 22 23522 3323326 32 33
4 22 23522 332332632 33
2()2 0,
2 ()2
xx
xx
ffq p ff
ffr p ff
qffqffffqff
rff rffffrff

  
++
  

  

+ ++

=

+ ++

22 232223
22 23
32 333233
32 33
(0, )(0, )10
.
(0, )(0, )01
f ffxfx
ff
f ffxfx
ff

 

= =

 


 

The n
21 2222
31 3233
x
Fffyfz
ffyfz


=++


++

is the RF of system (3).
Besides this, if the system (3) is
2 ω
--periodic with re-
spect to
t
, then its solution
(( ),(),())xtyt zt
define d on
the interval
[ ,] ωω −
with initial condition ( (),
x
ω −
( ),())yz
ω ωχ
− −=
is 2
ω
--periodic if and only if
( ,)
F
ωχ
−
. χ =
Proof. By checkout of the BR it is proved that the
function
212223 313233
(,, )
T
F xffyfzffyfz= ++++
is
the RF of system (3). At this moment, the Poincare map-
ping of periodic system (3) is (,,)Txyz =( ,,,F xy ω −
)z. By the previous introduction the assertions of the
present theorem is hold. The proof is finished.
Under the hypotheses of Theorem 5, the first compo-
nent of solution of system (3) is even function.
Example: Differential system

Z. X Zhou
Copyright © 2010 SciR es. AM
80
sin 32
6 232sin3 2
11
22
2252sin 42
4sin3236 2
11
22
26 2
'(1(1sin ))sin ,
'cos (sin1)cos(1sinsin)
(3sinsin)(sin2sin ),
'cos(1sinsin )cos(1sin )
( sin
t
t
t
t
xext yxzt
yy txtxzxtetxt
yxt xtyzextxt
zyxtet xtztxxt
yx t
−
−
=−− −
=+−−++
+ −+−
=− ++−−
−
3sin25 2
4sin)(3sinsin)
t
xteyzxtxt







− −−

has RF
sin 32
4sin 3
( ,,,)(1sin)sin
sin(1sin )
t
t
x
Ftxyzexty xzt
xyt extz
−


=−+


− ++

.
Since this system is a 2
π
--periodic system, and
(,,,) (,,)
T
Fxyzxyz
π −≡
, by Theorem 5, all the solutions
of the considered system defined
[ ,] ππ −
are2
π
--periodic.
3. References
[1] V. I. Arnold, “Ordinary Differential Equation,” Science
Press, Moscow, 1971, pp. 198-240.
[2] V. I. Mironenko, “Reflective Function and Periodic Solution
of the Differential System,” University Press, Minsk, 1986,
pp. 12-26.
[3] V. I. Mironenko, “The Reflecting Function and Integral
Manifolds of Differential Systems,” Differential Equations,
Vol. 28, No. 6, 1992, pp. 984-991.
[4] V. V. Mironenko, “Time Symmetry Preserving Perturbations
of Differential Systems,” Differential Equations, Vol. 40, No.
20, 2004, pp. 1395-1403.
[5] P. P. Verecovich, “Nonautonomous Second Order Quadric
Wystem Equivalent to Linear System,” Differential Equa-
tions, Vol. 34, No. 12, 1998, pp. 2257-2259.
[6] E. V. Musafirov, “Differential Systems, the Mapping over
Period for Which is Represented by a Product of Three Ex-
ponential Matrixes,” Journal of Mathematical Analysis and
Applications, Vol. 329, No.1, 2007, pp. 647-654.
[7] Z. X. Zhou, “On the Reflective Function of Polynomial Dif-
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