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Stability of a Class of Nonlinear Difference Equations
Journal of Mathematical Analysis and Applications 230, 211]222 Ž1999. Article ID jmaa.1998.6194, available online at http:rrwww.idealibrary.com on
Stability of a Class of Nonlinear Difference Equations G. Papaschinopoulos* Department of Electrical and Computer Engineering, Democritus Uni¨ ersity of Thrace, 67100 Xanthi, Greece
C. J. Schinas Department of Applied Informatics, Uni¨ ersity of Macedonia, 54006 Thessaloniki, Greece Submitted by Gerry Ladas Received April 2, 1998
In this paper using KAM theory we investigate the stability nature of the zero equilibrium of the system of two nonlinear difference equations x nq 1 s a1 x n q b1 yn q f Ž c1 x n q c2 yn . ynq 1 s a2 x n q b 2 yn q f Ž c1 x n q c 2 yn .
5
,
n s 0, 1, . . . ,
where a i , bi , c i , i s 1, 2, are real constants and f : R ª R is a C` function. Q 1999 Academic Press
Key Words: nonlinear difference equations; stability; area preserving map; KAM theory.
1. INTRODUCTION The main goal in this paper is to generalize the results concerning the stability nature of the zero equilibrium of the systems considered in w2x ] w4x. So we consider the system of two nonlinear difference equations x nq 1 s a1 x n q b1 yn q f Ž c1 x n q c 2 yn . s F1 Ž x n , yn . ynq 1 s a2 x n q b 2 yn q f Ž c1 x n q c 2 yn . s F2 Ž x n , yn .
5
,
n s 0, 1, . . . ,
Ž 1. *E-mail address: [email protected]. 211 0022-247Xr99 $30.00 Copyright Q 1999 by Academic Press All rights of reproduction in any form reserved.
212
PAPASCHINOPOULOS AND SCHINAS
where a i , bi , c i , i s 1, 2, are real constants and f : R ª R is a C` function such that f / 0,
f Ž 0 . s 0,
f 9 Ž 0 . s 0.
Ž 2.
Moreover, the map F s Ž F1 , F2 . satisfies the following condition: ‘‘F is an area preserving map with an elliptic fixed point at the origin and eigenvalues l , l such that l k / 1,
k s 3, 4.’’
Ž H.
In the first proposition of this paper we prove that F satisfies the condition ŽH. if and only if < a1 q b 2 < - 2,
a1 b 2 y a2 b1 s 1,
a1 q b 2 / y1,
Ž a1 y a2 . c2 s Ž b1 y b 2 . c1 .
Ž 3.
It is known Žsee w1]6x. that if F satisfies the condition ŽH., KAM theory can be applied to study the stability of the zero equilibrium of Ž1.. In Proposition 2 we find conditions on a i , bi , c i , i s 1, 2, and the function f so that the zero equilibrium of Ž1. is stable.
2. MAIN RESULTS We now prove our main results. In the first proposition we find necessary and sufficient conditions so that the map F s Ž F1 , F2 . satisfies the condition ŽH.. PROPOSITION 1. Consider the map F s Ž F1 , F2 . where F1 , F2 are defined in Ž1. where f : R ª R is a C` function such that Ž2. are satisfied. Then F satisfies the condition ŽH. if and only if Ž3. hold. Proof. Suppose that F satisfies the condition ŽH.. Then from Ž2. it is obvious that the point Ž0, 0. is a fixed point for F. Let, for any x, y g R,
DF s
F1
F1
x F2
y F2
x
y
0
s
ž
a1 q c1 f 9 a 2 q c1 f 9
b1 q c2 f 9 . b2 q c2 f 9
/
Ž 4.
STABILITY OF NONLINEAR DIFFERENCE EQUATIONS
213
Then, using Ž2., we can see that the characteristic equation of DF with x s y s 0 is the following:
l2 y lŽ a1 q b 2 . q a1 b 2 y a2 b1 s 0.
Ž 5.
From the condition ŽH. we have that Ž5. has roots l and l where l k / 1, k s 3, 4. Then we have a1 b 2 y a2 b1 s 1,
< a1 q b 2 < - 2,
a1 q b 2 / y1.
Ž 6.
Moreover, since from the condition ŽH., F is an area preserving map Žsee w1x. we have for all x, y g R det DF Ž x, y . s 1.
Ž 7.
Therefore, relations Ž4., Ž6., and Ž7. imply that f 9 Ž u . Ž Ž a1 y a2 . c 2 y Ž b1 y b 2 . c1 . s 0
Ž 8.
for all u g R. Since Ž2. holds, it is obvious that there exists a u g R such that f 9Ž u. / 0. Then, from Ž8.,
Ž a1 y a2 . c2 s Ž b1 y b 2 . c1 .
Ž 9.
Hence from Ž6. and Ž9., relations Ž3. are satisfied. Suppose now that Ž3. holds. Then we can easily prove that F satisfies the condition ŽH.. This completes the proof of the proposition. In the following proposition using KAM theory we study the stability of the zero equilibrium of Ž1. where Ž2. and Ž3. hold. PROPOSITION 2. Consider the system of two nonlinear difference equations Ž1. where f : R ª R is a C` function such that Ž2. hold and a i , bi , c i , i s 1, 2, are real constants such that Ž3. are satisfied. Moreo¨ er, suppose that c2 / 0
if b1 / b 2 ,
c1 / 0
if b1 s b 2 .
Ž 10 .
214
PAPASCHINOPOULOS AND SCHINAS
Then the zero equilibrium of Ž1. is stable if the following conditions are satisfied: 2
¡
f - Ž 0. / p Ž f 0 Ž 0. . , c 2 Ž 2 c q 1 . g Ž a1 y a2 .
Ž 2 y c . Ž c q 1. Ž b p s~
1
if b1 / b 2 ,
,
y b 2 . a2
Ž 11 .
yc1 Ž 1 q 2 c .
¢ Ž 2 y c . Ž c q 1. ,
if b1 s b 2 ,
where c s a1 q b 2 and g: R ª R is a function such that g Ž ¨ . s ¨ 2 y c¨ q 1.
Ž 12 .
Proof. We may proceed to find the Birkhoff Normal Form of Ž5. by using the transforms of KAM theory described in w2x. First Transformation. First we find the matrix P such that
Py1AP s diag Ž l , l . ,
As
ž
a1 a2
b1 , b2
/
where l, l are the eigenvalues of the matrix A which satisfy Ž12.. Since l satisfies Ž12., we get
Ps
a1 y l
a1 y l
a2
a2
1
1
0
,
Py1 s
Now the change of variables xn u sP n yn ¨n
a2
lyl
1
l y a1 a2
0 y1
a1 y l a2
.
STABILITY OF NONLINEAR DIFFERENCE EQUATIONS
215
transforms Ž1. into the system u nq 1 s l u n q ¨ nq 1 s l¨ n q
a2 y a1 q l
lyl a2 y a1 q l lyl
¦
f Ž m u n q m¨ n . f Ž m u n q m¨ n .
¥,
§
n s 0, 1, . . . , Ž 13 .
where
ms
c1 Ž a1 y l . a2
q c2 .
Ž 14 .
Since Ž2. holds, by writing the Taylor expansion for f Ž m u n q m¨ n . about Ž0, 0. and keeping terms up to order 3, from Ž13. we take u nq 1 s l u n q =
ž
A 2
¨ nq 1 s l¨ n q
=
ž
A 2
a2 y a1 q l
lyl 2 Ž m u n q m¨ n . q
B 6
Ž m u n q m¨ n .
3
/
q O4 ,
/
q O4 ,
Ž 15 .
a2 y a1 q l
lyl 2 Ž m u n q m¨ n . q
B 6
Ž m u n q m¨ n .
3
where n s 0, 1, . . . and A s f 0 Ž0., B s f -Ž0.. Second Transformation. In Ž15. we make the change of variables u n s j n q f 2 Ž j n , hn . q f 3 Ž j n , hn . , ¨ n s hn q c 2 Ž j n , hn . q c 3 Ž j n , hn . ,
Ž 16 .
with k
fk Ž j , h . s
Ý a k j j ky jh j , js0
k s 2, 3
k
ck Ž j , h . s
Ý ak j j h j
js0
kyj
,
Ž 17 .
216
PAPASCHINOPOULOS AND SCHINAS
to obtain
j nq 1 s lj n q a 2 j n2hn q O4 ,
hnq1 s lhn q a 2 j nhn2 q O4 . Ž 18 .
In order to calculate the coefficients a k j , a 2 we substitute Ž16., Ž17., Ž18. to Ž15.. Then the resulting equations will show an equality of two power series in j n and hn whose coefficients can be identified recursively up to order 3. Hence we get
a 20 s a 21 s a 22 s a 30 s
A Ž a2 y a1 q l . m2 2 Ž l y l . Ž l2 y l .
,
A Ž a2 y a1 q l . mm
Ž l y l. Ž 1 y l. A Ž a2 y a1 q l . m2 2 Ž l y l . Ž l2 y l .
,
,
m2 B Ž a2 y a1 q l . m A a m q a m q , Ž . 20 22 6 Ž l3 y l . Ž l y l .
ž
/
Ž 19 .
a 31 s arbitrary, a 32 s
a1 y a2 y l
Ž l y l.
2
ž
A mm Ž a 21 q a 20 . q A m2a 21
qA m2a 22 q
a 33 s
Bmm 2 2
/
,
m2 B Ž a2 y a1 q l . m A a m q a m q , Ž . 22 20 6 Ž l3 y l . Ž l y l .
ž
/
and
a2 s
A mma 20 Ž a2 y a1 q l .
lyl q
q
A m 2a 21 Ž a2 y a1 q l .
q
lyl Bm m Ž a2 y a1 q l .
A mma 21 Ž a2 y a1 q l .
lyl A m2a 22 Ž a2 y a1 q l .
lyl
2
q
2 Ž l y l.
.
Ž 20 .
217
STABILITY OF NONLINEAR DIFFERENCE EQUATIONS
We suppose first that b1 / b 2 . Then from Ž3., Ž14., and since l satisfies Ž12., we get
Ž a1 y l . Ž a1 y a2 . c2
ms s s s
a2 Ž b1 y b 2 .
q c2
c 2 Ž Ž a1 y l . Ž a1 y a2 . q b 2 Ž a1 y a2 . y 1 . a2 Ž b1 y b 2 . c 2 Ž Ž a1 y a2 . Ž a1 y l q b 2 . y 1 . a2 Ž b1 y b 2 . c 2 Ž Ž a1 y a2 . l y 1 . a2 Ž b1 y b 2 .
.
Ž 21 .
Then relations Ž19., Ž20., and Ž21. imply that
a2 s
c 23 < Ž a1 y a2 . l y 1 < 2 a23 Ž b1 y b 2 . =
ž
3
2
c 2 A2
ž
a2 Ž b1 y b 2 .
Ž a2 y a1 q l . Ž Ž a1 y a2 . l y 1 .
2
2
2 Ž l y l . Ž l2 y l . q
< a2 y a1 q l < 2 < Ž a1 y a2 . l y 1 < 2 2
Ž l y l. Ž l y 1. 2
q
q
q
Ž a2 y a1 q l . Ž Ž a1 y a2 . l y 1 .
2
2
Ž l y l. Ž 1 y l. < a2 y a1 q l < 2 < Ž a1 y a2 . l y 1 < 2 2
2 Ž l y l . Ž l y l2 .
B Ž a2 y a1 q l . Ž Ž a1 y a2 . l y 1 . 2 Ž l y l.
/
/
.
Then, for b1 / b 2 we take R Ž a2 . s
c 23 Ž g Ž a1 y a2 . . 4 a32 Ž b1 y b 2 .
3
2
ž
c 2 A2 Ž 2 c q 1 . g Ž a1 y a2 .
Ž 2 y c . Ž c q 1 . a2 Ž b1 y b 2 .
/
y B . Ž 22 .
218
PAPASCHINOPOULOS AND SCHINAS
Suppose now that b1 s b 2 . Then using Ž9. and since from Ž6., a1 / a2 , we have c 2 s 0. So, from Ž14., it is obvious that
ms
c1 Ž a1 y l . a2
.
Ž 23 .
Therefore, from Ž19., Ž20., and Ž23., we get
a2 s
c13 < a1 y l < 2 a23 =
c 1 A2
2 Ž a2 y a1 q l . Ž a1 y l .
a2
2 Ž l y l . Ž l2 y l .
2
q
q
2
q
2 Ž a2 y a1 q l . Ž a1 y l . 2
Ž l y l. Ž 1 y l.
B Ž a2 y a1 q l . Ž a1 y l . 2 Ž l y l.
0
< a2 y a1 q l < 2 < a1 y l < 2
2
q
2
Ž l y l. Ž l y 1. < a2 y a1 q l < 2 < a1 y l < 2 2
2 Ž l y l . Ž l y l2 .
0
.
Then, since l q l s a1 q b 2 , a2 s Ž a1 b 2 y 1.rb 2 , we have R Ž a2 . s
c13 b 22 4 Ž 1 y a1 b 2 .
ž
c 1 A2 Ž 1 q 2 c .
Ž2 y c. Ž1 q c.
qB .
/
Ž 24 .
Third Transformation. The change of variables
j n s rn q isn ,
hn s rn y isn ,
transforms Ž18. into the system rnq 1 s R Ž l . rn y I Ž l . sn q rn Ž rn2 q sn2 . R Ž a 2 . y sn Ž rn2 q sn2 . I Ž a 2 . , snq 1 s I Ž l . rn q R Ž l . sn q sn Ž rn2 q sn2 . R Ž a 2 . q rn Ž rn2 q sn2 . I Ž a 2 . .
Ž 25 .
STABILITY OF NONLINEAR DIFFERENCE EQUATIONS
219
Let w s g 0 q g 1Ž rn2 q sn2 ., g 0 , g 1 g R. Then cos w s cos g 0 cos Ž g 1 Ž rn2 q sn2 . . y sin g 0 sin Ž g 1 Ž rn2 q sn2 . . s 1y
ž
y
ž
g 12 Ž rn2 q sn2 .
2
2!
g 1 rn2
Ž
q
sn2
.y
q ??? cos g 0
/
g 13 Ž rn2 q sn2 .
3
q ??? sin g 0
/
3!
s cos g 0 y g 1 Ž rn2 q sn2 . sin g 0 q O4 .
Ž 26 .
Working similarly, we can find sin w s sin g 0 q g 1 Ž rn2 q sn2 . cos g 0 q O4 .
Ž 27 .
We write now Ž25. into the Birkhoff Normal Form ysin w cos w
rnq 1 cos w s snq 1 sin w
ž / ž
O4 rn q . sn O4
ž /
/ž /
Ž 28 .
Then, from Ž25., Ž26., Ž27., and Ž28., we take R Ž l . s cos g 0 ,
g1 s y
R Ž a2 . sin g 0
.
Ž 29 .
Hence, since l is a root of Ž12., and Ž22., Ž24., Ž29. hold, it follows that cos g 0 s
¡
y
g1 s
~
c 23 Ž g Ž a1 y a2 . .
¢
y
ž
2
,
2
2'4 y c 2 a32 Ž b1 y b 2 .
=
c
3
c 2 A2 Ž 2 c q 1 . g Ž a1 y a2 .
Ž 2 y c . Ž c q 1 . a2 Ž b1 y b 2 . c13 b 22
2'4 y c 2 Ž 1 y a1 b 2 .
ž
/
yB ,
c 1 A2 Ž 1 q 2 c .
Ž2 y c. Ž1 q c.
if b1 / b 2 , qB ,
/
if b1 s b 2 .
Ž 30 .
220
PAPASCHINOPOULOS AND SCHINAS
Since a1 y a2 g R and y2 - c - 2 hold, it follows that a1 y a2 does not satisfy Ž12.. Hence if Ž11. is satisfied, from Ž10. and Ž30. we get g 0 , g 1 / 0. Therefore, from Theorem 2.13 of w5x, the zero equilibrium of Ž1. is stable. This completes the proof of the proposition. Remark 1. We see now that our system Ž1. includes systems considered in w2x, w3x, and w4x and Proposition 2 holds for all of them. First consider the system in w3x: u nq 1 s 2 u n y ¨ n q ln ¨ nq 1 s u n
ž
a 1 q Ž a y 1. e u n
/
¦ ¥ §,
a)1
,
n s 0, 1, . . . .
Ž 31 . Then by adding both sides of Ž31. and setting u n s x n , yn s u n q ¨ n , f Ž x . s lnŽ arw1 q Ž a y 1. e x x. y wŽ1 y a.rax x, we take the following system: x nq 1 s ynq 1 s
2a q 1 a 3a q 1 a
¦
x n y yn q f Ž x n . x n y yn q f Ž x n .
¥,
n s 0, 1, . . . ,
§
Ž 32 .
which is equivalent to Ž31.. It is obvious that Ž32. is included in Ž1. where a1 s Ž2 a q 1.ra, a2 s Ž3a q 1.ra, b1 s b 2 s y1, c1 s 1, c 2 s 0. We can easily prove that Ž2., Ž3., and Ž10. are satisfied. Moreover, f 0 Ž 0. s
1ya a
2
f - Ž 0. s
,
psy
Ž 1 y a. Ž 2 y a. a3
a Ž 3a q 2 .
Ž a y 1. Ž 2 a q 1.
,
Ž 33 .
,
where the constant p is defined in Proposition 2. Therefore, from Ž33. and since a / 0, relation Ž11. is satisfied. Hence, from Proposition 2, the zero equilibrium of Ž31. is stable. Next consider the system defined in w2x: u nq 1 s y¨ n q ln e u n q ¨ nq 1 s u n
ž
b E
/
¦ ¥ §,
y ln E
n s 0, 1, . . . ,
Ž 34 .
221
STABILITY OF NONLINEAR DIFFERENCE EQUATIONS
where b g Ž0, `., b / 1, and E is the positive solution of the equation E 2 y E y b s 0. Working similarly, we can prove that Ž34. is equivalent to the system
x nq 1 s ynq 1 s
Eq1
x n y yn q f Ž x n .
E 2Eq 1 E
¦
¥,
n s 0, 1, . . . ,
§
Ž 35 .
x n y yn q f Ž x n .
where f Ž x . s lnŽ e x q brE . y ln E y Ž1rE . x. It is obvious that Ž35. is included in Ž1. where a1 s Ž E q 1.rE, a2 s Ž2 E q 1.rE, b1 s b 2 s y1, c1 s 1, c 2 s 0. We can prove that Ž2., Ž3., and Ž10. hold. Furthermore,
f 0 Ž 0. s
Ey1 E
2
f - Ž 0. s
,
psy
Ž E y 1. Ž E y 2. E3
E Ž E q 2.
Ž 2 E y 1. Ž E q 1.
,
Ž 36 .
,
where p is defined in Proposition 2. Then since b g Ž0, `. and b / 1, relations Ž36. imply that Ž11. holds and so the zero equilibrium of Ž34. is stable. Finally, consider the system defined in w4x: x nq 1 s x n y ayn q a Ž 1 y e y n q yn . ynq 1 s x n q Ž 1 y a . yn q a Ž 1 y e y n q yn .
5
,
n s 0, 1, . . . , Ž 37 .
where a is a constant, 1 - a - 4, a / 2, 3. It is obvious that Ž37. is included in Ž1. where a1 s a2 s 1, b1 s ya, b 2 s 1 y a, c1 s 0, c 2 s 1. We can easily prove that Ž2., Ž3., and Ž10. are satisfied. Moreover,
f 0 Ž 0 . s ya,
f - Ž 0 . s ya,
ps
5 y 2a aŽ a y 3.
,
Ž 38 .
where p is defined in Proposition 2. Then since a / 2, relations Ž38. imply that Ž11. holds and so the zero equilibrium of Ž37. is stable.
222
PAPASCHINOPOULOS AND SCHINAS
REFERENCES 1. J. K. Hale and H. Kocak, ‘‘Dynamics and Bifurcations,’’ Springer-Verlag, New York, Berlin, 1991. 2. V. L. Kocic, G. Ladas, G. Tzanetopoulos, and E. Thomas, On the stability of Lyness’s equation, Dynamics of Continuous, Discrete, and Impulsi¨ e Systems 1 Ž1995., 245]254. 3. G. Ladas, G. Tzanetopoulos, and A. Torbis, On May’s host parasitoid model, J. Difference Equations Appl. 2 Ž1996., 195]204. 4. C. Meier, W. Senn, R. Hauser, and M. Zimmermann, Strange limits of stability in host-parasitoid systems, J. Math. Biol. 32 Ž1994., 563]572. 5. J. Moser, Stable and random motions in dynamical systems, Ann. Math. 77 Ž1973., 50]59. 6. C. L. Siegel and J. K. Moser, ‘‘Lectures on Celestial Mechanics,’’ Springer-Verlag, New York, Berlin, 1971.
Stability of a Class of Nonlinear Difference Equations G. Papaschinopoulos* Department of Electrical and Computer Engineering, Democritus Uni¨ ersity of Thrace, 67100 Xanthi, Greece
C. J. Schinas Department of Applied Informatics, Uni¨ ersity of Macedonia, 54006 Thessaloniki, Greece Submitted by Gerry Ladas Received April 2, 1998
In this paper using KAM theory we investigate the stability nature of the zero equilibrium of the system of two nonlinear difference equations x nq 1 s a1 x n q b1 yn q f Ž c1 x n q c2 yn . ynq 1 s a2 x n q b 2 yn q f Ž c1 x n q c 2 yn .
5
,
n s 0, 1, . . . ,
where a i , bi , c i , i s 1, 2, are real constants and f : R ª R is a C` function. Q 1999 Academic Press
Key Words: nonlinear difference equations; stability; area preserving map; KAM theory.
1. INTRODUCTION The main goal in this paper is to generalize the results concerning the stability nature of the zero equilibrium of the systems considered in w2x ] w4x. So we consider the system of two nonlinear difference equations x nq 1 s a1 x n q b1 yn q f Ž c1 x n q c 2 yn . s F1 Ž x n , yn . ynq 1 s a2 x n q b 2 yn q f Ž c1 x n q c 2 yn . s F2 Ž x n , yn .
5
,
n s 0, 1, . . . ,
Ž 1. *E-mail address: [email protected]. 211 0022-247Xr99 $30.00 Copyright Q 1999 by Academic Press All rights of reproduction in any form reserved.
212
PAPASCHINOPOULOS AND SCHINAS
where a i , bi , c i , i s 1, 2, are real constants and f : R ª R is a C` function such that f / 0,
f Ž 0 . s 0,
f 9 Ž 0 . s 0.
Ž 2.
Moreover, the map F s Ž F1 , F2 . satisfies the following condition: ‘‘F is an area preserving map with an elliptic fixed point at the origin and eigenvalues l , l such that l k / 1,
k s 3, 4.’’
Ž H.
In the first proposition of this paper we prove that F satisfies the condition ŽH. if and only if < a1 q b 2 < - 2,
a1 b 2 y a2 b1 s 1,
a1 q b 2 / y1,
Ž a1 y a2 . c2 s Ž b1 y b 2 . c1 .
Ž 3.
It is known Žsee w1]6x. that if F satisfies the condition ŽH., KAM theory can be applied to study the stability of the zero equilibrium of Ž1.. In Proposition 2 we find conditions on a i , bi , c i , i s 1, 2, and the function f so that the zero equilibrium of Ž1. is stable.
2. MAIN RESULTS We now prove our main results. In the first proposition we find necessary and sufficient conditions so that the map F s Ž F1 , F2 . satisfies the condition ŽH.. PROPOSITION 1. Consider the map F s Ž F1 , F2 . where F1 , F2 are defined in Ž1. where f : R ª R is a C` function such that Ž2. are satisfied. Then F satisfies the condition ŽH. if and only if Ž3. hold. Proof. Suppose that F satisfies the condition ŽH.. Then from Ž2. it is obvious that the point Ž0, 0. is a fixed point for F. Let, for any x, y g R,
DF s
F1
F1
x F2
y F2
x
y
0
s
ž
a1 q c1 f 9 a 2 q c1 f 9
b1 q c2 f 9 . b2 q c2 f 9
/
Ž 4.
STABILITY OF NONLINEAR DIFFERENCE EQUATIONS
213
Then, using Ž2., we can see that the characteristic equation of DF with x s y s 0 is the following:
l2 y lŽ a1 q b 2 . q a1 b 2 y a2 b1 s 0.
Ž 5.
From the condition ŽH. we have that Ž5. has roots l and l where l k / 1, k s 3, 4. Then we have a1 b 2 y a2 b1 s 1,
< a1 q b 2 < - 2,
a1 q b 2 / y1.
Ž 6.
Moreover, since from the condition ŽH., F is an area preserving map Žsee w1x. we have for all x, y g R det DF Ž x, y . s 1.
Ž 7.
Therefore, relations Ž4., Ž6., and Ž7. imply that f 9 Ž u . Ž Ž a1 y a2 . c 2 y Ž b1 y b 2 . c1 . s 0
Ž 8.
for all u g R. Since Ž2. holds, it is obvious that there exists a u g R such that f 9Ž u. / 0. Then, from Ž8.,
Ž a1 y a2 . c2 s Ž b1 y b 2 . c1 .
Ž 9.
Hence from Ž6. and Ž9., relations Ž3. are satisfied. Suppose now that Ž3. holds. Then we can easily prove that F satisfies the condition ŽH.. This completes the proof of the proposition. In the following proposition using KAM theory we study the stability of the zero equilibrium of Ž1. where Ž2. and Ž3. hold. PROPOSITION 2. Consider the system of two nonlinear difference equations Ž1. where f : R ª R is a C` function such that Ž2. hold and a i , bi , c i , i s 1, 2, are real constants such that Ž3. are satisfied. Moreo¨ er, suppose that c2 / 0
if b1 / b 2 ,
c1 / 0
if b1 s b 2 .
Ž 10 .
214
PAPASCHINOPOULOS AND SCHINAS
Then the zero equilibrium of Ž1. is stable if the following conditions are satisfied: 2
¡
f - Ž 0. / p Ž f 0 Ž 0. . , c 2 Ž 2 c q 1 . g Ž a1 y a2 .
Ž 2 y c . Ž c q 1. Ž b p s~
1
if b1 / b 2 ,
,
y b 2 . a2
Ž 11 .
yc1 Ž 1 q 2 c .
¢ Ž 2 y c . Ž c q 1. ,
if b1 s b 2 ,
where c s a1 q b 2 and g: R ª R is a function such that g Ž ¨ . s ¨ 2 y c¨ q 1.
Ž 12 .
Proof. We may proceed to find the Birkhoff Normal Form of Ž5. by using the transforms of KAM theory described in w2x. First Transformation. First we find the matrix P such that
Py1AP s diag Ž l , l . ,
As
ž
a1 a2
b1 , b2
/
where l, l are the eigenvalues of the matrix A which satisfy Ž12.. Since l satisfies Ž12., we get
Ps
a1 y l
a1 y l
a2
a2
1
1
0
,
Py1 s
Now the change of variables xn u sP n yn ¨n
a2
lyl
1
l y a1 a2
0 y1
a1 y l a2
.
STABILITY OF NONLINEAR DIFFERENCE EQUATIONS
215
transforms Ž1. into the system u nq 1 s l u n q ¨ nq 1 s l¨ n q
a2 y a1 q l
lyl a2 y a1 q l lyl
¦
f Ž m u n q m¨ n . f Ž m u n q m¨ n .
¥,
§
n s 0, 1, . . . , Ž 13 .
where
ms
c1 Ž a1 y l . a2
q c2 .
Ž 14 .
Since Ž2. holds, by writing the Taylor expansion for f Ž m u n q m¨ n . about Ž0, 0. and keeping terms up to order 3, from Ž13. we take u nq 1 s l u n q =
ž
A 2
¨ nq 1 s l¨ n q
=
ž
A 2
a2 y a1 q l
lyl 2 Ž m u n q m¨ n . q
B 6
Ž m u n q m¨ n .
3
/
q O4 ,
/
q O4 ,
Ž 15 .
a2 y a1 q l
lyl 2 Ž m u n q m¨ n . q
B 6
Ž m u n q m¨ n .
3
where n s 0, 1, . . . and A s f 0 Ž0., B s f -Ž0.. Second Transformation. In Ž15. we make the change of variables u n s j n q f 2 Ž j n , hn . q f 3 Ž j n , hn . , ¨ n s hn q c 2 Ž j n , hn . q c 3 Ž j n , hn . ,
Ž 16 .
with k
fk Ž j , h . s
Ý a k j j ky jh j , js0
k s 2, 3
k
ck Ž j , h . s
Ý ak j j h j
js0
kyj
,
Ž 17 .
216
PAPASCHINOPOULOS AND SCHINAS
to obtain
j nq 1 s lj n q a 2 j n2hn q O4 ,
hnq1 s lhn q a 2 j nhn2 q O4 . Ž 18 .
In order to calculate the coefficients a k j , a 2 we substitute Ž16., Ž17., Ž18. to Ž15.. Then the resulting equations will show an equality of two power series in j n and hn whose coefficients can be identified recursively up to order 3. Hence we get
a 20 s a 21 s a 22 s a 30 s
A Ž a2 y a1 q l . m2 2 Ž l y l . Ž l2 y l .
,
A Ž a2 y a1 q l . mm
Ž l y l. Ž 1 y l. A Ž a2 y a1 q l . m2 2 Ž l y l . Ž l2 y l .
,
,
m2 B Ž a2 y a1 q l . m A a m q a m q , Ž . 20 22 6 Ž l3 y l . Ž l y l .
ž
/
Ž 19 .
a 31 s arbitrary, a 32 s
a1 y a2 y l
Ž l y l.
2
ž
A mm Ž a 21 q a 20 . q A m2a 21
qA m2a 22 q
a 33 s
Bmm 2 2
/
,
m2 B Ž a2 y a1 q l . m A a m q a m q , Ž . 22 20 6 Ž l3 y l . Ž l y l .
ž
/
and
a2 s
A mma 20 Ž a2 y a1 q l .
lyl q
q
A m 2a 21 Ž a2 y a1 q l .
q
lyl Bm m Ž a2 y a1 q l .
A mma 21 Ž a2 y a1 q l .
lyl A m2a 22 Ž a2 y a1 q l .
lyl
2
q
2 Ž l y l.
.
Ž 20 .
217
STABILITY OF NONLINEAR DIFFERENCE EQUATIONS
We suppose first that b1 / b 2 . Then from Ž3., Ž14., and since l satisfies Ž12., we get
Ž a1 y l . Ž a1 y a2 . c2
ms s s s
a2 Ž b1 y b 2 .
q c2
c 2 Ž Ž a1 y l . Ž a1 y a2 . q b 2 Ž a1 y a2 . y 1 . a2 Ž b1 y b 2 . c 2 Ž Ž a1 y a2 . Ž a1 y l q b 2 . y 1 . a2 Ž b1 y b 2 . c 2 Ž Ž a1 y a2 . l y 1 . a2 Ž b1 y b 2 .
.
Ž 21 .
Then relations Ž19., Ž20., and Ž21. imply that
a2 s
c 23 < Ž a1 y a2 . l y 1 < 2 a23 Ž b1 y b 2 . =
ž
3
2
c 2 A2
ž
a2 Ž b1 y b 2 .
Ž a2 y a1 q l . Ž Ž a1 y a2 . l y 1 .
2
2
2 Ž l y l . Ž l2 y l . q
< a2 y a1 q l < 2 < Ž a1 y a2 . l y 1 < 2 2
Ž l y l. Ž l y 1. 2
q
q
q
Ž a2 y a1 q l . Ž Ž a1 y a2 . l y 1 .
2
2
Ž l y l. Ž 1 y l. < a2 y a1 q l < 2 < Ž a1 y a2 . l y 1 < 2 2
2 Ž l y l . Ž l y l2 .
B Ž a2 y a1 q l . Ž Ž a1 y a2 . l y 1 . 2 Ž l y l.
/
/
.
Then, for b1 / b 2 we take R Ž a2 . s
c 23 Ž g Ž a1 y a2 . . 4 a32 Ž b1 y b 2 .
3
2
ž
c 2 A2 Ž 2 c q 1 . g Ž a1 y a2 .
Ž 2 y c . Ž c q 1 . a2 Ž b1 y b 2 .
/
y B . Ž 22 .
218
PAPASCHINOPOULOS AND SCHINAS
Suppose now that b1 s b 2 . Then using Ž9. and since from Ž6., a1 / a2 , we have c 2 s 0. So, from Ž14., it is obvious that
ms
c1 Ž a1 y l . a2
.
Ž 23 .
Therefore, from Ž19., Ž20., and Ž23., we get
a2 s
c13 < a1 y l < 2 a23 =
c 1 A2
2 Ž a2 y a1 q l . Ž a1 y l .
a2
2 Ž l y l . Ž l2 y l .
2
q
q
2
q
2 Ž a2 y a1 q l . Ž a1 y l . 2
Ž l y l. Ž 1 y l.
B Ž a2 y a1 q l . Ž a1 y l . 2 Ž l y l.
0
< a2 y a1 q l < 2 < a1 y l < 2
2
q
2
Ž l y l. Ž l y 1. < a2 y a1 q l < 2 < a1 y l < 2 2
2 Ž l y l . Ž l y l2 .
0
.
Then, since l q l s a1 q b 2 , a2 s Ž a1 b 2 y 1.rb 2 , we have R Ž a2 . s
c13 b 22 4 Ž 1 y a1 b 2 .
ž
c 1 A2 Ž 1 q 2 c .
Ž2 y c. Ž1 q c.
qB .
/
Ž 24 .
Third Transformation. The change of variables
j n s rn q isn ,
hn s rn y isn ,
transforms Ž18. into the system rnq 1 s R Ž l . rn y I Ž l . sn q rn Ž rn2 q sn2 . R Ž a 2 . y sn Ž rn2 q sn2 . I Ž a 2 . , snq 1 s I Ž l . rn q R Ž l . sn q sn Ž rn2 q sn2 . R Ž a 2 . q rn Ž rn2 q sn2 . I Ž a 2 . .
Ž 25 .
STABILITY OF NONLINEAR DIFFERENCE EQUATIONS
219
Let w s g 0 q g 1Ž rn2 q sn2 ., g 0 , g 1 g R. Then cos w s cos g 0 cos Ž g 1 Ž rn2 q sn2 . . y sin g 0 sin Ž g 1 Ž rn2 q sn2 . . s 1y
ž
y
ž
g 12 Ž rn2 q sn2 .
2
2!
g 1 rn2
Ž
q
sn2
.y
q ??? cos g 0
/
g 13 Ž rn2 q sn2 .
3
q ??? sin g 0
/
3!
s cos g 0 y g 1 Ž rn2 q sn2 . sin g 0 q O4 .
Ž 26 .
Working similarly, we can find sin w s sin g 0 q g 1 Ž rn2 q sn2 . cos g 0 q O4 .
Ž 27 .
We write now Ž25. into the Birkhoff Normal Form ysin w cos w
rnq 1 cos w s snq 1 sin w
ž / ž
O4 rn q . sn O4
ž /
/ž /
Ž 28 .
Then, from Ž25., Ž26., Ž27., and Ž28., we take R Ž l . s cos g 0 ,
g1 s y
R Ž a2 . sin g 0
.
Ž 29 .
Hence, since l is a root of Ž12., and Ž22., Ž24., Ž29. hold, it follows that cos g 0 s
¡
y
g1 s
~
c 23 Ž g Ž a1 y a2 . .
¢
y
ž
2
,
2
2'4 y c 2 a32 Ž b1 y b 2 .
=
c
3
c 2 A2 Ž 2 c q 1 . g Ž a1 y a2 .
Ž 2 y c . Ž c q 1 . a2 Ž b1 y b 2 . c13 b 22
2'4 y c 2 Ž 1 y a1 b 2 .
ž
/
yB ,
c 1 A2 Ž 1 q 2 c .
Ž2 y c. Ž1 q c.
if b1 / b 2 , qB ,
/
if b1 s b 2 .
Ž 30 .
220
PAPASCHINOPOULOS AND SCHINAS
Since a1 y a2 g R and y2 - c - 2 hold, it follows that a1 y a2 does not satisfy Ž12.. Hence if Ž11. is satisfied, from Ž10. and Ž30. we get g 0 , g 1 / 0. Therefore, from Theorem 2.13 of w5x, the zero equilibrium of Ž1. is stable. This completes the proof of the proposition. Remark 1. We see now that our system Ž1. includes systems considered in w2x, w3x, and w4x and Proposition 2 holds for all of them. First consider the system in w3x: u nq 1 s 2 u n y ¨ n q ln ¨ nq 1 s u n
ž
a 1 q Ž a y 1. e u n
/
¦ ¥ §,
a)1
,
n s 0, 1, . . . .
Ž 31 . Then by adding both sides of Ž31. and setting u n s x n , yn s u n q ¨ n , f Ž x . s lnŽ arw1 q Ž a y 1. e x x. y wŽ1 y a.rax x, we take the following system: x nq 1 s ynq 1 s
2a q 1 a 3a q 1 a
¦
x n y yn q f Ž x n . x n y yn q f Ž x n .
¥,
n s 0, 1, . . . ,
§
Ž 32 .
which is equivalent to Ž31.. It is obvious that Ž32. is included in Ž1. where a1 s Ž2 a q 1.ra, a2 s Ž3a q 1.ra, b1 s b 2 s y1, c1 s 1, c 2 s 0. We can easily prove that Ž2., Ž3., and Ž10. are satisfied. Moreover, f 0 Ž 0. s
1ya a
2
f - Ž 0. s
,
psy
Ž 1 y a. Ž 2 y a. a3
a Ž 3a q 2 .
Ž a y 1. Ž 2 a q 1.
,
Ž 33 .
,
where the constant p is defined in Proposition 2. Therefore, from Ž33. and since a / 0, relation Ž11. is satisfied. Hence, from Proposition 2, the zero equilibrium of Ž31. is stable. Next consider the system defined in w2x: u nq 1 s y¨ n q ln e u n q ¨ nq 1 s u n
ž
b E
/
¦ ¥ §,
y ln E
n s 0, 1, . . . ,
Ž 34 .
221
STABILITY OF NONLINEAR DIFFERENCE EQUATIONS
where b g Ž0, `., b / 1, and E is the positive solution of the equation E 2 y E y b s 0. Working similarly, we can prove that Ž34. is equivalent to the system
x nq 1 s ynq 1 s
Eq1
x n y yn q f Ž x n .
E 2Eq 1 E
¦
¥,
n s 0, 1, . . . ,
§
Ž 35 .
x n y yn q f Ž x n .
where f Ž x . s lnŽ e x q brE . y ln E y Ž1rE . x. It is obvious that Ž35. is included in Ž1. where a1 s Ž E q 1.rE, a2 s Ž2 E q 1.rE, b1 s b 2 s y1, c1 s 1, c 2 s 0. We can prove that Ž2., Ž3., and Ž10. hold. Furthermore,
f 0 Ž 0. s
Ey1 E
2
f - Ž 0. s
,
psy
Ž E y 1. Ž E y 2. E3
E Ž E q 2.
Ž 2 E y 1. Ž E q 1.
,
Ž 36 .
,
where p is defined in Proposition 2. Then since b g Ž0, `. and b / 1, relations Ž36. imply that Ž11. holds and so the zero equilibrium of Ž34. is stable. Finally, consider the system defined in w4x: x nq 1 s x n y ayn q a Ž 1 y e y n q yn . ynq 1 s x n q Ž 1 y a . yn q a Ž 1 y e y n q yn .
5
,
n s 0, 1, . . . , Ž 37 .
where a is a constant, 1 - a - 4, a / 2, 3. It is obvious that Ž37. is included in Ž1. where a1 s a2 s 1, b1 s ya, b 2 s 1 y a, c1 s 0, c 2 s 1. We can easily prove that Ž2., Ž3., and Ž10. are satisfied. Moreover,
f 0 Ž 0 . s ya,
f - Ž 0 . s ya,
ps
5 y 2a aŽ a y 3.
,
Ž 38 .
where p is defined in Proposition 2. Then since a / 2, relations Ž38. imply that Ž11. holds and so the zero equilibrium of Ž37. is stable.
222
PAPASCHINOPOULOS AND SCHINAS
REFERENCES 1. J. K. Hale and H. Kocak, ‘‘Dynamics and Bifurcations,’’ Springer-Verlag, New York, Berlin, 1991. 2. V. L. Kocic, G. Ladas, G. Tzanetopoulos, and E. Thomas, On the stability of Lyness’s equation, Dynamics of Continuous, Discrete, and Impulsi¨ e Systems 1 Ž1995., 245]254. 3. G. Ladas, G. Tzanetopoulos, and A. Torbis, On May’s host parasitoid model, J. Difference Equations Appl. 2 Ž1996., 195]204. 4. C. Meier, W. Senn, R. Hauser, and M. Zimmermann, Strange limits of stability in host-parasitoid systems, J. Math. Biol. 32 Ž1994., 563]572. 5. J. Moser, Stable and random motions in dynamical systems, Ann. Math. 77 Ž1973., 50]59. 6. C. L. Siegel and J. K. Moser, ‘‘Lectures on Celestial Mechanics,’’ Springer-Verlag, New York, Berlin, 1971.