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Integrability of fractional order generalized systems with p:-q resonance
Chaos, Solitons & Fractals 67 (2014) 82–86
Contents lists available at ScienceDirect
Chaos, Solitons & Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos
Integrability of fractional order generalized systems with p : q resonance Wentao Huang a,b,⇑, Tianlong Gu a, Huili Li a a b
Guangxi Key Laboratory of Trusted Software, School of Computing Science and Mathematics, Guilin University of Electronic Technology, Guilin 541004, China Department of Mathematics, Hezhou University, Hezhou 542800, China
a r t i c l e
i n f o
Article history: Received 11 May 2014 Accepted 17 June 2014
a b s t r a c t This paper is devoted to studying integrability for fractional order systems with p : q resonance. We develop some methods to transform such systems into corresponding poly nomial ones. As an application, we discuss the integrability of a class of 2; 14 -order system 1 with 1 : 2 resonance and a class of 3; 5 -order system with 1 : 3 resonance. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction Consider the system
8 n X dz > > > ¼ pz þ ðzwÞk ani;i zni wi ; > < dT i¼0 n X > dw > k > > bni;i wni zi ; : dT ¼ qw ðzwÞ
ð1Þ
i¼0
where n is a given positive integer, ani;i ; bni;i are complex parameters, p; q 2 Z þ ; ðp; qÞ ¼ 1, and k is a real constant satisfying k P 0. We call system (1) the ðn; kÞ-order system with p : q resonance. Notice that a ðn; 0Þ-order system is a usual polynomial system of degree n having a p : q resonance singular point at the origin. In this paper we study integrability of system (1). The problem of integrability of planar differential systems has attracted much attention in few last decades. Starting from the work of Dulac [5] on the integrability of a 1 : 1 resonant system many studies have been devoted to the ⇑ Corresponding author at: School of Computing Science and Mathematics, Guilin University of Electronic Technology, Guilin 541004, China. E-mail addresses: [email protected] (W. Huang), [email protected] (T. Gu), [email protected] (H. Li). http://dx.doi.org/10.1016/j.chaos.2014.06.006 0960-0779/Ó 2014 Elsevier Ltd. All rights reserved.
investigation of local integrability of such polynomial systems, see, for instance [1,3,7,11], and references therein. Many authors also investigated the integrability problem for systems with 1 : q resonance, see for instance, [2,4,9,10,13]. In particular, the authors of [6] have solved the integrability problem for the quadratic system with 1 : 2 resonance. Systems with p : q resonant singular points were considered in [3,8,15,14]. We mention that the authors of [12] have obtained sufficient conditions for integrability of non-linearizable Lotka–Volterra systems and have given the integrability conditions for Lotka–Volterra systems with 3 : q resonance. In [16], Xiao gave the definition of generalized singular point value and obtained the integrability conditions for some quadratic systems with p : q resonance. Recently, the authors of [17] generalized the integrability problem to the systems with degenerate resonant singular point, further research in this direction has be performed in [10,18]. In this paper we investigate the integrability problem for ðn; kÞ-order systems with p : q resonance, that is, for systems of the form (1). With a suitable change of the coordinates we transform the system under consideration to the usual polynomial system which admits a certain ~Þ resonance. ~; q ðp This paper is organized as follows. In Section 2 an approach that allows to transform a fractional order system with a p : q resonant singular point at the origin into
83
W. Huang et al. / Chaos, Solitons & Fractals 67 (2014) 82–86
a usual polynomial systems is introduced. Then an algorithm to calculate the singular point values of the system is presented. As an application of our method in Sections 3 and 4, respectively, using computer algebra systemMathematica to compute, we obtain the integrability con 1 ditions 1 for 2; 4 -order system with 1 : 2 resonance and 3; 5 order system with 1 : 3 resonance. As far as we know, this is the first time that the integrability of fractional order systems with p : q resonance is investigated.
8 1 X dz > > ¼ pz þ aab za wb ¼ Zðz; wÞ; > > < dT aþb¼2
ð7Þ
1 X > dw > > ¼ qw bab wa zb ¼ Wðz; wÞ; > : dT
aþb¼2
we can derive successively the following series 1 X
Fðz; wÞ ¼
cab za wb ;
aþb¼2
2. A homeomorphic transformation and computation of singular point values In order to investigate the integrability of system (1) we first prove the following theorem. Theorem 2.1. Under the transformation nþk
nþk 1þn
k1
v ¼w
u ¼ z1þn w1þn ;
k1
z1þn ;
dt ¼
1 dT 1þn
ð2Þ
such that
X1 dF @F @F ¼ l ðzwÞmþ1 ; Z W¼ m¼1 m dT @z @w
ð8Þ
For all pairs ða; bÞ, if a þ b 6 p þ q, then ca;b ¼ 1, except cpq ¼ 1; if pa ¼ qb, then ca;b ¼ 0; if pa qb – 0, then ca;b is computed by the recursive formula cab ¼
aX þbþ2 1 ½ða k þ 1Þak;j1 ðb j þ 1Þbj;k1 cakþ1;bjþ1 : ð9Þ qb pa kþj¼3
For any positive integer m; lm is given by
system (1) is changed into
8 n n X X du > > > ¼ ððn þ kÞp þ ð1 kÞqÞu þ ðn þ kÞ ani;i u1þni v iþ1 þ ð1 kÞ bni;i u2þi v ni ; > < dt i¼0 i¼0
ð3Þ
n n X X > dv > 1þni iþ1 > > u ð1 kÞ ani;i v 2þi uni : : dt ¼ ððn þ kÞq þ ð1 kÞpÞv ðn þ kÞ bni;i v i¼0
i¼0
Proof. Applying transformation (2) we obtain
(
nþk k1 nþk 1þn z 1 w1þn dz 1þn dt
nþk k1 k1 1þn z w1þn1 dw ; 1þn dt
du dt
¼
dv dt
nþk k1 ¼ 1þn w1þn1 z1þn dw þ 1þn w1þn z1þn1 dz ; dt dt
nþk
þ
lm ¼
nþk
k1
n ln uþk ln uþln v k ln v 1þnþ2k
;
w¼e
ð4Þ
k1
ln uk ln uþn ln v þk ln v 1þnþ2k
ð5Þ
;
and
dz dz dT ¼ ; dt dT dt
dw dw dT ¼ ; dt dT dt
dT ¼1þn dt
Now it is easy to see that the theorem holds.
½ðqm þ q k þ 1Þak;j1 ðpm þ p j þ 1Þbj;k1
kþj¼3
On the other hand, it is clear that
z¼e
1 X
ð6Þ h
cqmþqkþ1;pmþpjþ1
ð10Þ
where 8ðk; jÞ, when k < 0 or j < 0, we take ak;j ¼ bk;j ¼ ck;j ¼ 0. In Lemma 2.1, lm is called the mth singular point quantity of the origin of system (7). Evidently, F is a first integral of system (7) if and only if all lm ¼ 0. For system (1), according to Lemma 2.1, we have the following recursive formulas to compute the singular point values at the origin.
Since the transformation (2) is a homeomorphism in a deleted neighborhood of the origin, system (1) and system (3) have the same topological structure in a neighborhood of the origin. Thus, the following theorem holds.
Theorem 2.3. For system (1), the singular point value lm (m ¼ 1; 2; . . .) at the origin is given by the recursive formulas as follows: if a < 0 or b < 0 or ððn þ kÞp þ ð1 kÞqÞa ¼ ððn þ kÞq þ ð1 kÞpÞb or a þ b < p þ q, then ca;b ¼ 0; else:
Theorem 2.2. System (1) is integrable at the origin if and only if system (3) is integrable at the origin. Theorems 2.1 and 2.2 give a method to study the integrability of a fractional order generalized system (1) with p : q resonance. That is, one can transform system (1) into a polynomial system (3) with p : q resonance and then apply the methods of investigation of integrability of system (3) to study the integrability of system (1). Xiao gave the following lemma in [16].
ca;b ¼
Lemma 2.1 [16]. For the system
1 ððn þ kÞq þ ð1 kÞpÞb ððn þ kÞp þ ð1 kÞqÞa
X ½ððn þ kÞa ð1 kÞb þ ðn þ q þ 1 kÞðn þ 1ÞÞaj;k1 k;j
ððn þ kÞb ð1 kÞa þ ðn þ p þ 1 jÞðn þ 1ÞÞbj;k1 cað1kÞjðnþkÞkþðnþ1Þð1kÞ;bð1kÞkðnþkÞjþðnþ1Þð1kÞ ; for any positive integer m,
ð11Þ
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W. Huang et al. / Chaos, Solitons & Fractals 67 (2014) 82–86
X ½ðn þ q þ qm k þ 1Þak;j1
lm ¼ ðn þ 1Þ
Using the recursive formulas of Theorem 3.1 the singular point values of system (16) at the origin can be calculated using a computer algebra system, for instance, Mathematica.
k;j
ðn þ p þ pm j þ 1Þbj;k1 cðð1kÞpþðnþkÞqÞmð1kÞjðnþkÞkþð1kÞðnþ1Þ ;ððn þ kÞp þ ð1 kÞqÞm ðn þ kÞj ð1 kÞk þ ð1 kÞðn þ 1Þ:
ð12Þ
As an application of our method, we study the integrability for a 14 order generalized quadratic system with 1 : 2 resonance and a 15 order generalized cubic system with 1 : 3 resonance in the following sections.
Here for all lk , we have applied the conditions l1 ¼ l2 ¼ ¼ lk1 ¼ 0; k ¼ 2; 3; 4; 5.
Consider the following fractional order generalized quadratic system with 1 : 2 resonance:
ð13Þ
where p ¼ 1; q ¼ 2; k ¼ 14 ; a02 ¼ b02 ¼ 0; b20 ¼ 5a11 , that is
8 1 dz > > < ¼ z þ ðzwÞ4 ða20 z2 þ a11 zwÞ; dT > > dw ¼ 2w ðzwÞ14 ðb w2 þ b zwÞ; : 20 11 dT
ð14Þ
3
1
3 4
1 4
v ¼ w z ;
dt ¼
1 dT: 3
ð15Þ
that is transformation (2) with jm¼2;k¼1 system (14) can be 4 reduced to
8 du > > ¼ 5u þ ð3a20 þ b11 Þu3 v þ 8a11 u2 v 2 ; < dt > dv > : ¼ 7v ða20 þ 3b11 Þu2 v 2 16a11 uv 3 : dt
Theorem 3.1. For system (16) the singular point value lm (m ¼ 1; 2; ) at the origin is given by the recursive formulas as follows: if a < 0 or b < 0 or 5a ¼ 7b or a þ b < 12, then ca;b ¼ 0; else
1 ½ðða 1Þð3a11 þ b20 Þ 5a þ 7b
ðb 2Þða11 þ 3b20 ÞÞca1;b2 þ ðða 2Þð3a20 þ b11 Þ ðb 1Þða20 þ 3b11 ÞÞca2;b1 ; ð17Þ For any positive integer m,
lm ¼ ½8a11 ð7m þ 6Þ 16a11 ð5m þ 3Þc7mþ6;5mþ3 ½ð7m þ 5Þð3a20 þ b11 Þ ð5m þ 4Þða20 þ 3b11 ÞÞc7mþ5;5mþ4 :
Theorem 3.3. For system (16) the first five singular point values are equal to zero if and only if one of the following four conditions holds:
ðIÞ a11 ¼ 0;
ð19Þ
ðIIÞ 11a20 ¼ 7b11 ;
ð20Þ
ðIIIÞ a20 ¼ 5b11 ;
ð21Þ
ðIVÞ a20 ¼ b11 :
ð22Þ
Theorem 3.4. For system (16) all singular point values at the origin are zero if and only if the first five singular point values at the origin are zero, that is, one of four conditions of Theorem 3.3 holds. That is, the four conditions of Theorem 3.3 are the conditions for integrability of system (16).
ð16Þ
Thus, the investigation of the fractional order generalized system (14) with 1 : 2 resonance is reduced to the investigation of the polynomial system (16) with 5 : 7 resonance. By Theorem 2.3 we have.
ca;b ¼
From Theorem 3.2 we have.
Using the above theorem we obtain the integrability conditions of system (16).
By the transformation,
u ¼ z4 w4 ;
8 9
l1 ¼ a11 ð11a20 7b11 Þða20 5b11 Þða20 b11 Þ; l2 ¼ l3 ¼ l4 ¼ l5 ¼ 0:
3. Integrability of a 2; 14 order system with 1 : 2 resonance
dz ¼ pz þ ðzwÞk ða20 z2 þ a11 zw þ a02 w2 Þ; dT dw ¼ qw ðzwÞk ðb20 w2 þ b11 wz þ b02 z2 Þ; dT
Theorem 3.2. The first five singular point values of system (16) at the origin are given as follows:
ð18Þ
Proof. The necessity is obvious. We need only to prove the sufficiency. If condition (I) holds, then system (16) has the inte
grating factor J 1 ¼ u
3ð17a20 5b11 Þ 8ð2a20 b11 Þ
41a20 13b11 8ð2a20 b11 Þ
v
, when 2a20 ¼ b11 , 7
system (16) has the first integral Fðu; v Þ ¼ u5 v . If condition (II) holds, then system (16) has the integrating factor J 2 ¼ u8 v 6 . If condition (III) holds, then system (16) has the integrating factor J 3 ¼ u8 v 6 ð1 þ 8b11 u2 v Þ. If condition (IV) holds, then system (16) has the 2 integrating factor J 4 ¼ u8 v 6 ð1 þ 43 a20 u2 v Þ . Thus, in each case the corresponding system is integrable. h Since transformation (15) is a homeomorphism in a deleted neighborhood of the origin, the integrability conditions of system (14) are the same as the integrability conditions of system (16). Thus, we have the following theorem. Theorem 3.5. System (14) is integrable if and only if one of the four conditions in Theorem 3.3 holds.
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4. Integrability of a 3; 15 order system with 1 : 3 resonance
l2 ¼ l3 ¼ l4 ¼ l5 ¼ 0
1
Consider a 3; 5 order system with 1 : 3 resonance: 3 X dz ¼ pz þ ðzwÞk a3i;i z3i wi ; dT i¼0 3 X dw ¼ qw ðzwÞk b3i;i w3i zi ; dT i¼0
ð23Þ
where p ¼ 1; q ¼ 3; k ¼ 15 ; a03 ¼ b03 ¼ 0; a21 ¼ b21 ¼ 0; b30 ¼ 11a12 , that is, the system
8 < dz ¼ z þ ðzwÞ15 ða30 z3 þ a12 zw2 Þ; dT : dw dT
1
ð24Þ
¼ 2w ðzwÞ5 ðb30 w3 þ b12 z2 wÞ:
By the transformation 4
1
u ¼ z5 w5 ;
4 5
1 5
v ¼ w z ;
dt ¼
1 dT 4
ð25Þ
(which is transformation (2) with jm¼3;k¼1 ) system (24) can 5 be reduced to
(
du dt dv dt
¼ 7u þ ð4a30 þ b12 Þu4 v þ ð4a12 þ b30 Þu2 v 3 ; ¼ 13v ða30 þ 4b12 Þu3 v 2 ða12 þ 4b30 Þuv 4 :
ð26Þ
Thus, the investigation of the fractional order generalized system (24) with 1 : 3 resonance is reduced to the study of the integrability of the polynomial system (26) with 7 : 13 resonance. Similar as in the previous section we have
For all lk , we have applied the conditions l1 ¼ l2 ¼ ¼ lk1 ¼ 0; k ¼ 2; 3; 4; 5. From Theorem 4.2 we have the following statement. Theorem 4.3. For system (26) the first five singular point values are zeros if and only if one of the following five conditions holds:
ðIÞ b30 ¼ 0;
ð29Þ
ðIIÞ 23a30 ¼ 13b12 ;
ð30Þ
ðIIIÞ a30 ¼ 11b12 ;
ð31Þ
ðIVÞ 17a30 ¼ 7b12 ;
ð32Þ
ðVÞ a30 ¼ b12 :
ð33Þ
Using the conditions (29)–(33) we obtain the following theorem. Theorem 4.4. For system (26) all the singular point values at the origin are zero if and only if the first five singular point values at the origin are zero, i.e., one of the five conditions of Theorem 4.3 holds. That is, the five conditions of Theorem 4.3 are the integrability conditions for the system (26). Proof. The necessity is obvious. Thus we have only to prove the existence of first integrals of integrating factors. If condition (I) holds, then system (26) has the inte4ð47a
Theorem 4.1. For system (26) the singular point value lm (m ¼ 1; 2; . . .) at the origin is given by the recursive formulas: if a < 0 or b < 0 or 7a ¼ 13b or a þ b < 20, then ca;b ¼ 0; else
ca;b ¼
1 ½ðða 1Þð4a12 þ b30 Þ 7a þ 13b ðb 3Þða12 þ 4b30 ÞÞca1;b3
grating factor J 1 ¼ u
7b12 Þ 30 b12 Þ
15ð3a30
v
2ð61a30 11b12 Þ 15ð3a30 b12 Þ
, when 2a20 ¼ b11 , 13
system (26) has the first integral F½u; v ¼ Cð1Þðu 7 v Þ, where Cð1Þ is a constant. If condition (II) holds, then system (26) has the 3 integrating factor J 2 ¼ u14 v 8 ð1 þ 15 23 b12 u v Þ. If condition (III) holds, then system (26) has the 3 integrating factor J 3 ¼ u14 v 8 ð1 þ 15b12 u3 v Þ .
þ ðða 3Þð4a30 þ b12 Þ ðb 1Þða30 þ 4b12 ÞÞca3;b1 ; ð27Þ
If condition (IV) holds, then system (26) has the integrating factor J 4 ¼ u14 v 8 . If condition (V) holds, then system (26) has the 2
For any positive integer m,
integrating factor J 5 ¼ u14 v 8 ð1 þ 54 b12 u3 v Þ .
lm ¼ ½ð4a30 þ b12 Þð13m þ 10Þ
From these observations we conclude the correctness of the theorem. h
ða30 þ 4b12 Þð7m þ 6Þc13mþ10;7mþ6 ½ð13m þ 12Þð4a12 þ b30 Þ ð7m þ 4Þða12 þ 4b30 ÞÞc13mþ12;7mþ4 :
From Theorem 4.4 we obtain.
ð28Þ
Applying the recursive formulas of Theorem 4.1 the singular point values of system (26) at the origin are easily calculated.
Theorem 4.5. System (24) is integrable if and only if one of the five conditions in Theorem 4.3 holds. Acknowledgments
Theorem 4.2. The first five singular point values of system (26) at the origin are given as follows:
15 b30 ð23a30 13b12 Þða30 11b12 Þ 4096 ð17a30 7b12 Þða30 b12 Þ;
l1 ¼
We would like to give our thanks to Prof. Jiazhong Yang and Prof. Valery Romanovski for their helpful suggestions. This research are supported by Nature Science Foundation of China (No. 11261013, 11361017), Nature Science Foundation of Guangxi (2012GXNSFAA053003) and Guangxi
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education department Key Laboratory of Symbolic Computation and Engineering Data Processing. References [1] Algaba A, García C, Giné J. On the formal integrability problem for planar differential systems. Abstr Appl Anal 2013; (Article ID 482305;10 pp.). [2] Chen X, Gine J, Romanovski V, Shafer D. The 1:-q resonant center problem for certain cubic Lotka–Volterra systems. Appl Math Comput 2012;218:11620–33. [3] Christopher C, Mardešic P, Rousseau C. Normalizable, integrable and linearizable saddle points for complex quadratic systems in C2 . J Dyn Control Syst 2003;9:311–63. [4] Dolic´anin D, Giné J, Oliveira R, Romanovski V. The resonant center problem for a 2 : 3 resonant cubic Lotka–Volterra system. Appl Math Comput 2013;220:12–9. [5] Dulac H. Détermination et intégration d’une certaine classe d’équations différentielles ayant pour point singulier un centre. Bull Sci Math 1908;32:230–52. [6] Fronville A, Sadovski A, Zoladek H. Solution of the 1 : 2 resonant center problem in the quadratic case. Fund Math 1998;157:191–207. [7] Gine J, Romanovski V. Integrability conditions for Lotka–Volterra planar complex quintic systems. Nonlinear Anal-Real 2010;11:2100–5. [8] Gravel S, Thibault P. Integrability and linearizablity of the Lotka– Volterra system with a saddle points with rational hyperbolicity ratio. Diff Equ 2002;184:20–47.
[9] Hu Z, Romanovski V, Shafer D. 1 : 3 resonant centers on C2 with homogeneous cubic nonlinearities. Comput Math Appl 2008;56:1927–40. [10] Li H, Huang W. Generalized singular point quantity and integrability of the degenerate resonant singular point of a 1 : 2 system. J Guangxi Acad Sci 2011;27:182–3 [193 (in Chinese)]. [11] Liu C, Chen G, Chen G. Integrability of Lotka–Volterra type systems of degree 4. J Math Anal Appl 2012;388:1107–16. [12] Liu C, Chen G, Li C. Integrability and linearizability of the Lotka– Volterra systems. Diff Equ 2004;198:301–20. [13] Romanovski V, Prešern M. An approach to solving systems of polynomials via modular arithmetic with applications. J Comput Appl Math 2011;236:196–208. [14] Romanovski V, Shafer D. On the center problem for p : q resonant polynomial vector fields. Bull Belg Math Soc Simon Stevin 2008;15:871–87. [15] Wang Q, Liu Y. Linearizability of the polynomial differential systems with a resonant singular point. Bull Sci math 2008;132:97–111. [16] Xiao P. Critical point quantities and integrability conditions for complex planar resonant polynomial differential systems. Central South University Doctoral Dissertation 2005 [in Chinese]. [17] Zhang Q, Gui W, Liu Y. The generalized center problem of degenerate resonant singular point. Appl Math Comput 2009;215:1507–12. [18] Zhang Q, Liu Y, Gui W. Generalized singular point quantity and integrability of degenerate resonant singular point. Bull Sci Math 2009;133:198–204.
Contents lists available at ScienceDirect
Chaos, Solitons & Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos
Integrability of fractional order generalized systems with p : q resonance Wentao Huang a,b,⇑, Tianlong Gu a, Huili Li a a b
Guangxi Key Laboratory of Trusted Software, School of Computing Science and Mathematics, Guilin University of Electronic Technology, Guilin 541004, China Department of Mathematics, Hezhou University, Hezhou 542800, China
a r t i c l e
i n f o
Article history: Received 11 May 2014 Accepted 17 June 2014
a b s t r a c t This paper is devoted to studying integrability for fractional order systems with p : q resonance. We develop some methods to transform such systems into corresponding poly nomial ones. As an application, we discuss the integrability of a class of 2; 14 -order system 1 with 1 : 2 resonance and a class of 3; 5 -order system with 1 : 3 resonance. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction Consider the system
8 n X dz > > > ¼ pz þ ðzwÞk ani;i zni wi ; > < dT i¼0 n X > dw > k > > bni;i wni zi ; : dT ¼ qw ðzwÞ
ð1Þ
i¼0
where n is a given positive integer, ani;i ; bni;i are complex parameters, p; q 2 Z þ ; ðp; qÞ ¼ 1, and k is a real constant satisfying k P 0. We call system (1) the ðn; kÞ-order system with p : q resonance. Notice that a ðn; 0Þ-order system is a usual polynomial system of degree n having a p : q resonance singular point at the origin. In this paper we study integrability of system (1). The problem of integrability of planar differential systems has attracted much attention in few last decades. Starting from the work of Dulac [5] on the integrability of a 1 : 1 resonant system many studies have been devoted to the ⇑ Corresponding author at: School of Computing Science and Mathematics, Guilin University of Electronic Technology, Guilin 541004, China. E-mail addresses: [email protected] (W. Huang), [email protected] (T. Gu), [email protected] (H. Li). http://dx.doi.org/10.1016/j.chaos.2014.06.006 0960-0779/Ó 2014 Elsevier Ltd. All rights reserved.
investigation of local integrability of such polynomial systems, see, for instance [1,3,7,11], and references therein. Many authors also investigated the integrability problem for systems with 1 : q resonance, see for instance, [2,4,9,10,13]. In particular, the authors of [6] have solved the integrability problem for the quadratic system with 1 : 2 resonance. Systems with p : q resonant singular points were considered in [3,8,15,14]. We mention that the authors of [12] have obtained sufficient conditions for integrability of non-linearizable Lotka–Volterra systems and have given the integrability conditions for Lotka–Volterra systems with 3 : q resonance. In [16], Xiao gave the definition of generalized singular point value and obtained the integrability conditions for some quadratic systems with p : q resonance. Recently, the authors of [17] generalized the integrability problem to the systems with degenerate resonant singular point, further research in this direction has be performed in [10,18]. In this paper we investigate the integrability problem for ðn; kÞ-order systems with p : q resonance, that is, for systems of the form (1). With a suitable change of the coordinates we transform the system under consideration to the usual polynomial system which admits a certain ~Þ resonance. ~; q ðp This paper is organized as follows. In Section 2 an approach that allows to transform a fractional order system with a p : q resonant singular point at the origin into
83
W. Huang et al. / Chaos, Solitons & Fractals 67 (2014) 82–86
a usual polynomial systems is introduced. Then an algorithm to calculate the singular point values of the system is presented. As an application of our method in Sections 3 and 4, respectively, using computer algebra systemMathematica to compute, we obtain the integrability con 1 ditions 1 for 2; 4 -order system with 1 : 2 resonance and 3; 5 order system with 1 : 3 resonance. As far as we know, this is the first time that the integrability of fractional order systems with p : q resonance is investigated.
8 1 X dz > > ¼ pz þ aab za wb ¼ Zðz; wÞ; > > < dT aþb¼2
ð7Þ
1 X > dw > > ¼ qw bab wa zb ¼ Wðz; wÞ; > : dT
aþb¼2
we can derive successively the following series 1 X
Fðz; wÞ ¼
cab za wb ;
aþb¼2
2. A homeomorphic transformation and computation of singular point values In order to investigate the integrability of system (1) we first prove the following theorem. Theorem 2.1. Under the transformation nþk
nþk 1þn
k1
v ¼w
u ¼ z1þn w1þn ;
k1
z1þn ;
dt ¼
1 dT 1þn
ð2Þ
such that
X1 dF @F @F ¼ l ðzwÞmþ1 ; Z W¼ m¼1 m dT @z @w
ð8Þ
For all pairs ða; bÞ, if a þ b 6 p þ q, then ca;b ¼ 1, except cpq ¼ 1; if pa ¼ qb, then ca;b ¼ 0; if pa qb – 0, then ca;b is computed by the recursive formula cab ¼
aX þbþ2 1 ½ða k þ 1Þak;j1 ðb j þ 1Þbj;k1 cakþ1;bjþ1 : ð9Þ qb pa kþj¼3
For any positive integer m; lm is given by
system (1) is changed into
8 n n X X du > > > ¼ ððn þ kÞp þ ð1 kÞqÞu þ ðn þ kÞ ani;i u1þni v iþ1 þ ð1 kÞ bni;i u2þi v ni ; > < dt i¼0 i¼0
ð3Þ
n n X X > dv > 1þni iþ1 > > u ð1 kÞ ani;i v 2þi uni : : dt ¼ ððn þ kÞq þ ð1 kÞpÞv ðn þ kÞ bni;i v i¼0
i¼0
Proof. Applying transformation (2) we obtain
(
nþk k1 nþk 1þn z 1 w1þn dz 1þn dt
nþk k1 k1 1þn z w1þn1 dw ; 1þn dt
du dt
¼
dv dt
nþk k1 ¼ 1þn w1þn1 z1þn dw þ 1þn w1þn z1þn1 dz ; dt dt
nþk
þ
lm ¼
nþk
k1
n ln uþk ln uþln v k ln v 1þnþ2k
;
w¼e
ð4Þ
k1
ln uk ln uþn ln v þk ln v 1þnþ2k
ð5Þ
;
and
dz dz dT ¼ ; dt dT dt
dw dw dT ¼ ; dt dT dt
dT ¼1þn dt
Now it is easy to see that the theorem holds.
½ðqm þ q k þ 1Þak;j1 ðpm þ p j þ 1Þbj;k1
kþj¼3
On the other hand, it is clear that
z¼e
1 X
ð6Þ h
cqmþqkþ1;pmþpjþ1
ð10Þ
where 8ðk; jÞ, when k < 0 or j < 0, we take ak;j ¼ bk;j ¼ ck;j ¼ 0. In Lemma 2.1, lm is called the mth singular point quantity of the origin of system (7). Evidently, F is a first integral of system (7) if and only if all lm ¼ 0. For system (1), according to Lemma 2.1, we have the following recursive formulas to compute the singular point values at the origin.
Since the transformation (2) is a homeomorphism in a deleted neighborhood of the origin, system (1) and system (3) have the same topological structure in a neighborhood of the origin. Thus, the following theorem holds.
Theorem 2.3. For system (1), the singular point value lm (m ¼ 1; 2; . . .) at the origin is given by the recursive formulas as follows: if a < 0 or b < 0 or ððn þ kÞp þ ð1 kÞqÞa ¼ ððn þ kÞq þ ð1 kÞpÞb or a þ b < p þ q, then ca;b ¼ 0; else:
Theorem 2.2. System (1) is integrable at the origin if and only if system (3) is integrable at the origin. Theorems 2.1 and 2.2 give a method to study the integrability of a fractional order generalized system (1) with p : q resonance. That is, one can transform system (1) into a polynomial system (3) with p : q resonance and then apply the methods of investigation of integrability of system (3) to study the integrability of system (1). Xiao gave the following lemma in [16].
ca;b ¼
Lemma 2.1 [16]. For the system
1 ððn þ kÞq þ ð1 kÞpÞb ððn þ kÞp þ ð1 kÞqÞa
X ½ððn þ kÞa ð1 kÞb þ ðn þ q þ 1 kÞðn þ 1ÞÞaj;k1 k;j
ððn þ kÞb ð1 kÞa þ ðn þ p þ 1 jÞðn þ 1ÞÞbj;k1 cað1kÞjðnþkÞkþðnþ1Þð1kÞ;bð1kÞkðnþkÞjþðnþ1Þð1kÞ ; for any positive integer m,
ð11Þ
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X ½ðn þ q þ qm k þ 1Þak;j1
lm ¼ ðn þ 1Þ
Using the recursive formulas of Theorem 3.1 the singular point values of system (16) at the origin can be calculated using a computer algebra system, for instance, Mathematica.
k;j
ðn þ p þ pm j þ 1Þbj;k1 cðð1kÞpþðnþkÞqÞmð1kÞjðnþkÞkþð1kÞðnþ1Þ ;ððn þ kÞp þ ð1 kÞqÞm ðn þ kÞj ð1 kÞk þ ð1 kÞðn þ 1Þ:
ð12Þ
As an application of our method, we study the integrability for a 14 order generalized quadratic system with 1 : 2 resonance and a 15 order generalized cubic system with 1 : 3 resonance in the following sections.
Here for all lk , we have applied the conditions l1 ¼ l2 ¼ ¼ lk1 ¼ 0; k ¼ 2; 3; 4; 5.
Consider the following fractional order generalized quadratic system with 1 : 2 resonance:
ð13Þ
where p ¼ 1; q ¼ 2; k ¼ 14 ; a02 ¼ b02 ¼ 0; b20 ¼ 5a11 , that is
8 1 dz > > < ¼ z þ ðzwÞ4 ða20 z2 þ a11 zwÞ; dT > > dw ¼ 2w ðzwÞ14 ðb w2 þ b zwÞ; : 20 11 dT
ð14Þ
3
1
3 4
1 4
v ¼ w z ;
dt ¼
1 dT: 3
ð15Þ
that is transformation (2) with jm¼2;k¼1 system (14) can be 4 reduced to
8 du > > ¼ 5u þ ð3a20 þ b11 Þu3 v þ 8a11 u2 v 2 ; < dt > dv > : ¼ 7v ða20 þ 3b11 Þu2 v 2 16a11 uv 3 : dt
Theorem 3.1. For system (16) the singular point value lm (m ¼ 1; 2; ) at the origin is given by the recursive formulas as follows: if a < 0 or b < 0 or 5a ¼ 7b or a þ b < 12, then ca;b ¼ 0; else
1 ½ðða 1Þð3a11 þ b20 Þ 5a þ 7b
ðb 2Þða11 þ 3b20 ÞÞca1;b2 þ ðða 2Þð3a20 þ b11 Þ ðb 1Þða20 þ 3b11 ÞÞca2;b1 ; ð17Þ For any positive integer m,
lm ¼ ½8a11 ð7m þ 6Þ 16a11 ð5m þ 3Þc7mþ6;5mþ3 ½ð7m þ 5Þð3a20 þ b11 Þ ð5m þ 4Þða20 þ 3b11 ÞÞc7mþ5;5mþ4 :
Theorem 3.3. For system (16) the first five singular point values are equal to zero if and only if one of the following four conditions holds:
ðIÞ a11 ¼ 0;
ð19Þ
ðIIÞ 11a20 ¼ 7b11 ;
ð20Þ
ðIIIÞ a20 ¼ 5b11 ;
ð21Þ
ðIVÞ a20 ¼ b11 :
ð22Þ
Theorem 3.4. For system (16) all singular point values at the origin are zero if and only if the first five singular point values at the origin are zero, that is, one of four conditions of Theorem 3.3 holds. That is, the four conditions of Theorem 3.3 are the conditions for integrability of system (16).
ð16Þ
Thus, the investigation of the fractional order generalized system (14) with 1 : 2 resonance is reduced to the investigation of the polynomial system (16) with 5 : 7 resonance. By Theorem 2.3 we have.
ca;b ¼
From Theorem 3.2 we have.
Using the above theorem we obtain the integrability conditions of system (16).
By the transformation,
u ¼ z4 w4 ;
8 9
l1 ¼ a11 ð11a20 7b11 Þða20 5b11 Þða20 b11 Þ; l2 ¼ l3 ¼ l4 ¼ l5 ¼ 0:
3. Integrability of a 2; 14 order system with 1 : 2 resonance
dz ¼ pz þ ðzwÞk ða20 z2 þ a11 zw þ a02 w2 Þ; dT dw ¼ qw ðzwÞk ðb20 w2 þ b11 wz þ b02 z2 Þ; dT
Theorem 3.2. The first five singular point values of system (16) at the origin are given as follows:
ð18Þ
Proof. The necessity is obvious. We need only to prove the sufficiency. If condition (I) holds, then system (16) has the inte
grating factor J 1 ¼ u
3ð17a20 5b11 Þ 8ð2a20 b11 Þ
41a20 13b11 8ð2a20 b11 Þ
v
, when 2a20 ¼ b11 , 7
system (16) has the first integral Fðu; v Þ ¼ u5 v . If condition (II) holds, then system (16) has the integrating factor J 2 ¼ u8 v 6 . If condition (III) holds, then system (16) has the integrating factor J 3 ¼ u8 v 6 ð1 þ 8b11 u2 v Þ. If condition (IV) holds, then system (16) has the 2 integrating factor J 4 ¼ u8 v 6 ð1 þ 43 a20 u2 v Þ . Thus, in each case the corresponding system is integrable. h Since transformation (15) is a homeomorphism in a deleted neighborhood of the origin, the integrability conditions of system (14) are the same as the integrability conditions of system (16). Thus, we have the following theorem. Theorem 3.5. System (14) is integrable if and only if one of the four conditions in Theorem 3.3 holds.
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4. Integrability of a 3; 15 order system with 1 : 3 resonance
l2 ¼ l3 ¼ l4 ¼ l5 ¼ 0
1
Consider a 3; 5 order system with 1 : 3 resonance: 3 X dz ¼ pz þ ðzwÞk a3i;i z3i wi ; dT i¼0 3 X dw ¼ qw ðzwÞk b3i;i w3i zi ; dT i¼0
ð23Þ
where p ¼ 1; q ¼ 3; k ¼ 15 ; a03 ¼ b03 ¼ 0; a21 ¼ b21 ¼ 0; b30 ¼ 11a12 , that is, the system
8 < dz ¼ z þ ðzwÞ15 ða30 z3 þ a12 zw2 Þ; dT : dw dT
1
ð24Þ
¼ 2w ðzwÞ5 ðb30 w3 þ b12 z2 wÞ:
By the transformation 4
1
u ¼ z5 w5 ;
4 5
1 5
v ¼ w z ;
dt ¼
1 dT 4
ð25Þ
(which is transformation (2) with jm¼3;k¼1 ) system (24) can 5 be reduced to
(
du dt dv dt
¼ 7u þ ð4a30 þ b12 Þu4 v þ ð4a12 þ b30 Þu2 v 3 ; ¼ 13v ða30 þ 4b12 Þu3 v 2 ða12 þ 4b30 Þuv 4 :
ð26Þ
Thus, the investigation of the fractional order generalized system (24) with 1 : 3 resonance is reduced to the study of the integrability of the polynomial system (26) with 7 : 13 resonance. Similar as in the previous section we have
For all lk , we have applied the conditions l1 ¼ l2 ¼ ¼ lk1 ¼ 0; k ¼ 2; 3; 4; 5. From Theorem 4.2 we have the following statement. Theorem 4.3. For system (26) the first five singular point values are zeros if and only if one of the following five conditions holds:
ðIÞ b30 ¼ 0;
ð29Þ
ðIIÞ 23a30 ¼ 13b12 ;
ð30Þ
ðIIIÞ a30 ¼ 11b12 ;
ð31Þ
ðIVÞ 17a30 ¼ 7b12 ;
ð32Þ
ðVÞ a30 ¼ b12 :
ð33Þ
Using the conditions (29)–(33) we obtain the following theorem. Theorem 4.4. For system (26) all the singular point values at the origin are zero if and only if the first five singular point values at the origin are zero, i.e., one of the five conditions of Theorem 4.3 holds. That is, the five conditions of Theorem 4.3 are the integrability conditions for the system (26). Proof. The necessity is obvious. Thus we have only to prove the existence of first integrals of integrating factors. If condition (I) holds, then system (26) has the inte4ð47a
Theorem 4.1. For system (26) the singular point value lm (m ¼ 1; 2; . . .) at the origin is given by the recursive formulas: if a < 0 or b < 0 or 7a ¼ 13b or a þ b < 20, then ca;b ¼ 0; else
ca;b ¼
1 ½ðða 1Þð4a12 þ b30 Þ 7a þ 13b ðb 3Þða12 þ 4b30 ÞÞca1;b3
grating factor J 1 ¼ u
7b12 Þ 30 b12 Þ
15ð3a30
v
2ð61a30 11b12 Þ 15ð3a30 b12 Þ
, when 2a20 ¼ b11 , 13
system (26) has the first integral F½u; v ¼ Cð1Þðu 7 v Þ, where Cð1Þ is a constant. If condition (II) holds, then system (26) has the 3 integrating factor J 2 ¼ u14 v 8 ð1 þ 15 23 b12 u v Þ. If condition (III) holds, then system (26) has the 3 integrating factor J 3 ¼ u14 v 8 ð1 þ 15b12 u3 v Þ .
þ ðða 3Þð4a30 þ b12 Þ ðb 1Þða30 þ 4b12 ÞÞca3;b1 ; ð27Þ
If condition (IV) holds, then system (26) has the integrating factor J 4 ¼ u14 v 8 . If condition (V) holds, then system (26) has the 2
For any positive integer m,
integrating factor J 5 ¼ u14 v 8 ð1 þ 54 b12 u3 v Þ .
lm ¼ ½ð4a30 þ b12 Þð13m þ 10Þ
From these observations we conclude the correctness of the theorem. h
ða30 þ 4b12 Þð7m þ 6Þc13mþ10;7mþ6 ½ð13m þ 12Þð4a12 þ b30 Þ ð7m þ 4Þða12 þ 4b30 ÞÞc13mþ12;7mþ4 :
From Theorem 4.4 we obtain.
ð28Þ
Applying the recursive formulas of Theorem 4.1 the singular point values of system (26) at the origin are easily calculated.
Theorem 4.5. System (24) is integrable if and only if one of the five conditions in Theorem 4.3 holds. Acknowledgments
Theorem 4.2. The first five singular point values of system (26) at the origin are given as follows:
15 b30 ð23a30 13b12 Þða30 11b12 Þ 4096 ð17a30 7b12 Þða30 b12 Þ;
l1 ¼
We would like to give our thanks to Prof. Jiazhong Yang and Prof. Valery Romanovski for their helpful suggestions. This research are supported by Nature Science Foundation of China (No. 11261013, 11361017), Nature Science Foundation of Guangxi (2012GXNSFAA053003) and Guangxi
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