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Limit cycle bifurcations in a class of perturbed piecewise smooth systems
Applied Mathematics and Computation 242 (2014) 47–64
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Limit cycle bifurcations in a class of perturbed piecewise smooth systems q Yanqin Xiong, Maoan Han ⇑ Department of Mathematics, Shanghai Normal University, Shanghai 200234, China
a r t i c l e
i n f o
a b s t r a c t In this paper, we concern with the number of limit cycles in a piecewise polynomial system. First, we give 42 different phase portraits of the unperturbed system with at least one closed orbit. Then, we perturb one phase portrait of them by piecewise polynomials, and consider lower bounds for the maximal numbers of limit cycles emerging from the origin and generalized homoclinic loop, respectively. Ó 2014 Published by Elsevier Inc.
Keywords: Piecewise perturbed Hamiltonian system Hopf bifurcation Generalized homoclinic loop Phase portrait
1. Introduction and main results Recently, there has been increasing interest in studying dynamic behaviors of smooth and non-smooth systems due to their powerful potential applications in electrical engineering, physical sciences and so on, see [1,3,8,2,5– 7,10,13,14,11,12,9]. As we have seen, one of the phenomena is limit cycle bifurcation originated from Hilbert [4]. From Han [6] and Liu and Han [10], one can find that the first order Melnikov function plays an important role in studying the number of limit cycles of smooth and piecewise smooth near-Hamiltonian systems in local and global bifurcations. Many researchers use this method to discuss this problem for various systems and get interesting results, see [11–13,9]. In this paper, we study the following system
x_ ¼ y þ pðx; y; dÞ;
y_ ¼ gðxÞ þ qðx; y; dÞ;
ð1:1Þ
where 0 < 1,
gðxÞ ¼
ax þ b; x < 0; cx2 þ dx þ e; x P 0;
ð1:2Þ
with jaj þ jbj > 0; c – 0,
( Pn pðx; y; dÞ ¼
i j iþj¼0 aij x y ;
x < 0;
þ i j iþj¼0 aij x y ;
x P 0;
Pn
( Pn qðx; y; dÞ ¼
and d ¼ a 2 D Rðnþ1Þðnþ2Þ with D bounded. For ij ; bij below
i j iþj¼0 bij x y ; þ i j iþj¼0 bij x y ;
Pn
¼ 0,
x < 0; x P 0;
ð1:3Þ
system (1.1) has the piecewise-Hamiltonian function given
q The project was supported by National Natural Science Foundation of China (11271261), a Grant from Ministry of Education of China (20103127110001) and FP7-PEOPLE-2012-IRSES-316338. ⇑ Corresponding author. E-mail address: [email protected] (M. Han).
http://dx.doi.org/10.1016/j.amc.2014.05.035 0096-3003/Ó 2014 Published by Elsevier Inc.
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Y. Xiong, M. Han / Applied Mathematics and Computation 242 (2014) 47–64
Hðx; yÞ ¼
1 2 y þ 2
Z
x
gðsÞds ¼
H ðx; yÞ;
x < 0;
þ
H ðx; yÞ;
0
ð1:4Þ
x P 0;
where
1 2 1 2 y þ ax þ bx; 2 2
H ðx; yÞ ¼
Hþ ðx; yÞ ¼
1 2 1 3 1 2 y þ cx þ dx þ ex: 2 3 2
Figs. 1.1 and 1.2 show the possible graphs of the level curves of ðx; yÞjH ðx; yÞ ¼ h on the plane, respectively. We remark that in Figs. 1.1 and 1.2, A ¼ ðaA ; 0Þ; C ¼ ðaC ; 0Þ; S ¼ ðaS ; 0Þ; J ¼ ðaJ ; 0Þ, in which aA ¼ ba (when a – 0), pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi 2 2 aC ¼ dþ 2cd 4ec ; aS ¼ d 2cd 4ec ; aJ ¼ 2cd . Using Figs. 1.1 and 1.2, one can obtain Table 1.1, which presents the possible phase portraits of (1.1)j¼0 together with the corresponding coefficient conditions when it has a family of closed curves. From Fig. 1.3, we can obtain 42 different phase portraits when (1.1)j¼0 has at least one closed orbit. Now, we turn to study limit cycle bifurcation of system (1.1). For simplicity, we mainly investigate Fig. 1.3(27) with a ¼ 0; b < 0 or a > 0; b ¼ 0. That is, a; b; c; d and e in (1.2) satisfy 2
c > 0;
e¼
d ; 4c
a > b;
ab ¼ 0:
ð1:5Þ
Obviously, system (1.1)j¼0 satisfying (1.5) has two families of closed orbits fLh j0 < h < h0 g and Lh ¼ fhjh > h0 g, where 3
Lh ¼ Lh [ Lþh ;
h 2 ð0; h0 Þ [ ðh0 ; þ1Þ , I;
h0 ¼
d > 0; 24c2
with
x < 0g; Lþh ¼ ðx; yÞjHþ ðx; yÞ ¼ h;
Lh ¼ fðx; yÞjH ðx; yÞ ¼ h;
xP0 :
It is easy to see that limh!0þ Lh ¼ 0; limh!h0 Lh ¼ L and the limit L denotes the generalized homoclinic loop with a cusp. Then, by [10], associated to Lh , system (1.1) has the first order Melnikov function written as
MðhÞ ¼ M ðhÞ þ M þ ðhÞ;
h 2 I;
ð1:6Þ
where
M ðhÞ ¼
Z
n X
L h iþj¼0
bij xi yj dx
n X
aij xi yj dy:
ð1:7Þ
iþj¼0
About system (1.1), let Z Hopf and Z Loop denote the maximal number of limit cycles generated near the origin and near L respectively for 0 < 1. Then, the main result can be stated as follows. Theorem 1.1. Let (1.2), (1.3) and (1.5) hold. Then,
Z Hopf P n þ
nþ1 ; 2
Z Loop P 2n þ
nþ1 ; 2
where n P 1. In the following, we divide into two sections to prove our main result. In Section 2, we mainly derive the expression of M ðhÞ in (1.7). In Section 3, we study Hopf and generalized homoclinic loop bifurcations, proveing Theorem 1.1. 2. Preliminary lemmas In this section, we deduce the expansion of M ðhÞ in (1.7). The following lemma is direct from Lemma 4 of [13].
Fig. 1.1. The level curves of fðx; yÞjH ðx; yÞ ¼ hg on the plane.
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Y. Xiong, M. Han / Applied Mathematics and Computation 242 (2014) 47–64
Fig. 1.2. The level curves of ðx; yÞjHþ ðx; yÞ ¼ h on the plane.
Table 1.1 Conditions of existing periodic orbits and their correlated phase portraits. Coefficient conditions c<0
a < 0; b < 0
a P 0; b 0; a > b
No closed orbits
No closed orbits
No closed orbits
Fig. 1.3(1)
Fig. 1.3(2)–(9)
Fig. 1.3(10)–(12)
Fig. 1.3(13)–(16)
Fig. 1.3(1), (17)–(20)
2
Fig. 1.3(21)
No closed orbits
Fig. 1.3(22)
Fig. 1.3(23)
2
Fig. 1.3(21), (24)–(26)
No closed orbits
Fig. 1.3(22), (27)
Fig. 1.3(23), (28)
2
Fig. 1.3(21), (29)–(33)
Fig. 1.3(34)–(36)
Fig. 1.3(22), (37)–(39)
Fig. 1.3(23), (40)–(42)
d < 4ec d ¼ 4ec d > 4ec
Lemma 2.1. Suppose that (1.2), (1.3) and (1.5) hold. Then,
M ðhÞ ¼
8 n pffiffiffi X > > ~i;2k hiþk > h q > <
for a ¼ 0; b < 0;
n pffiffiffiiþ2k > pffiffiffi X > > > qi;2k h : h
for a > 0; b ¼ 0;
iþ2k¼0
h 2 I;
iþ2k¼0
where
~0;2k ¼ q0;2k ¼ q
qi;2k
2
kþ1þ12
pffiffiffi 2kþ1 2ð 2Þ a0;2k
2k þ 1
;
2k þ 1 bi1;2kþ1 i
Z
p 2
2k
sin h cos2iþ1 hdh; i P 1; i 0 b pffiffiffi 2kþ1þi
Z 0 2ð 2Þ i i1 ¼ b þ a sin h cos2kþ2 hdh; i P 1: pffiffiffi i i1;2kþ1 2k þ 1 i;2k p2 ð aÞ
~i;2k ¼ q
a > 0; b > 0
2
d 4ec d > 4ec
c>0
a 0; b P 0; a < b
2
ai;2k þ
About the expression of M þ ðhÞ, we have Lemma 2.2. Let the assumption of Lemma 2.1 hold. Then, for h 2 I,
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Y. Xiong, M. Han / Applied Mathematics and Computation 242 (2014) 47–64
pffiffiffi h
Mþ ðhÞ ¼
n1 X
qþiþ1;2k /þik ðhÞ þ
iþ2k¼0
n X
! k
qþ0;2k h
n1 X
þ
2k¼0
qþiþ1;2k
i X T rk ðhÞ;
where each /þ ik ðhÞ is a polynomial in h of degree k þ
iþ1 3
ð2:1Þ
r¼0
iþ2k¼0
with /ik ð0Þ ¼ 0, and
kþ1þ12
2 i þ aþ ; qþi;2k ¼ bi1;2kþ1 þ aþ ; i P 1; 2k þ 1 i;2k 2k þ 1 0;2k 8 kþm ð0Þ > d3 > ; r ¼ 3m; > aimk T 00 ðhÞ 2h þ 12c 2 > > < kþm T rk ðhÞ ¼ d3 > að1Þ T ðhÞ 2h þ 12c ; r ¼ 3m þ 1; 2 > > imk 10 > > : 0; r ¼ 3m þ 2
qþ0;2k ¼
satisfying
T i0 ðhÞ ¼
Z
ð2hþ
ð
1 d 3 Þ3 12c2
1 d 3 Þ3 12c2
3
t
d 2h þ t3 12c2
i
!12 dt;
i ¼ 0; 1
d i 3 13 ð1Þ d i1 3 23 ð0Þ ð1Þ ð0Þ and aimk > 0; aimk > 0 with ai00 ¼ 2 2c ; ai00 ¼ 2i 2c . 2c 2c Proof. Following the idea of (2.9) in [11] or Lemma 2 of [13], by Green formula, M þ ðhÞ can be rewritten as
Mþ ðhÞ ¼ NðhÞ þ
1 n X 2kþ1þ2 þ kþ12 a h ; 2k þ 1 0;2k 2k¼0
h 2 I;
ð2:2Þ
where
NðhÞ ¼
n1 X
qþiþ1;j
iþj¼0
Z
xi yjþ1 dx;
Lþ h
þ
qþiþ1;j ¼ bi;jþ1 þ
iþ1 þ a : j þ 1 iþ1;j
ð2:3Þ
þ d 3 d3 By (1.5), rewrite Hþ ðx; yÞ in (1.4) as Hþ ðx; yÞ ¼ 12 y2 þ 13 c x þ 2c 24c 2 . Thus, along the curve Lh
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
3 3 d 2 d y ¼ 2h þ c x þ ; 2c 12c2 3 which intersects the positive x-axis at a point
NðhÞ ¼
n1 X
3 h c
3
d þ 8c 3
13
d 2c ; 0 . Then, NðhÞ in (2.3) becomes
n1 h i X qþiþ1;j Iij þ ð1Þj Iij ¼ 2qþiþ1;2k Ii;2k ðhÞ;
iþj¼0
ð2:4Þ
iþ2k¼0
where
Iij ðhÞ ¼
Z
1
3
d ð3c hþ d 3 Þ3 2c 8c
0
"
3 #jþ1 2 3 d 2 d x 2h þ c x þ dx: 2c 12c2 3 i
1 3 13 d Let t ¼ 23 c 3 x þ 2c dt and Ii;2k ðhÞ in (2.4) can be transformed into . Then, dx ¼ 2c
Ii;2k ðhÞ ¼
1 d 3 Þ3 12c2
Z
ð2hþ
ð
d3 Þ 12c2
"
3 2c
1 3
13
d t 2c
#i
3
d 2h þ t3 12c2
!kþ12
3 2c
13
ir 3 i X d 3 C ri T rk ðhÞ; 2c 2c r¼0 rþ1
dt ¼
ð2:5Þ
where
T rk ðhÞ ¼
Z ð
ð2hþ
1 d 3 Þ3 12c2
1 d3 Þ3 12c2
3
t
r
d 2h þ t3 12c2
!kþ12 dt:
Note that
Z
3
tr 2h þ
d t3 12c2
!kþ12
3
dt ¼ N1 ðkÞt rþ1 2h þ
d t3 12c2
!kþ12
3
þ N2 ðkÞ 2h þ
d 12c2
!Z
3
tr 2h þ
d t3 12c2
!k12 dt;
k P 1;
Y. Xiong, M. Han / Applied Mathematics and Computation 242 (2014) 47–64
Fig. 1.3. The possible phase portraits of system (1.1)j¼0 .
in which
N1 ðkÞ ¼
1 ; 3ðk þ 12Þ þ r þ 1
N2 ðkÞ ¼
3ðk þ 12Þ : 3ðk þ 12Þ þ r þ 1
51
52
Y. Xiong, M. Han / Applied Mathematics and Computation 242 (2014) 47–64
Fig. 1.3 (continued)
Then, one obtains 3
d T rk ðhÞ ¼ 12c2
!rþ1 3 N1 ðkÞð2hÞ
kþ12
! 3 d T r;k1 ðhÞ; þ N 2 ðkÞ 2h þ 12c2
k P 1;
Y. Xiong, M. Han / Applied Mathematics and Computation 242 (2014) 47–64
53
Fig. 1.3 (continued)
which leads to 3
d T rk ðhÞ ¼ 12c2
!rþ1 3
3 pffiffiffiffiffiffi d k 2h þ k ðhÞ þ a 2hu 12c2
!k T r0 ðhÞ;
k P 1;
where
k ðhÞ ¼ N 1 ðkÞð2hÞk þ u
a k ¼
k1 Y
3 j1 k1 X Y d kj N1 ðk jÞ N2 ðk iÞð2hÞ 2h þ 12c2 j¼1 i¼0
!j ;
N2 ðk iÞ:
i¼0
Hence, 3
d T rk ðhÞ ¼ 12c2
!rþ1 3
3 pffiffiffiffiffiffi d 2huk ðhÞ þ ak 2h þ 12c2
!k T r0 ðhÞ;
k P 0;
ð2:6Þ
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Y. Xiong, M. Han / Applied Mathematics and Computation 242 (2014) 47–64
in which
uk ðhÞ ¼
0; k ¼ 0; k ðhÞ; k P 1; u
ak ¼
1; k ¼ 0; a k ; k P 1;
with uk ð0Þ ¼ 0 and ak > 0. Using the formula
Z
3
t
r
d 2h þ t3 12c2
!12
3
dt ¼ N3 ðrÞt
d 2h þ t3 12c2
r2
!32
3
d þ N4 ðrÞ 2h þ 12c2
!Z
3
t
r3
d 2h þ t3 12c2
!12 dt;
ð2:7Þ
we obtain 3
T r0 ðhÞ ¼ N5 ðrÞð2hÞ2 þ N4 ðrÞ 2h þ
! 3 d T r3;0 ðhÞ; 12c2
r P 2;
where
1 N3 ðrÞ ¼ ; r þ 52
3
r2 N 4 ðrÞ ¼ ; r þ 52
d 12c2
N 5 ðrÞ ¼
!r2 3 N3 ðrÞ:
Thus, by induction, one can prove 3
3
T r0 ðhÞ ¼ ð2hÞ2 w½rþ11 ðhÞ þ b½rþ1 2h þ 3
3
d 12c2
!½3r T r3½ r ;0 ðhÞ;
r P 0;
3
ð2:8Þ
where
w½rþ11 ðhÞ ¼ 3
b½rþ1 ¼ 3
8 0; > > > <
r ¼ 0; 1;
> > > : N5 ðrÞ þ
½rþ1 1 3
X
j1 j Y d3 N 4 ðr 3iÞ 2h þ 12c ; r P 2; 2 i¼0
j¼1
8 < 1; :Q
N5 ðr 3jÞ r ¼ 0; 1;
½rþ1 1 3 i¼0
N4 ðr 3iÞ; r P 2:
Since N 4 ðrÞ > 0; r > 2 and N 4 ð2Þ ¼ 0, we have from the above
b½rþ1 3
r¼3m;3mþ1
b½rþ1
> 0;
3
r¼3mþ2
¼ 0:
ð2:9Þ
Inserting (2.8) into (2.6) leads to 3 pffiffiffiffiffiffi d T rk ðhÞ ¼ 2hvkþ½rþ1 ðhÞ þ ak b½rþ1 2h þ 3 3 12c2
!kþ½3r T r3½ r ;0 ðhÞ;
r P 0;
3
k P 0;
ð2:10Þ
where 3
d vkþ½rþ13 ðhÞ ¼ 12c2
!rþ1 3
3
d uk ðhÞ þ 2ak h 2h þ 12c2
which is a polynomial in h of degree k þ
rþ1 3
with
!k w½rþ11 ðhÞ; 3
vkþ½rþ13 ð0Þ ¼ 0.
Combining (2.4), (2.5) and (2.10) yields that
NðhÞ ¼
n1 X
2qþiþ1;2k
iþ2k¼0
¼
n1 X iþ2k¼0
qþiþ1;2k
2 3 !kþ½3r
ir rþ1 3 i X d 3 3 4pffiffiffiffiffiffi d r Ci T r3½ r ;0 ðhÞ5 2hvkþ½rþ1 ðhÞ þ ak b½rþ1 2h þ 3 3 3 2c 2c 12c2 r¼0 X
063mþj6i; j¼0;1;2
3
2h þ
d 12c2
!kþm
n1 pffiffiffi X
aðjÞ h imk T j0 ðhÞ þ
iþ2k¼0
qþiþ1;2k /þik ðhÞ;
ð2:11Þ
Y. Xiong, M. Han / Applied Mathematics and Computation 242 (2014) 47–64
55
where, together with (2.9)
3m að0Þ imk ¼ 2C i
d 2c
i3m
3 2c
3mþ1 3
ak b½rþ13
> 0;
3 ak b½rþ13 > 0; 2c r¼3mþ1
i3m2 mþ1 d 3 3mþ2 að2Þ ¼ 2C a b ¼ 0; rþ1 k i ½ 3 imk 2c 2c r¼3mþ2
ir rþ1 i pffiffiffiX d 3 3 vkþ½rþ13 ðhÞ /þik ðhÞ ¼ 2 2 C ri 2c 2c r¼0 3mþ1 að1Þ imk ¼ 2C i
d 2c
i3m1
r¼3m
3mþ2 3
and /þ ik ðhÞ are polynomials in h of degree k þ proof. h
iþ1 3
with /þ ik ð0Þ ¼ 0. Then, substituting (2.11) into (2.2) gives (2.1). This ends the
3. Proof of the main result In this section, we split into two subsections to discuss our problem. 3.1. Hopf bifurcation In this subsection, we study the limit cycle bifurcations of (1.1) in the case (27) of Fig. 1.3 near the origin. For the purpose, we introduce
Ak ¼ ð1Þk
A0 ¼ 1;
k1 Y j1 þ ji i¼0
j2 þ ji
k P 1;
;
ð3:1Þ
where j1 ; j2 ; j are positive constants with ðm n þ 1Þ ðm n þ 1Þ matrix
0
Am B A B mþ1 B B Am;n ðj1 ; j2 ; jÞ ¼ B Amþ2 B . B . @ . A2mn
j1 – j2 . Then, for any m P n P 0, we use Ak to construct an
Am1
Am2
Am
Am1
Amþ1 .. .
Am .. .
A2mn1
A2mn2
An
1
C
Anþ1 C C
Anþ2 C C: .. .. C C . . A
Am
ð3:2Þ
Then, denote
Bp ¼
Bp1
Bp2
Bp3
0
;
p ¼ 1; . . . ; m n;
where Bp1 is a p ðm n þ 1 pÞ matrix,
0
Bp2
B B B B ¼B B B B @
Cn Dn C nþ1 Dnþ2 C nþ2 Dnþ4
q C nþp1 Dnþ2ðp1Þ
0
1 C C C C C; C C C A
0
Bp3
Cm Dmþp
B B C mþ1 BD B mþpþ1 ¼B B .. B . @
C 2mnp D2mn
C m1 Dmþp1
C nþp Dnþ2p
Cm Dmþp
C nþpþ1 Dnþ2pþ1
.. . C 2mnp1 D2mn1
..
.
.. . Cm Dmþp
1 C C C C C C C A
and
C 0 ¼ D0 ¼ 1;
Ck ¼
k1 Y
ðj1 þ jiÞ;
i¼0
Dk ¼
k1 Y
ðj2 þ jiÞ;
k P 1:
i¼0
Then, for the relationship between Am;n ðj1 ; j2 ; jÞ and Bp , we have Lemma 3.1. For each Bp , we have
Bp Am;n ðj1 ; j2 ; jÞ: In other words, for each p, there must exist a non-singular ðm n þ 1Þ ðm n þ 1Þ matrix T p such that
Am;n ðj1 ; j2 ; jÞ ¼ T 1 p Bp T p :
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Y. Xiong, M. Han / Applied Mathematics and Computation 242 (2014) 47–64
Proof. For Am;n ðj1 ; j2 ; jÞ, by multiplying the ith column by ð1Þi1 ; i ¼ 2; . . . ; m n þ 1, and dividing the ith row by ð1Þmþi1 ; i ¼ 1; . . . ; m n þ 1, one has
0
Cm Dm
BC B mþ1 B Dmþ1 Am;n ðj1 ; j2 ; jÞ B0 ¼ B B . B .. @
C 2mn D2mn
C m1 Dm1
Cn Dn
Cm Dm
C nþ1 Dnþ1
.. .
..
C 2mn1 D2mn1
C C C C: .. C . C A
.
1
Cm Dm
Thus, it suffices to prove
p ¼ 0; . . . ; m n 1:
Bp Bpþ1 ;
ð3:3Þ
In fact, for p ¼ 0; . . . ; m n 1, make elementary transformations to Bp as follows: ðnþi1Þ (i) Multiply the ith row by jj2 þ1 þjjðnþiþp1Þ , then add the result to the ði þ 1Þth row, i ¼ m n; . . . ; p þ 1.
(ii) Multiply the ith row by j2 þ jðn þ i þ p 2Þ; i ¼ p þ 2; . . . ; m n þ 1. (iii) Divide the ith column by jðm n þ 1 p iÞðjp þ j2 j1 Þ; i ¼ 1; . . . ; m n p. Then, Bp becomes Bpþ1 . Thus, (3.3) holds. The proof is finished. h Therefore, we obtain Lemma 3.2. For matrix Am;n ðj1 ; j2 ; jÞ in (3.2), we have
detðAm;n ðj1 ; j2 ; jÞÞ – 0: Proof. By setting p ¼ m n in Lemma 3.1, we have
Am;n ðj1 ; j2 ; jÞ ¼ T 1 mn B mn T mn : Note that
detðBmn Þ ¼
m n Y i¼0
C nþi – 0: Dnþ2i
Thus, we obtain the conclusion. This finishes the proof.
h
The following lemma presents a property of T 00 ðhÞ near the origin. Lemma 3.3. The function T 00 ðhÞ in Lemma 2.2 has the expansion for 0 < h 1
T 00 ðhÞ ¼
pffiffiffi
2
k 4 2 12c2 3 pffiffiffiX ~ 24c2 k h h; A h k 3 3 9 d d kP0
ð3:4Þ
where
~ 0 ¼ 1; A
~ k ¼ ð1Þk A
k1 Y 2 þ 3i ; 5 þ 52 þ 3i i¼0
Proof. Perform the transformation
pffiffiffiffiffiffi Z T 00 ðhÞ ¼ 2h 2h
1
0
k P 1:
3
3
t d v 3 ¼ 2h 24hc
2
ð3:5Þ
to T 00 ðhÞ to obtain
2 1 pffiffiffi 12c2 3 pffiffiffi Z 1 v 2 ð1 v 3 Þ2 d v ¼ 2 h 2 h 23 2 dv : 3 2 3 3 3 0 d d3 2hv þ 12c 1 þ 24hcd3 v 2 1 2
v 2 ð1 v 3 Þ
ð3:6Þ
Notice that for 0 < h 1
1þ
24hc
2
3
d
v3
!23 ¼1þ
X ð1Þk Qk1 ð2 þ 3iÞ 24c2 h k i¼0
kP1
Inserting (3.7) into (3.6) yields that
3k k!
3
d
v 3k :
ð3:7Þ
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Y. Xiong, M. Han / Applied Mathematics and Computation 242 (2014) 47–64
" # 2 X ð1Þk Qk1 ð2 þ 3iÞ 24c2 h k pffiffiffi 12c2 3 pffiffiffi i¼0 T 00 ðhÞ ¼ 2 2 h h J0 þ Jk ; 3 3 3k k! d d kP1
ð3:8Þ
where
Jk ¼
Z
1
1 2
v 3kþ2 ð1 v 3 Þ dv ;
k P 0:
0
Using (2.7), we get
J0 ¼
2 ; 9
Jk ¼
3k J ; 3k þ 2 þ 52 k1
k P 1;
ð3:9Þ
which implies that
Jk ¼
2 3k k! ; Qk 9 i¼1 ð3i þ 2 þ 5Þ
k P 1:
ð3:10Þ
2
Combining (3.8), (3.9) and (3.10) gives (3.4). The proof is completed. h Applying Lemmas 2.1, 2.2, 3.2 and 3.3, we can obtain the theorem below. Theorem 3.1. Suppose (1.2), (1.3) and (1.5) hold. Then, for any n P 1 and 0 < 1, system (1.1) can have at least n þ limit cycles near the origin.
nþ1 2
Proof. For convenience, take þ
aij ¼ 0; j P 1 and aþij ¼ bij ¼ 0:
bij ¼ 0; i P 1;
Then, in view of (1.6), Lemmas 2.1 and 2.2, one obtains
! !k 3 n1 n1 X pffiffiffi X d þ þ ð0Þ þ MðhÞ ¼ h b0;2kþ1 /0k ðhÞ þ GðhÞ þ b0;2kþ1 a00k 2h þ T 00 ðhÞ; 12c2 2k¼0 2k¼0
ð3:11Þ
where
GðhÞ ¼
8 Pn ~i0 hi ; < i¼0 q : Pn
i¼0 qi0 ð
a ¼ 0; b < 0;
pffiffiffi i hÞ ; a > 0; b ¼ 0;
with
~i0 ¼ q
q00
pffiffiffi 2 2 b
i
ai0
Z
2
cos2iþ1 hdh ¼
0
pffiffiffi ¼ 2 2a00 ;
gi ¼
p
qi0
pffiffiffi 2 2ð2iÞ!! i
ð2i þ 1Þ!!b
ai0 ;
pffiffiffi iþ1 Z 0 2ð 2Þ i1 ¼ pffiffiffi i iai0 sin h cos2 hdh ¼ gi ai0 ; p2 ð aÞ
i P 1;
ð3:12Þ
pffiffiffi iþ1 pffiffiffi iþ1 2ð 2Þ i!! p 2ð 2Þ i!! ; i P 0 even; g ¼ pffiffiffi i ði þ 1Þ!! pffiffiffi i ði þ 1Þ!! ; i P 1 odd: i 2 ð aÞ ð aÞ
Then, for h > 0 small, by (3.4) and (3.11), one can find that
2 pffiffiffi 3 !ir 2 ½n1
½n1 3 i 2 2 2 j X X X pffiffiffi 4 2 12c2 3 X d 24c þ rþ1 j þ ð0Þ ~j MðhÞ ¼ h4 b0;2iþ1 a00i C ri 2r h h þ GðhÞ þ b0;2iþ1 /þ0i ðhÞ5; A 3 3 12c2 9 d d r¼0 jP0 i¼0 i¼0 where
GðhÞ ¼
8 n X 2pffiffi2ð2iÞ!! > > i < a ¼ 0; b < 0; i ai0 h ; i¼0
> > : Pn
i¼0
Rewrite (3.13) as
ð2iþ1Þ!!b
pffiffiffi
i
gi ai0 ð hÞ ;
a > 0; b ¼ 0:
ð3:13Þ
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Y. Xiong, M. Han / Applied Mathematics and Computation 242 (2014) 47–64
8 pffiffiffiX i h v i h ; a ¼ 0; b < 0; > > > > iP0 > < 0 1 MðhÞ ¼ pffiffiffi X ½n2 ½n1 2 X X pffiffiffi > B i i iC > > v i h h þ v~ i h A; h@ v~ i h þ > > : i¼0 i¼0 iP½2nþ1
ð3:14Þ a > 0;
b ¼ 0;
where
vi ¼ v nþi ¼
pffiffiffi 2 2ð2iÞ!! i
ð2i þ 1Þ!!b
i ¼ 0; . . . ; n;
2
pffiffiffi
2
n1 X ½n1
j 2 2 ij X 4 2 12c2 3 24c2 þ ð0Þ j 24c ~ nþir1 ; 2 b a C rj A 0;2jþ1 00j 3 3 3 9 d d d r¼0 j¼0
v~ i ¼ g2i a2i;0 þ O
þ þ b01 ; b03 ; . . . ; b0;2½n1þ1 ;
v i ¼ g2iþ1 a2iþ1;0 ; v~ ½n2þi ¼
þ þ ai0 þ O b01 ; b03 ; . . . ; b0;2½n1þ1 ;
2
i ¼ 0; . . . ;
i ¼ 0; . . . ;
hn i 2
n1 ; 2
iP1
;
pffiffiffi
2
½n1 ½n1 ij
j 2 X 4 2 12c2 3 24c2 2 X 24c2 þ ð0Þ ~ ½nþir1 ; 2j b0;2jþ1 a00j C rj A 3 3 3 2 9 d d d r¼0 j¼0
i P 1:
Now, we only consider the case a ¼ 0; b < 0 since the other case can be similarly verified. From the above formulas, we get
det
@ðv 0 ; v 1 ; . . . ; v nþ½nþ1 Þ 2
þ þ @ða00 ; . . . ; an0 ; b01 ; . . . ; b0;2½n1þ1 Þ
¼ det
C11
C12
0
C22
2
where C12 is a ðn þ 1Þ
C11
nþ1 2
, detðCÞ;
ð3:15Þ
matrix,
pffiffiffi 2!! 4!! ð2nÞ!! ¼ 2 2diag 1; ; ;...; n 2 3!!b 5!!b ð2n þ 1Þ!!b
and
pffiffiffi
2
n1
4 2 12c2 3 24c2 ð0Þ ð0Þ ½n1 ð0Þ 2 a a . ; 2 a . ; . . . ; 2 . nþ1 ½ 2 ; 000 1 001 2 3 3 00;½n1 9 2 d d 0 1i 1 Pi r~ 24c2 A C nþ1r1 i r¼0 C B d3 B 2i C C B Pi r~ C B 24c2 3 C B r¼0 C i Anþ2r1 d C; i ¼ 0; . . . ; n 1 : ¼B C B 2 C B .. C B . C B @P ½nþ1 i A 2 i r~ 24c2 3 r¼0 C i Anþ½nþ1 r1 d 2
C22 ¼
.iþ1
It follows that
" pffiffiffi #½nþ1 2 ½n1 n 2 pffiffiffi nþ1 4 2 12c2 3 24c2 n1 2 Y ð2iÞ!! Y ð0Þ detðCÞ ¼ detðC11 Þ detðC22 Þ ¼ ð2 2Þ 2i a00i detðC1 Þ; 3 3 i 9 d d ð2i þ 1Þ!!b i¼0 i¼0 where
C1 ¼
.1 ; .2 ; . . . ; .½nþ1 : 2
ð3:16Þ
For C1 , make elementary transformations as follows: i1 2 (1) Multiply the ith column by 24c ; i ¼ 2; . . . ; nþ1 . 3 2 i d
2 (2) Divide the ith row by 24c ; i ¼ 1; . . . ; nþ1 . 2 d3 (3) Add the jth column multiplying by C ji1 to the ith column, j ¼ 1; . . . ;
n1 ; i ¼ j þ 1; . . . ; nþ1 . 2 2
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Y. Xiong, M. Han / Applied Mathematics and Computation 242 (2014) 47–64
Then, C1 becomes An;n½n1 2; 5 þ 52 ; 3 , i. e., C1 An;n½n1 2; 5 þ 52 ; 3 . By Lemma 3.2, we obtain det An;n½n1 2 2 2 2; 5 þ 52 ; 3 – 0. Thus, one has detðCÞ – 0. Then, by (3.15), we conclude that v 0 ; v 1 ; . . . ; v nþ½nþ1 are independent with 2 each other such that MðhÞ in (3.14) can have n þ nþ1 v satisfying zeros for 0 < h 1 by varying j 2 nþ1 nþ1 0 < v 0 v 1 ð1Þnþ½ 2 1 v nþ½nþ11 ð1Þnþ½ 2 v nþ½nþ1 1: 2
2
This ends the proof. h 3.2. Limit cycles near the generalized homoclinic loop L In this subsection, we investigate the bifurcation of limit cycles near the generalized homoclinic loop L by the first order Melnikov function. For the purpose, we need to know the expressions of M ðhÞ near L. The following lemma presents the the expansion of T i0 ðhÞ in Lemma 2.2 near L. 3
d Lemma 3.4. Let u ¼ 2h þ 12c 2 . Then for 0 < juj 1 and i ¼ 0; 1, we have
T i0 ðhÞ ¼ ð1Þi
X 5þ2i 5þ2i Bki ðuÞk jujkþ 6 akþ 6 þ Di ;
ð3:17Þ
kP0
3 13 d where a ¼ 12c and 2
B0i ¼
Di ¼
1 ; i þ 52 ( 0;
B1i ¼
1 ; 2i 1
Bki ¼
ð2k 3Þ!! ; ð2kÞ!!ð3k þ i þ 52Þ
k P 2;
0 < u 1; 5þ2i
xi u 6 ; 0 < u 1;
with xi a constant. 3
d Proof. Note that u ¼ 2h þ 12c 2 . Then, T i0 ðhÞ in Lemma 2.2 is reduced to
T i0 ðhÞ ¼
Z
1 u3 1 3 ð d 2 Þ3 12c
1 t i u t 3 2 dt ¼ T i1 ðuÞ þ T i2 ðuÞ;
ð3:18Þ
where
T i1 ðuÞ ¼ Since
d3 12c2
u t3
Z
13
12
1
juj3
1 3 ð d 2 Þ3 12c
i
t ut
3
12
T i2 ðuÞ ¼
dt;
< 0, then for t 2
d3 12c2
13
Z
1
u3 1 juj3
1
1 ti u t3 2 dt:
ð3:19Þ
; juj3 , I1 , we have t3 > 0. Thus, for 0 < juj 1 and t 2 I1 ,
X ð1Þk1 ð2k 3Þ!! 1 1
1=2 1 1 kþ1 ¼ ðt 3 Þ2 1 þ ðt3 uÞ ¼ ðt3 Þ2 þ ðt 3 Þ 2 u þ ðt 3 Þ 2 uk : 2 ð2kÞ!! kP2
ð3:20Þ
Note that
Z
1
juj3 1 3
3 ð d 2Þ 12c
kþ12
t i ðt 3 Þ
3 13 d where a ¼ 12c . 2
dt ¼
ð1Þiþ1 kþ5þ2i 5þ2i 6 akþ 6 juj ; 5 3k þ i þ 2
k P 0;
ð3:21Þ
Obviously, (3.20) is a power series in t3 u and its domain of convergence is ½1; 1. Thus, (3.20) is uniformly convergent i 1 for t 2 1; juj3 , I2 . Note that I1 I2 . Thus, on account of (3.19), (3.20) and (3.21), together with the property of term by term integration of power series, we derive
ð1Þi T i1 ðuÞ ¼
5þ2i 1 5þ2i 1 5þ2i 5þ2i juj 6 a 6 þ ðuÞ juj1þ 6 a1þ 6 5 2i 1 iþ2 X 5þ2i ð2k 3Þ!! 5þ2i ðuÞk jujkþ 6 akþ 6 : þ 5 ð2kÞ!!ð3k þ i þ 2Þ kP2
For 0 < u 1, it is easy to see that
T i0 ðhÞ ¼ T i1 ðuÞ:
ð3:22Þ
60
Y. Xiong, M. Han / Applied Mathematics and Computation 242 (2014) 47–64
Thus, in this case, the conclusion follows from (3.22) and the above. For 0 < u 1, we have 1 T i2 ðuÞ ¼ u2
1 t3 2 i t 1 dt 1 u u3
Z
Z 1 t3 1 i 5 1 i2 5þ2i ¼ u3þ6 Let x ¼ x 3 ð1 xÞ2 dx ¼ xi u 6 ; 3 u 1
1
u3
ð3:23Þ
where xi is a constant satisfying
xi ¼
1 3
Z
1
1
i2
x 3 ð1 xÞ2 dx:
1
By (3.18), (3.19), (3.22) and (3.23), we get that (3.17) for 0 < u 1. This ends the proof. h For convenience, denote
ri ¼ ð1Þi
X Bki ;
di ¼ ð1Þi
kP0
X ð1Þk Bki þ xi ;
i ¼ 0; 1:
ð3:24Þ
kP0
Then, we have Lemma 3.5. It holds that
ri < 0; di > 0; i ¼ 0; 1:
ð3:25Þ
Proof. Denote by U a disk of diameter Then, for
0 > 0 with its center at J. See Fig. 3.1.
0 > 0 small, rewrite Mþ ðhÞ in (1.7)
Mþ ðhÞ ¼ I1 ðhÞ þ I2 ðhÞ; where
Ik ðhÞ ¼
Z
n X
þ
bij xi yj dx
Lþ kh iþj¼0
Lþ1h ¼ Lþh \ U;
n X iþj¼0
aþij xi yj dy;
k ¼ 1; 2;
þ
Lþ2h ¼ Cl: Lþh L1h :
Note that I2 ðhÞ is a C 1 function in u for juj > 0 small. And for I1 ðhÞ, we claim that
8 > 0 b0 juj56 þ B 1 b1 juj76 þ o juj76 ; 0 < u 1; b1 juj76 þ o juj76 ; 0 < u 1; b0 juj56 þ B :B 0 1
ð3:26Þ
0 > 0; B 1 > 0; B < 0; B < 0, and where B 0 1
i n1
pffiffiffi c 13 X
d þ b0 ¼ 2 2 bi1 þ ði þ 1Þaiþ1;0 ; 3 2c i¼0 pffiffiffi i1 h 5 n1
i 2 2 c 3 X d þ c i bi1 þ ði þ 1Þaþiþ1;0 : b1 ¼ 3 2c 3 i¼1
ð3:27Þ
In fact, make a variable transformation
d ~x ¼ x þ ; 2c
~ ¼ y; y
t ¼ s:
ð3:28Þ
Then, the right system of (1.1) becomes
i d ~j ; aþij ð1Þi ~x þ y 2c iþj¼0
i n X d þ ~j : ~_ ¼ c~x2 þ bij ð1Þiþ1 ~x þ y y 2c iþj¼0
~x_ ¼ y ~þ
n X
ð3:29Þ
From Theorem 3.1 of [7], Remark 3.14 in [5], (3.28) and (3.29), one can obtain (3.26), (3.27). Thus, the claim holds. Hence, by above discussion we obtain
8 > 0 b0 juj56 þ B 1 b1 juj76 þ o juj76 þ / ðuÞ; 0 < u 1; b1 juj6 þ o juj6 þ / ðuÞ; 0 < u 1; b0 juj6 þ B :B 2 0 1 i ; B ; bi ; i ¼ 0; 1 are given in (3.26), (3.27) and / ðuÞ; / ðuÞ are C 1 functions. where B 1 2 i
ð3:30Þ
61
Y. Xiong, M. Han / Applied Mathematics and Computation 242 (2014) 47–64
On the other hand, by (1.6), (3.17) and Lemma 2.2, one can find that
0 M þ ðhÞ ¼ b
X
5 1 Bk0 juj6 b
kP0
0 M þ ðhÞ ¼ b
X
7 7 Bk1 juj6 þ o juj6 þ /3 ðuÞ;
for 0 < u 1;
kP0
! ! X X 5 1 ð1Þk B þ x1 juj76 þ o juj76 þ / ðuÞ; ð1Þk Bk0 þ x0 juj6 þ b k1 4 kP0
ð3:31Þ for 0 < u 1;
kP0
where /3 ðuÞ and /4 ðuÞ are C 1 functions and
13 X i n1
i d h þ 0 ¼ 2 3 b bi1 þ ði þ 1Þaþiþ1;0 ; 2c i¼0 2c
23 X i1 h n1
i d þ 1 ¼ 2 3 i bi1 þ ði þ 1Þaþiþ1;0 : b 2c i¼1 2c Comparing the coefficients in (3.30) and (3.31), together with (3.27) and (3.24), we find
5þ2i 5þ2i Bb Bb ri ¼ i i ¼ 2 6 Bi < 0; di ¼ i i ¼ 2 6 Bi > 0; bi bi
which implies (3.25). This finishes the proof. h Summarizing the conclusions of Lemmas 3.4 and 3.5, we obtain Lemma 3.6. The expansions of T i0 ðhÞ in Lemma 2.2 have the form
T i0 ðhÞ ¼
8 X 5þ2i 5þ2i iþ1 > 6 þ ð1Þ r Bki ðuÞk akþ 6 ; 0 < u 1; > i juj > > < kP0 X 5þ2i > 5þ2i iþ1 > 6 > Bki ðuÞk akþ 6 ; > : di juj þ ð1Þ
ð3:32Þ
0 < u 1;
kP0
where ri < 0; di > 0; i ¼ 0; 1. Therefore, one has Theorem 3.2. Suppose (1.2), (1.3) and (1.5) hold. Then for 0 < 1 and n P 1, system (1.1) can have 2n þ near the generalized homoclinic loop L.
nþ1 2
limit cycles
Proof. We only give the proof of the case a > 0; b ¼ 0 with n ¼ 2l þ 1, since the other cases can be proved in a similar way. For convenience, take þ
bij ¼ 0;
i P 2;
aij ¼ 0;
j P 1;
aþij ¼ bij ¼ 0:
Fig. 3.1. A disk U.
62
Y. Xiong, M. Han / Applied Mathematics and Computation 242 (2014) 47–64
Then, in view of (1.6) and Lemmas 2.1, 2.2, one can derive that 3 l X d þ ð0Þ MðhÞ ¼ b0;2iþ1 a00i 2h þ 12c2 i¼0
!i
3 l1 X d þ ð1Þ T 00 ðhÞ þ b1;2iþ1 a10i 2h þ 12c2 i¼0
!i
l l1 X pffiffiffi X þ þ T 10 ðhÞ þ h b0;2iþ1 /þ0i ðhÞ þ b1;2iþ1 /þ1i ðhÞ i¼0
!
i¼0
2lþ1 X pffiffiffi iþ1 qi0 ð hÞ : þ i¼0
From (3.12), we further have 3
l l1 l X X MðhÞ X T 00 ðhÞ T 10 ðhÞ u d þ þ þ ð0Þ ð1Þ pffiffiffi ¼ b0;2iþ1 a00i ui qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ b1;2iþ1 a10i ui qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ b0;2iþ1 /þ0i 2 3 3 2 24c u d u d h i¼0 i¼0 i¼0 24c2
2
3 l1 X u d þ b1;2iþ1 /þ1i þ 2 24c2 i¼0
2
! þ
l X
g2i a2i;0
i¼0
!
24c2
3
u d 2 24c2
!i þ
l X
g2iþ1 a2iþ1;0
i¼0
3
u d 2 24c2
!i sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 u d : 2 24c2
ð3:33Þ
Note that for 0 < juj 1
2 !12 !12 !k12 3 3 3 3 X pffiffiffi pffiffiffi 1 d d ð2k 1Þ!! d k qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 2 u ¼ 24 þ ð1Þ u k 5; ð2kÞ!! 12c2 12c2 12c2 u d3 kP1 24c2 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 !12 !12 !12 !kþ12 3 3 3 3 3 3 X ð1Þk ð2k 3Þ!! u d 1 d 1 4 d 1 d d ¼ pffiffiffi þ uþ u k 5: ¼ pffiffiffi u 2 24c2 2 ð2kÞ!! 12c2 12c2 12c2 12c2 2 2 kP2 Thus, in the light of Lemma 3.6 and the above, (3.33) can be rewritten as for 0 < u 1
X 5 1 5 MðhÞ pffiffiffi ¼ c0 þ ~c0 juj6 þ ci þ ci juj6 þ ~ci juj6 juji ; h iP1
ð3:34Þ
where
ci ¼
l X g2j j¼i
2
j
3
C ij
d 12c2
!ji
þ þ þ þ þ þ a2j;0 þ O b01 ; b03 ; . . . ; b0;2lþ1 ; b11 ; b13 ; . . . ; b1;2l1 ; a10 ; a30 ; . . . ; a2lþ1 ;
!12 3 d þ þ þ þ þ þ þ O b01 ; b03 ; . . . ; b0;2lþ1 ; b11 ; b13 ; . . . ; b1;2l1 ; a10 ; a30 ; . . . ; a2l1 ; p ffiffiffi lþ1 2 12c 2 2 !12 3 d ð0Þ þ ~c0 ¼ d0 a000 b01 ; 24c2 !12 3 d þ þ þ ð0Þ þ ~ci ¼ d0 a00i b0;2iþ1 þ O b01 ; b03 ; . . . ; b0;2i1 ; i ¼ 1; 2; . . . ; l; 2 24c !12 3 d ð1Þ þ c1 ¼ d1 a100 b11 ; 24c2 !12 3 d þ þ þ þ ð1Þ ci ¼ d1 a10;i1 b1;2i1 þ O b11 ; b13 ; . . . ; b1;2i3 ; i ¼ 2; 3; . . . ; l: 24c2 clþ1 ¼
g2lþ1 a2lþ1;0
Similar to (3.34), one has for 0 < u 1 5
MðhÞ ¼ v 0 þ v~ 0 juj6 þ
X
v i v i juj
1 6
X 5 ~ juj56 þ juj16 þ d ~ juj56 juji ; þ v~ i juj6 ui ¼ d0 þ d di þ d 0 i i
iP1
iP1
where
v~ i ¼
r0 ~
i ¼ 0; 1; . . . ; l;
v i ¼
r1
i ¼ 0; 1; . . . ; l 1;
vi
i ¼ 0; 1; . . . ; l;
d0
d1 ¼ ci ;
ci ; ci ;
i P 0; ¼ ð1Þi1 v ; di ¼ ð1Þ v i ; d i i i
~ ¼ ð1Þi v~ : d i i
ð3:35Þ
Y. Xiong, M. Han / Applied Mathematics and Computation 242 (2014) 47–64
63
Let
þ þ þ þ þ þ þ d ¼ aþ00 ; b01 ; aþ02 ; b11 ; b03 ; aþ04 ; b13 ; b05 ; . . . ; aþ0;2l ; b1;2l1 ; b0;2lþ1 ; a2lþ1;0 : Then, by (3.35), one can obtain that ðlþ1Þðlþ2Þ @ðc0 ; ~c0 ; c1 ; c1 ; ~c1 ; . . . ; cl ; cl ; ~cl ; clþ1 Þ @ð~c0 ; ~c1 ; . . . ; ~cl ; c1 ; c2 ; . . . ; cl ; c0 ; c1 ; . . . ; cl ; clþ1 Þ ¼ ð1Þ 2 þl det @d @d
D11 0 , detðDÞ; ¼ det D21 D22
det
where D21 is a ðl þ 2Þ ð2l þ 1Þ matrix, and
0
D11 ¼
B B B 1 ! 2 B B 3 B d B B 2 24c B B B B @ 0
D22
B B B B B ¼B B B B @
ð0Þ
d0 a000 ..
. ð0Þ
d0 a00l ð1Þ
d1 a100 ..
. ð1Þ
d1 a10;l1
1
g0 g2 2
..
1
0
. g2l 2l
g2lþ1
0
2lþ2
d3 24c 2
C C C C C C C; C C C C C A
C C C C C C: C C C 12 A
Thus,
detðDÞ ¼
2 Y
3
detðDii Þ ¼
i¼1
d 24c2
!l1
g2lþ1 2
lþ2
ðr0 Þlþ1 ðr1 Þl
l l1 Y g að0Þ Y i
i¼0
00i
2i
að1Þ 10i – 0;
i¼0
which means that c0 ; ~c0 ; c1 ; c1 ; ~c1 ; . . . ; cl ; cl ; ~cl ; clþ1 can be taken as free parameters. Summarizing the above, we obtain
8 X 5 1 5 > 6 ci þ ci juj6 þ ~ci juj6 juji ; 0 < u 1; > > c0 þ ~c0 juj þ < MðhÞ iP1 pffiffiffi ¼ X 5 1 5 > 6 h > ~ ci þ ci j rd11 jjuj6 ~ci j rd00 jjuj6 ð1Þi juji ; 0 < u 1; > : c0 þ c0 juj þ
ð3:36Þ
iP1
where c0 ; ~c0 ; c1 ; c1 ; ~c1 ; . . . ; cl ; cl ; ~cl ; clþ1 are independent with each other. Thus, we choose them satisfying
0 < c0 ~c0 ð1Þc1 ð1Þ2 c1 ð1Þ~c1 ð1Þl cl ð1Þlþ1 cl ð1Þl ~cl ð1Þlþ1 clþ1 1; pffiffi in (3.36) has already been changed l þ 1 þ l þ 1 þ l ¼ n þ l þ 1 and 2l þ 1 ¼ n times for 0 < u 1 such that the sign of MðhÞ h and 0 < u 1 respectively. Thus, MðhÞ can have 2n þ nþ1 zeros for 0 < juj 1 : n þ nþ1 zeros are positive and n zeros 2 2 are negative. Similarly, it is easy to prove that MðhÞ can have 2n þ nþ1 zeros for 0 < juj 1; n þ nþ1 of which are negative and n of 2 2 which are positive. This ends the proof. h
Acknowledgment The authors would like to thank the reviewers for their valuable comments and suggestions, which improve the presentation of the paper a lot. References [1] M. di Bernardo, C.J. Budd, A.R. Champneys, P. Kowalczyk, Piecewise Smooth Dynamical Systems, Theory and Applications, Springer-Verlag, London, 2008. [2] M.S. Branicky, Stability of switched and hybrid systems, in: Proceedings of the 33rd IEEE Conference on Decision Control, Lake Buena Vista, FL, 14–16, 1994, pp. 3498–3503. [3] A.F. Filipov, Differential Equations with Discontionuous Righthand Sides, Kluwer Academic, Netherlands, 1988.
64 [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
Y. Xiong, M. Han / Applied Mathematics and Computation 242 (2014) 47–64 D. Hilbert, Mathematical problems, Reprinted from Bull. Am. Math. Soc. 8 (1902) 437–479, in Bull. Am. Math. Soc. 37 (2000) 407–436. M. Han, Bifurcation Theory of Limit Cycles, Science press, Beijing, 2013. M. Han, Asymptotic expansions of Melnikov functions and limit cycle bifurcations, Int. J. Bifurcation Chaos 22 (2012) 1250296. 30 pp.. M. Han, H. Zang, J. Yang, Limit cycle bifurcations by perturbing a cuspidal loop in a Hmailtonian system, J. Differ. Equ. 246 (2009) 129–163. M. Kunze, Non-smooth Dynamical Systems, Springer-Verlag, Berlin, 2000. Y. Liu, Y. Xiong, Bifurcation of limit cycles by perturbing a piecewise linear Hamiltonian system with one or two saddles, Chaos, Solitons & Fractals, accepted for publication. X. Liu, M. Han, Bifurcation of limit cycles by perturbing piecewise Haniltionian systems, Int. J. Bifurcation Chaos Appl. Sci. Eng. 5 (2010) 1–12. F. Liang, M. Han, Limit cycles near generalized homoclinic and double homoclinic loops in piecewise smooth systems, Chaos Soliton Fractals 45 (2012) 454–464. F. Liang, M. Han, V.G. Romanovski, Bifurcation of limit cycles by perturbing a piecewise linear Hamiltonian system with a homoclinic loop, Nonlinear Anal. 75 (2012) 4355–4374. Y. Xiong, M. Han, Bifurcation of limit cycles by perturbing a piecewise linear Hamiltonian system, Abstr. Appl. Anal. 2013 (2013) (Article ID 575390), 19 pp.. Y. Xiong, M. Han, Hopf bifurcation of limit cycles in discontinuous Liénard systems, Abstr. Appl. Anal. http://dx.doi.org/10.1155/2012/690453.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Limit cycle bifurcations in a class of perturbed piecewise smooth systems q Yanqin Xiong, Maoan Han ⇑ Department of Mathematics, Shanghai Normal University, Shanghai 200234, China
a r t i c l e
i n f o
a b s t r a c t In this paper, we concern with the number of limit cycles in a piecewise polynomial system. First, we give 42 different phase portraits of the unperturbed system with at least one closed orbit. Then, we perturb one phase portrait of them by piecewise polynomials, and consider lower bounds for the maximal numbers of limit cycles emerging from the origin and generalized homoclinic loop, respectively. Ó 2014 Published by Elsevier Inc.
Keywords: Piecewise perturbed Hamiltonian system Hopf bifurcation Generalized homoclinic loop Phase portrait
1. Introduction and main results Recently, there has been increasing interest in studying dynamic behaviors of smooth and non-smooth systems due to their powerful potential applications in electrical engineering, physical sciences and so on, see [1,3,8,2,5– 7,10,13,14,11,12,9]. As we have seen, one of the phenomena is limit cycle bifurcation originated from Hilbert [4]. From Han [6] and Liu and Han [10], one can find that the first order Melnikov function plays an important role in studying the number of limit cycles of smooth and piecewise smooth near-Hamiltonian systems in local and global bifurcations. Many researchers use this method to discuss this problem for various systems and get interesting results, see [11–13,9]. In this paper, we study the following system
x_ ¼ y þ pðx; y; dÞ;
y_ ¼ gðxÞ þ qðx; y; dÞ;
ð1:1Þ
where 0 < 1,
gðxÞ ¼
ax þ b; x < 0; cx2 þ dx þ e; x P 0;
ð1:2Þ
with jaj þ jbj > 0; c – 0,
( Pn pðx; y; dÞ ¼
i j iþj¼0 aij x y ;
x < 0;
þ i j iþj¼0 aij x y ;
x P 0;
Pn
( Pn qðx; y; dÞ ¼
and d ¼ a 2 D Rðnþ1Þðnþ2Þ with D bounded. For ij ; bij below
i j iþj¼0 bij x y ; þ i j iþj¼0 bij x y ;
Pn
¼ 0,
x < 0; x P 0;
ð1:3Þ
system (1.1) has the piecewise-Hamiltonian function given
q The project was supported by National Natural Science Foundation of China (11271261), a Grant from Ministry of Education of China (20103127110001) and FP7-PEOPLE-2012-IRSES-316338. ⇑ Corresponding author. E-mail address: [email protected] (M. Han).
http://dx.doi.org/10.1016/j.amc.2014.05.035 0096-3003/Ó 2014 Published by Elsevier Inc.
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Y. Xiong, M. Han / Applied Mathematics and Computation 242 (2014) 47–64
Hðx; yÞ ¼
1 2 y þ 2
Z
x
gðsÞds ¼
H ðx; yÞ;
x < 0;
þ
H ðx; yÞ;
0
ð1:4Þ
x P 0;
where
1 2 1 2 y þ ax þ bx; 2 2
H ðx; yÞ ¼
Hþ ðx; yÞ ¼
1 2 1 3 1 2 y þ cx þ dx þ ex: 2 3 2
Figs. 1.1 and 1.2 show the possible graphs of the level curves of ðx; yÞjH ðx; yÞ ¼ h on the plane, respectively. We remark that in Figs. 1.1 and 1.2, A ¼ ðaA ; 0Þ; C ¼ ðaC ; 0Þ; S ¼ ðaS ; 0Þ; J ¼ ðaJ ; 0Þ, in which aA ¼ ba (when a – 0), pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi 2 2 aC ¼ dþ 2cd 4ec ; aS ¼ d 2cd 4ec ; aJ ¼ 2cd . Using Figs. 1.1 and 1.2, one can obtain Table 1.1, which presents the possible phase portraits of (1.1)j¼0 together with the corresponding coefficient conditions when it has a family of closed curves. From Fig. 1.3, we can obtain 42 different phase portraits when (1.1)j¼0 has at least one closed orbit. Now, we turn to study limit cycle bifurcation of system (1.1). For simplicity, we mainly investigate Fig. 1.3(27) with a ¼ 0; b < 0 or a > 0; b ¼ 0. That is, a; b; c; d and e in (1.2) satisfy 2
c > 0;
e¼
d ; 4c
a > b;
ab ¼ 0:
ð1:5Þ
Obviously, system (1.1)j¼0 satisfying (1.5) has two families of closed orbits fLh j0 < h < h0 g and Lh ¼ fhjh > h0 g, where 3
Lh ¼ Lh [ Lþh ;
h 2 ð0; h0 Þ [ ðh0 ; þ1Þ , I;
h0 ¼
d > 0; 24c2
with
x < 0g; Lþh ¼ ðx; yÞjHþ ðx; yÞ ¼ h;
Lh ¼ fðx; yÞjH ðx; yÞ ¼ h;
xP0 :
It is easy to see that limh!0þ Lh ¼ 0; limh!h0 Lh ¼ L and the limit L denotes the generalized homoclinic loop with a cusp. Then, by [10], associated to Lh , system (1.1) has the first order Melnikov function written as
MðhÞ ¼ M ðhÞ þ M þ ðhÞ;
h 2 I;
ð1:6Þ
where
M ðhÞ ¼
Z
n X
L h iþj¼0
bij xi yj dx
n X
aij xi yj dy:
ð1:7Þ
iþj¼0
About system (1.1), let Z Hopf and Z Loop denote the maximal number of limit cycles generated near the origin and near L respectively for 0 < 1. Then, the main result can be stated as follows. Theorem 1.1. Let (1.2), (1.3) and (1.5) hold. Then,
Z Hopf P n þ
nþ1 ; 2
Z Loop P 2n þ
nþ1 ; 2
where n P 1. In the following, we divide into two sections to prove our main result. In Section 2, we mainly derive the expression of M ðhÞ in (1.7). In Section 3, we study Hopf and generalized homoclinic loop bifurcations, proveing Theorem 1.1. 2. Preliminary lemmas In this section, we deduce the expansion of M ðhÞ in (1.7). The following lemma is direct from Lemma 4 of [13].
Fig. 1.1. The level curves of fðx; yÞjH ðx; yÞ ¼ hg on the plane.
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Y. Xiong, M. Han / Applied Mathematics and Computation 242 (2014) 47–64
Fig. 1.2. The level curves of ðx; yÞjHþ ðx; yÞ ¼ h on the plane.
Table 1.1 Conditions of existing periodic orbits and their correlated phase portraits. Coefficient conditions c<0
a < 0; b < 0
a P 0; b 0; a > b
No closed orbits
No closed orbits
No closed orbits
Fig. 1.3(1)
Fig. 1.3(2)–(9)
Fig. 1.3(10)–(12)
Fig. 1.3(13)–(16)
Fig. 1.3(1), (17)–(20)
2
Fig. 1.3(21)
No closed orbits
Fig. 1.3(22)
Fig. 1.3(23)
2
Fig. 1.3(21), (24)–(26)
No closed orbits
Fig. 1.3(22), (27)
Fig. 1.3(23), (28)
2
Fig. 1.3(21), (29)–(33)
Fig. 1.3(34)–(36)
Fig. 1.3(22), (37)–(39)
Fig. 1.3(23), (40)–(42)
d < 4ec d ¼ 4ec d > 4ec
Lemma 2.1. Suppose that (1.2), (1.3) and (1.5) hold. Then,
M ðhÞ ¼
8 n pffiffiffi X > > ~i;2k hiþk > h q > <
for a ¼ 0; b < 0;
n pffiffiffiiþ2k > pffiffiffi X > > > qi;2k h : h
for a > 0; b ¼ 0;
iþ2k¼0
h 2 I;
iþ2k¼0
where
~0;2k ¼ q0;2k ¼ q
qi;2k
2
kþ1þ12
pffiffiffi 2kþ1 2ð 2Þ a0;2k
2k þ 1
;
2k þ 1 bi1;2kþ1 i
Z
p 2
2k
sin h cos2iþ1 hdh; i P 1; i 0 b pffiffiffi 2kþ1þi
Z 0 2ð 2Þ i i1 ¼ b þ a sin h cos2kþ2 hdh; i P 1: pffiffiffi i i1;2kþ1 2k þ 1 i;2k p2 ð aÞ
~i;2k ¼ q
a > 0; b > 0
2
d 4ec d > 4ec
c>0
a 0; b P 0; a < b
2
ai;2k þ
About the expression of M þ ðhÞ, we have Lemma 2.2. Let the assumption of Lemma 2.1 hold. Then, for h 2 I,
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Y. Xiong, M. Han / Applied Mathematics and Computation 242 (2014) 47–64
pffiffiffi h
Mþ ðhÞ ¼
n1 X
qþiþ1;2k /þik ðhÞ þ
iþ2k¼0
n X
! k
qþ0;2k h
n1 X
þ
2k¼0
qþiþ1;2k
i X T rk ðhÞ;
where each /þ ik ðhÞ is a polynomial in h of degree k þ
iþ1 3
ð2:1Þ
r¼0
iþ2k¼0
with /ik ð0Þ ¼ 0, and
kþ1þ12
2 i þ aþ ; qþi;2k ¼ bi1;2kþ1 þ aþ ; i P 1; 2k þ 1 i;2k 2k þ 1 0;2k 8 kþm ð0Þ > d3 > ; r ¼ 3m; > aimk T 00 ðhÞ 2h þ 12c 2 > > < kþm T rk ðhÞ ¼ d3 > að1Þ T ðhÞ 2h þ 12c ; r ¼ 3m þ 1; 2 > > imk 10 > > : 0; r ¼ 3m þ 2
qþ0;2k ¼
satisfying
T i0 ðhÞ ¼
Z
ð2hþ
ð
1 d 3 Þ3 12c2
1 d 3 Þ3 12c2
3
t
d 2h þ t3 12c2
i
!12 dt;
i ¼ 0; 1
d i 3 13 ð1Þ d i1 3 23 ð0Þ ð1Þ ð0Þ and aimk > 0; aimk > 0 with ai00 ¼ 2 2c ; ai00 ¼ 2i 2c . 2c 2c Proof. Following the idea of (2.9) in [11] or Lemma 2 of [13], by Green formula, M þ ðhÞ can be rewritten as
Mþ ðhÞ ¼ NðhÞ þ
1 n X 2kþ1þ2 þ kþ12 a h ; 2k þ 1 0;2k 2k¼0
h 2 I;
ð2:2Þ
where
NðhÞ ¼
n1 X
qþiþ1;j
iþj¼0
Z
xi yjþ1 dx;
Lþ h
þ
qþiþ1;j ¼ bi;jþ1 þ
iþ1 þ a : j þ 1 iþ1;j
ð2:3Þ
þ d 3 d3 By (1.5), rewrite Hþ ðx; yÞ in (1.4) as Hþ ðx; yÞ ¼ 12 y2 þ 13 c x þ 2c 24c 2 . Thus, along the curve Lh
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
3 3 d 2 d y ¼ 2h þ c x þ ; 2c 12c2 3 which intersects the positive x-axis at a point
NðhÞ ¼
n1 X
3 h c
3
d þ 8c 3
13
d 2c ; 0 . Then, NðhÞ in (2.3) becomes
n1 h i X qþiþ1;j Iij þ ð1Þj Iij ¼ 2qþiþ1;2k Ii;2k ðhÞ;
iþj¼0
ð2:4Þ
iþ2k¼0
where
Iij ðhÞ ¼
Z
1
3
d ð3c hþ d 3 Þ3 2c 8c
0
"
3 #jþ1 2 3 d 2 d x 2h þ c x þ dx: 2c 12c2 3 i
1 3 13 d Let t ¼ 23 c 3 x þ 2c dt and Ii;2k ðhÞ in (2.4) can be transformed into . Then, dx ¼ 2c
Ii;2k ðhÞ ¼
1 d 3 Þ3 12c2
Z
ð2hþ
ð
d3 Þ 12c2
"
3 2c
1 3
13
d t 2c
#i
3
d 2h þ t3 12c2
!kþ12
3 2c
13
ir 3 i X d 3 C ri T rk ðhÞ; 2c 2c r¼0 rþ1
dt ¼
ð2:5Þ
where
T rk ðhÞ ¼
Z ð
ð2hþ
1 d 3 Þ3 12c2
1 d3 Þ3 12c2
3
t
r
d 2h þ t3 12c2
!kþ12 dt:
Note that
Z
3
tr 2h þ
d t3 12c2
!kþ12
3
dt ¼ N1 ðkÞt rþ1 2h þ
d t3 12c2
!kþ12
3
þ N2 ðkÞ 2h þ
d 12c2
!Z
3
tr 2h þ
d t3 12c2
!k12 dt;
k P 1;
Y. Xiong, M. Han / Applied Mathematics and Computation 242 (2014) 47–64
Fig. 1.3. The possible phase portraits of system (1.1)j¼0 .
in which
N1 ðkÞ ¼
1 ; 3ðk þ 12Þ þ r þ 1
N2 ðkÞ ¼
3ðk þ 12Þ : 3ðk þ 12Þ þ r þ 1
51
52
Y. Xiong, M. Han / Applied Mathematics and Computation 242 (2014) 47–64
Fig. 1.3 (continued)
Then, one obtains 3
d T rk ðhÞ ¼ 12c2
!rþ1 3 N1 ðkÞð2hÞ
kþ12
! 3 d T r;k1 ðhÞ; þ N 2 ðkÞ 2h þ 12c2
k P 1;
Y. Xiong, M. Han / Applied Mathematics and Computation 242 (2014) 47–64
53
Fig. 1.3 (continued)
which leads to 3
d T rk ðhÞ ¼ 12c2
!rþ1 3
3 pffiffiffiffiffiffi d k 2h þ k ðhÞ þ a 2hu 12c2
!k T r0 ðhÞ;
k P 1;
where
k ðhÞ ¼ N 1 ðkÞð2hÞk þ u
a k ¼
k1 Y
3 j1 k1 X Y d kj N1 ðk jÞ N2 ðk iÞð2hÞ 2h þ 12c2 j¼1 i¼0
!j ;
N2 ðk iÞ:
i¼0
Hence, 3
d T rk ðhÞ ¼ 12c2
!rþ1 3
3 pffiffiffiffiffiffi d 2huk ðhÞ þ ak 2h þ 12c2
!k T r0 ðhÞ;
k P 0;
ð2:6Þ
54
Y. Xiong, M. Han / Applied Mathematics and Computation 242 (2014) 47–64
in which
uk ðhÞ ¼
0; k ¼ 0; k ðhÞ; k P 1; u
ak ¼
1; k ¼ 0; a k ; k P 1;
with uk ð0Þ ¼ 0 and ak > 0. Using the formula
Z
3
t
r
d 2h þ t3 12c2
!12
3
dt ¼ N3 ðrÞt
d 2h þ t3 12c2
r2
!32
3
d þ N4 ðrÞ 2h þ 12c2
!Z
3
t
r3
d 2h þ t3 12c2
!12 dt;
ð2:7Þ
we obtain 3
T r0 ðhÞ ¼ N5 ðrÞð2hÞ2 þ N4 ðrÞ 2h þ
! 3 d T r3;0 ðhÞ; 12c2
r P 2;
where
1 N3 ðrÞ ¼ ; r þ 52
3
r2 N 4 ðrÞ ¼ ; r þ 52
d 12c2
N 5 ðrÞ ¼
!r2 3 N3 ðrÞ:
Thus, by induction, one can prove 3
3
T r0 ðhÞ ¼ ð2hÞ2 w½rþ11 ðhÞ þ b½rþ1 2h þ 3
3
d 12c2
!½3r T r3½ r ;0 ðhÞ;
r P 0;
3
ð2:8Þ
where
w½rþ11 ðhÞ ¼ 3
b½rþ1 ¼ 3
8 0; > > > <
r ¼ 0; 1;
> > > : N5 ðrÞ þ
½rþ1 1 3
X
j1 j Y d3 N 4 ðr 3iÞ 2h þ 12c ; r P 2; 2 i¼0
j¼1
8 < 1; :Q
N5 ðr 3jÞ r ¼ 0; 1;
½rþ1 1 3 i¼0
N4 ðr 3iÞ; r P 2:
Since N 4 ðrÞ > 0; r > 2 and N 4 ð2Þ ¼ 0, we have from the above
b½rþ1 3
r¼3m;3mþ1
b½rþ1
> 0;
3
r¼3mþ2
¼ 0:
ð2:9Þ
Inserting (2.8) into (2.6) leads to 3 pffiffiffiffiffiffi d T rk ðhÞ ¼ 2hvkþ½rþ1 ðhÞ þ ak b½rþ1 2h þ 3 3 12c2
!kþ½3r T r3½ r ;0 ðhÞ;
r P 0;
3
k P 0;
ð2:10Þ
where 3
d vkþ½rþ13 ðhÞ ¼ 12c2
!rþ1 3
3
d uk ðhÞ þ 2ak h 2h þ 12c2
which is a polynomial in h of degree k þ
rþ1 3
with
!k w½rþ11 ðhÞ; 3
vkþ½rþ13 ð0Þ ¼ 0.
Combining (2.4), (2.5) and (2.10) yields that
NðhÞ ¼
n1 X
2qþiþ1;2k
iþ2k¼0
¼
n1 X iþ2k¼0
qþiþ1;2k
2 3 !kþ½3r
ir rþ1 3 i X d 3 3 4pffiffiffiffiffiffi d r Ci T r3½ r ;0 ðhÞ5 2hvkþ½rþ1 ðhÞ þ ak b½rþ1 2h þ 3 3 3 2c 2c 12c2 r¼0 X
063mþj6i; j¼0;1;2
3
2h þ
d 12c2
!kþm
n1 pffiffiffi X
aðjÞ h imk T j0 ðhÞ þ
iþ2k¼0
qþiþ1;2k /þik ðhÞ;
ð2:11Þ
Y. Xiong, M. Han / Applied Mathematics and Computation 242 (2014) 47–64
55
where, together with (2.9)
3m að0Þ imk ¼ 2C i
d 2c
i3m
3 2c
3mþ1 3
ak b½rþ13
> 0;
3 ak b½rþ13 > 0; 2c r¼3mþ1
i3m2 mþ1 d 3 3mþ2 að2Þ ¼ 2C a b ¼ 0; rþ1 k i ½ 3 imk 2c 2c r¼3mþ2
ir rþ1 i pffiffiffiX d 3 3 vkþ½rþ13 ðhÞ /þik ðhÞ ¼ 2 2 C ri 2c 2c r¼0 3mþ1 að1Þ imk ¼ 2C i
d 2c
i3m1
r¼3m
3mþ2 3
and /þ ik ðhÞ are polynomials in h of degree k þ proof. h
iþ1 3
with /þ ik ð0Þ ¼ 0. Then, substituting (2.11) into (2.2) gives (2.1). This ends the
3. Proof of the main result In this section, we split into two subsections to discuss our problem. 3.1. Hopf bifurcation In this subsection, we study the limit cycle bifurcations of (1.1) in the case (27) of Fig. 1.3 near the origin. For the purpose, we introduce
Ak ¼ ð1Þk
A0 ¼ 1;
k1 Y j1 þ ji i¼0
j2 þ ji
k P 1;
;
ð3:1Þ
where j1 ; j2 ; j are positive constants with ðm n þ 1Þ ðm n þ 1Þ matrix
0
Am B A B mþ1 B B Am;n ðj1 ; j2 ; jÞ ¼ B Amþ2 B . B . @ . A2mn
j1 – j2 . Then, for any m P n P 0, we use Ak to construct an
Am1
Am2
Am
Am1
Amþ1 .. .
Am .. .
A2mn1
A2mn2
An
1
C
Anþ1 C C
Anþ2 C C: .. .. C C . . A
Am
ð3:2Þ
Then, denote
Bp ¼
Bp1
Bp2
Bp3
0
;
p ¼ 1; . . . ; m n;
where Bp1 is a p ðm n þ 1 pÞ matrix,
0
Bp2
B B B B ¼B B B B @
Cn Dn C nþ1 Dnþ2 C nþ2 Dnþ4
q C nþp1 Dnþ2ðp1Þ
0
1 C C C C C; C C C A
0
Bp3
Cm Dmþp
B B C mþ1 BD B mþpþ1 ¼B B .. B . @
C 2mnp D2mn
C m1 Dmþp1
C nþp Dnþ2p
Cm Dmþp
C nþpþ1 Dnþ2pþ1
.. . C 2mnp1 D2mn1
..
.
.. . Cm Dmþp
1 C C C C C C C A
and
C 0 ¼ D0 ¼ 1;
Ck ¼
k1 Y
ðj1 þ jiÞ;
i¼0
Dk ¼
k1 Y
ðj2 þ jiÞ;
k P 1:
i¼0
Then, for the relationship between Am;n ðj1 ; j2 ; jÞ and Bp , we have Lemma 3.1. For each Bp , we have
Bp Am;n ðj1 ; j2 ; jÞ: In other words, for each p, there must exist a non-singular ðm n þ 1Þ ðm n þ 1Þ matrix T p such that
Am;n ðj1 ; j2 ; jÞ ¼ T 1 p Bp T p :
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Y. Xiong, M. Han / Applied Mathematics and Computation 242 (2014) 47–64
Proof. For Am;n ðj1 ; j2 ; jÞ, by multiplying the ith column by ð1Þi1 ; i ¼ 2; . . . ; m n þ 1, and dividing the ith row by ð1Þmþi1 ; i ¼ 1; . . . ; m n þ 1, one has
0
Cm Dm
BC B mþ1 B Dmþ1 Am;n ðj1 ; j2 ; jÞ B0 ¼ B B . B .. @
C 2mn D2mn
C m1 Dm1
Cn Dn
Cm Dm
C nþ1 Dnþ1
.. .
..
C 2mn1 D2mn1
C C C C: .. C . C A
.
1
Cm Dm
Thus, it suffices to prove
p ¼ 0; . . . ; m n 1:
Bp Bpþ1 ;
ð3:3Þ
In fact, for p ¼ 0; . . . ; m n 1, make elementary transformations to Bp as follows: ðnþi1Þ (i) Multiply the ith row by jj2 þ1 þjjðnþiþp1Þ , then add the result to the ði þ 1Þth row, i ¼ m n; . . . ; p þ 1.
(ii) Multiply the ith row by j2 þ jðn þ i þ p 2Þ; i ¼ p þ 2; . . . ; m n þ 1. (iii) Divide the ith column by jðm n þ 1 p iÞðjp þ j2 j1 Þ; i ¼ 1; . . . ; m n p. Then, Bp becomes Bpþ1 . Thus, (3.3) holds. The proof is finished. h Therefore, we obtain Lemma 3.2. For matrix Am;n ðj1 ; j2 ; jÞ in (3.2), we have
detðAm;n ðj1 ; j2 ; jÞÞ – 0: Proof. By setting p ¼ m n in Lemma 3.1, we have
Am;n ðj1 ; j2 ; jÞ ¼ T 1 mn B mn T mn : Note that
detðBmn Þ ¼
m n Y i¼0
C nþi – 0: Dnþ2i
Thus, we obtain the conclusion. This finishes the proof.
h
The following lemma presents a property of T 00 ðhÞ near the origin. Lemma 3.3. The function T 00 ðhÞ in Lemma 2.2 has the expansion for 0 < h 1
T 00 ðhÞ ¼
pffiffiffi
2
k 4 2 12c2 3 pffiffiffiX ~ 24c2 k h h; A h k 3 3 9 d d kP0
ð3:4Þ
where
~ 0 ¼ 1; A
~ k ¼ ð1Þk A
k1 Y 2 þ 3i ; 5 þ 52 þ 3i i¼0
Proof. Perform the transformation
pffiffiffiffiffiffi Z T 00 ðhÞ ¼ 2h 2h
1
0
k P 1:
3
3
t d v 3 ¼ 2h 24hc
2
ð3:5Þ
to T 00 ðhÞ to obtain
2 1 pffiffiffi 12c2 3 pffiffiffi Z 1 v 2 ð1 v 3 Þ2 d v ¼ 2 h 2 h 23 2 dv : 3 2 3 3 3 0 d d3 2hv þ 12c 1 þ 24hcd3 v 2 1 2
v 2 ð1 v 3 Þ
ð3:6Þ
Notice that for 0 < h 1
1þ
24hc
2
3
d
v3
!23 ¼1þ
X ð1Þk Qk1 ð2 þ 3iÞ 24c2 h k i¼0
kP1
Inserting (3.7) into (3.6) yields that
3k k!
3
d
v 3k :
ð3:7Þ
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Y. Xiong, M. Han / Applied Mathematics and Computation 242 (2014) 47–64
" # 2 X ð1Þk Qk1 ð2 þ 3iÞ 24c2 h k pffiffiffi 12c2 3 pffiffiffi i¼0 T 00 ðhÞ ¼ 2 2 h h J0 þ Jk ; 3 3 3k k! d d kP1
ð3:8Þ
where
Jk ¼
Z
1
1 2
v 3kþ2 ð1 v 3 Þ dv ;
k P 0:
0
Using (2.7), we get
J0 ¼
2 ; 9
Jk ¼
3k J ; 3k þ 2 þ 52 k1
k P 1;
ð3:9Þ
which implies that
Jk ¼
2 3k k! ; Qk 9 i¼1 ð3i þ 2 þ 5Þ
k P 1:
ð3:10Þ
2
Combining (3.8), (3.9) and (3.10) gives (3.4). The proof is completed. h Applying Lemmas 2.1, 2.2, 3.2 and 3.3, we can obtain the theorem below. Theorem 3.1. Suppose (1.2), (1.3) and (1.5) hold. Then, for any n P 1 and 0 < 1, system (1.1) can have at least n þ limit cycles near the origin.
nþ1 2
Proof. For convenience, take þ
aij ¼ 0; j P 1 and aþij ¼ bij ¼ 0:
bij ¼ 0; i P 1;
Then, in view of (1.6), Lemmas 2.1 and 2.2, one obtains
! !k 3 n1 n1 X pffiffiffi X d þ þ ð0Þ þ MðhÞ ¼ h b0;2kþ1 /0k ðhÞ þ GðhÞ þ b0;2kþ1 a00k 2h þ T 00 ðhÞ; 12c2 2k¼0 2k¼0
ð3:11Þ
where
GðhÞ ¼
8 Pn ~i0 hi ; < i¼0 q : Pn
i¼0 qi0 ð
a ¼ 0; b < 0;
pffiffiffi i hÞ ; a > 0; b ¼ 0;
with
~i0 ¼ q
q00
pffiffiffi 2 2 b
i
ai0
Z
2
cos2iþ1 hdh ¼
0
pffiffiffi ¼ 2 2a00 ;
gi ¼
p
qi0
pffiffiffi 2 2ð2iÞ!! i
ð2i þ 1Þ!!b
ai0 ;
pffiffiffi iþ1 Z 0 2ð 2Þ i1 ¼ pffiffiffi i iai0 sin h cos2 hdh ¼ gi ai0 ; p2 ð aÞ
i P 1;
ð3:12Þ
pffiffiffi iþ1 pffiffiffi iþ1 2ð 2Þ i!! p 2ð 2Þ i!! ; i P 0 even; g ¼ pffiffiffi i ði þ 1Þ!! pffiffiffi i ði þ 1Þ!! ; i P 1 odd: i 2 ð aÞ ð aÞ
Then, for h > 0 small, by (3.4) and (3.11), one can find that
2 pffiffiffi 3 !ir 2 ½n1
½n1 3 i 2 2 2 j X X X pffiffiffi 4 2 12c2 3 X d 24c þ rþ1 j þ ð0Þ ~j MðhÞ ¼ h4 b0;2iþ1 a00i C ri 2r h h þ GðhÞ þ b0;2iþ1 /þ0i ðhÞ5; A 3 3 12c2 9 d d r¼0 jP0 i¼0 i¼0 where
GðhÞ ¼
8 n X 2pffiffi2ð2iÞ!! > > i < a ¼ 0; b < 0; i ai0 h ; i¼0
> > : Pn
i¼0
Rewrite (3.13) as
ð2iþ1Þ!!b
pffiffiffi
i
gi ai0 ð hÞ ;
a > 0; b ¼ 0:
ð3:13Þ
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Y. Xiong, M. Han / Applied Mathematics and Computation 242 (2014) 47–64
8 pffiffiffiX i h v i h ; a ¼ 0; b < 0; > > > > iP0 > < 0 1 MðhÞ ¼ pffiffiffi X ½n2 ½n1 2 X X pffiffiffi > B i i iC > > v i h h þ v~ i h A; h@ v~ i h þ > > : i¼0 i¼0 iP½2nþ1
ð3:14Þ a > 0;
b ¼ 0;
where
vi ¼ v nþi ¼
pffiffiffi 2 2ð2iÞ!! i
ð2i þ 1Þ!!b
i ¼ 0; . . . ; n;
2
pffiffiffi
2
n1 X ½n1
j 2 2 ij X 4 2 12c2 3 24c2 þ ð0Þ j 24c ~ nþir1 ; 2 b a C rj A 0;2jþ1 00j 3 3 3 9 d d d r¼0 j¼0
v~ i ¼ g2i a2i;0 þ O
þ þ b01 ; b03 ; . . . ; b0;2½n1þ1 ;
v i ¼ g2iþ1 a2iþ1;0 ; v~ ½n2þi ¼
þ þ ai0 þ O b01 ; b03 ; . . . ; b0;2½n1þ1 ;
2
i ¼ 0; . . . ;
i ¼ 0; . . . ;
hn i 2
n1 ; 2
iP1
;
pffiffiffi
2
½n1 ½n1 ij
j 2 X 4 2 12c2 3 24c2 2 X 24c2 þ ð0Þ ~ ½nþir1 ; 2j b0;2jþ1 a00j C rj A 3 3 3 2 9 d d d r¼0 j¼0
i P 1:
Now, we only consider the case a ¼ 0; b < 0 since the other case can be similarly verified. From the above formulas, we get
det
@ðv 0 ; v 1 ; . . . ; v nþ½nþ1 Þ 2
þ þ @ða00 ; . . . ; an0 ; b01 ; . . . ; b0;2½n1þ1 Þ
¼ det
C11
C12
0
C22
2
where C12 is a ðn þ 1Þ
C11
nþ1 2
, detðCÞ;
ð3:15Þ
matrix,
pffiffiffi 2!! 4!! ð2nÞ!! ¼ 2 2diag 1; ; ;...; n 2 3!!b 5!!b ð2n þ 1Þ!!b
and
pffiffiffi
2
n1
4 2 12c2 3 24c2 ð0Þ ð0Þ ½n1 ð0Þ 2 a a . ; 2 a . ; . . . ; 2 . nþ1 ½ 2 ; 000 1 001 2 3 3 00;½n1 9 2 d d 0 1i 1 Pi r~ 24c2 A C nþ1r1 i r¼0 C B d3 B 2i C C B Pi r~ C B 24c2 3 C B r¼0 C i Anþ2r1 d C; i ¼ 0; . . . ; n 1 : ¼B C B 2 C B .. C B . C B @P ½nþ1 i A 2 i r~ 24c2 3 r¼0 C i Anþ½nþ1 r1 d 2
C22 ¼
.iþ1
It follows that
" pffiffiffi #½nþ1 2 ½n1 n 2 pffiffiffi nþ1 4 2 12c2 3 24c2 n1 2 Y ð2iÞ!! Y ð0Þ detðCÞ ¼ detðC11 Þ detðC22 Þ ¼ ð2 2Þ 2i a00i detðC1 Þ; 3 3 i 9 d d ð2i þ 1Þ!!b i¼0 i¼0 where
C1 ¼
.1 ; .2 ; . . . ; .½nþ1 : 2
ð3:16Þ
For C1 , make elementary transformations as follows: i1 2 (1) Multiply the ith column by 24c ; i ¼ 2; . . . ; nþ1 . 3 2 i d
2 (2) Divide the ith row by 24c ; i ¼ 1; . . . ; nþ1 . 2 d3 (3) Add the jth column multiplying by C ji1 to the ith column, j ¼ 1; . . . ;
n1 ; i ¼ j þ 1; . . . ; nþ1 . 2 2
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Y. Xiong, M. Han / Applied Mathematics and Computation 242 (2014) 47–64
Then, C1 becomes An;n½n1 2; 5 þ 52 ; 3 , i. e., C1 An;n½n1 2; 5 þ 52 ; 3 . By Lemma 3.2, we obtain det An;n½n1 2 2 2 2; 5 þ 52 ; 3 – 0. Thus, one has detðCÞ – 0. Then, by (3.15), we conclude that v 0 ; v 1 ; . . . ; v nþ½nþ1 are independent with 2 each other such that MðhÞ in (3.14) can have n þ nþ1 v satisfying zeros for 0 < h 1 by varying j 2 nþ1 nþ1 0 < v 0 v 1 ð1Þnþ½ 2 1 v nþ½nþ11 ð1Þnþ½ 2 v nþ½nþ1 1: 2
2
This ends the proof. h 3.2. Limit cycles near the generalized homoclinic loop L In this subsection, we investigate the bifurcation of limit cycles near the generalized homoclinic loop L by the first order Melnikov function. For the purpose, we need to know the expressions of M ðhÞ near L. The following lemma presents the the expansion of T i0 ðhÞ in Lemma 2.2 near L. 3
d Lemma 3.4. Let u ¼ 2h þ 12c 2 . Then for 0 < juj 1 and i ¼ 0; 1, we have
T i0 ðhÞ ¼ ð1Þi
X 5þ2i 5þ2i Bki ðuÞk jujkþ 6 akþ 6 þ Di ;
ð3:17Þ
kP0
3 13 d where a ¼ 12c and 2
B0i ¼
Di ¼
1 ; i þ 52 ( 0;
B1i ¼
1 ; 2i 1
Bki ¼
ð2k 3Þ!! ; ð2kÞ!!ð3k þ i þ 52Þ
k P 2;
0 < u 1; 5þ2i
xi u 6 ; 0 < u 1;
with xi a constant. 3
d Proof. Note that u ¼ 2h þ 12c 2 . Then, T i0 ðhÞ in Lemma 2.2 is reduced to
T i0 ðhÞ ¼
Z
1 u3 1 3 ð d 2 Þ3 12c
1 t i u t 3 2 dt ¼ T i1 ðuÞ þ T i2 ðuÞ;
ð3:18Þ
where
T i1 ðuÞ ¼ Since
d3 12c2
u t3
Z
13
12
1
juj3
1 3 ð d 2 Þ3 12c
i
t ut
3
12
T i2 ðuÞ ¼
dt;
< 0, then for t 2
d3 12c2
13
Z
1
u3 1 juj3
1
1 ti u t3 2 dt:
ð3:19Þ
; juj3 , I1 , we have t3 > 0. Thus, for 0 < juj 1 and t 2 I1 ,
X ð1Þk1 ð2k 3Þ!! 1 1
1=2 1 1 kþ1 ¼ ðt 3 Þ2 1 þ ðt3 uÞ ¼ ðt3 Þ2 þ ðt 3 Þ 2 u þ ðt 3 Þ 2 uk : 2 ð2kÞ!! kP2
ð3:20Þ
Note that
Z
1
juj3 1 3
3 ð d 2Þ 12c
kþ12
t i ðt 3 Þ
3 13 d where a ¼ 12c . 2
dt ¼
ð1Þiþ1 kþ5þ2i 5þ2i 6 akþ 6 juj ; 5 3k þ i þ 2
k P 0;
ð3:21Þ
Obviously, (3.20) is a power series in t3 u and its domain of convergence is ½1; 1. Thus, (3.20) is uniformly convergent i 1 for t 2 1; juj3 , I2 . Note that I1 I2 . Thus, on account of (3.19), (3.20) and (3.21), together with the property of term by term integration of power series, we derive
ð1Þi T i1 ðuÞ ¼
5þ2i 1 5þ2i 1 5þ2i 5þ2i juj 6 a 6 þ ðuÞ juj1þ 6 a1þ 6 5 2i 1 iþ2 X 5þ2i ð2k 3Þ!! 5þ2i ðuÞk jujkþ 6 akþ 6 : þ 5 ð2kÞ!!ð3k þ i þ 2Þ kP2
For 0 < u 1, it is easy to see that
T i0 ðhÞ ¼ T i1 ðuÞ:
ð3:22Þ
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Y. Xiong, M. Han / Applied Mathematics and Computation 242 (2014) 47–64
Thus, in this case, the conclusion follows from (3.22) and the above. For 0 < u 1, we have 1 T i2 ðuÞ ¼ u2
1 t3 2 i t 1 dt 1 u u3
Z
Z 1 t3 1 i 5 1 i2 5þ2i ¼ u3þ6 Let x ¼ x 3 ð1 xÞ2 dx ¼ xi u 6 ; 3 u 1
1
u3
ð3:23Þ
where xi is a constant satisfying
xi ¼
1 3
Z
1
1
i2
x 3 ð1 xÞ2 dx:
1
By (3.18), (3.19), (3.22) and (3.23), we get that (3.17) for 0 < u 1. This ends the proof. h For convenience, denote
ri ¼ ð1Þi
X Bki ;
di ¼ ð1Þi
kP0
X ð1Þk Bki þ xi ;
i ¼ 0; 1:
ð3:24Þ
kP0
Then, we have Lemma 3.5. It holds that
ri < 0; di > 0; i ¼ 0; 1:
ð3:25Þ
Proof. Denote by U a disk of diameter Then, for
0 > 0 with its center at J. See Fig. 3.1.
0 > 0 small, rewrite Mþ ðhÞ in (1.7)
Mþ ðhÞ ¼ I1 ðhÞ þ I2 ðhÞ; where
Ik ðhÞ ¼
Z
n X
þ
bij xi yj dx
Lþ kh iþj¼0
Lþ1h ¼ Lþh \ U;
n X iþj¼0
aþij xi yj dy;
k ¼ 1; 2;
þ
Lþ2h ¼ Cl: Lþh L1h :
Note that I2 ðhÞ is a C 1 function in u for juj > 0 small. And for I1 ðhÞ, we claim that
8 > 0 b0 juj56 þ B 1 b1 juj76 þ o juj76 ; 0 < u 1; b1 juj76 þ o juj76 ; 0 < u 1; b0 juj56 þ B :B 0 1
ð3:26Þ
0 > 0; B 1 > 0; B < 0; B < 0, and where B 0 1
i n1
pffiffiffi c 13 X
d þ b0 ¼ 2 2 bi1 þ ði þ 1Þaiþ1;0 ; 3 2c i¼0 pffiffiffi i1 h 5 n1
i 2 2 c 3 X d þ c i bi1 þ ði þ 1Þaþiþ1;0 : b1 ¼ 3 2c 3 i¼1
ð3:27Þ
In fact, make a variable transformation
d ~x ¼ x þ ; 2c
~ ¼ y; y
t ¼ s:
ð3:28Þ
Then, the right system of (1.1) becomes
i d ~j ; aþij ð1Þi ~x þ y 2c iþj¼0
i n X d þ ~j : ~_ ¼ c~x2 þ bij ð1Þiþ1 ~x þ y y 2c iþj¼0
~x_ ¼ y ~þ
n X
ð3:29Þ
From Theorem 3.1 of [7], Remark 3.14 in [5], (3.28) and (3.29), one can obtain (3.26), (3.27). Thus, the claim holds. Hence, by above discussion we obtain
8 > 0 b0 juj56 þ B 1 b1 juj76 þ o juj76 þ / ðuÞ; 0 < u 1; b1 juj6 þ o juj6 þ / ðuÞ; 0 < u 1; b0 juj6 þ B :B 2 0 1 i ; B ; bi ; i ¼ 0; 1 are given in (3.26), (3.27) and / ðuÞ; / ðuÞ are C 1 functions. where B 1 2 i
ð3:30Þ
61
Y. Xiong, M. Han / Applied Mathematics and Computation 242 (2014) 47–64
On the other hand, by (1.6), (3.17) and Lemma 2.2, one can find that
0 M þ ðhÞ ¼ b
X
5 1 Bk0 juj6 b
kP0
0 M þ ðhÞ ¼ b
X
7 7 Bk1 juj6 þ o juj6 þ /3 ðuÞ;
for 0 < u 1;
kP0
! ! X X 5 1 ð1Þk B þ x1 juj76 þ o juj76 þ / ðuÞ; ð1Þk Bk0 þ x0 juj6 þ b k1 4 kP0
ð3:31Þ for 0 < u 1;
kP0
where /3 ðuÞ and /4 ðuÞ are C 1 functions and
13 X i n1
i d h þ 0 ¼ 2 3 b bi1 þ ði þ 1Þaþiþ1;0 ; 2c i¼0 2c
23 X i1 h n1
i d þ 1 ¼ 2 3 i bi1 þ ði þ 1Þaþiþ1;0 : b 2c i¼1 2c Comparing the coefficients in (3.30) and (3.31), together with (3.27) and (3.24), we find
5þ2i 5þ2i Bb Bb ri ¼ i i ¼ 2 6 Bi < 0; di ¼ i i ¼ 2 6 Bi > 0; bi bi
which implies (3.25). This finishes the proof. h Summarizing the conclusions of Lemmas 3.4 and 3.5, we obtain Lemma 3.6. The expansions of T i0 ðhÞ in Lemma 2.2 have the form
T i0 ðhÞ ¼
8 X 5þ2i 5þ2i iþ1 > 6 þ ð1Þ r Bki ðuÞk akþ 6 ; 0 < u 1; > i juj > > < kP0 X 5þ2i > 5þ2i iþ1 > 6 > Bki ðuÞk akþ 6 ; > : di juj þ ð1Þ
ð3:32Þ
0 < u 1;
kP0
where ri < 0; di > 0; i ¼ 0; 1. Therefore, one has Theorem 3.2. Suppose (1.2), (1.3) and (1.5) hold. Then for 0 < 1 and n P 1, system (1.1) can have 2n þ near the generalized homoclinic loop L.
nþ1 2
limit cycles
Proof. We only give the proof of the case a > 0; b ¼ 0 with n ¼ 2l þ 1, since the other cases can be proved in a similar way. For convenience, take þ
bij ¼ 0;
i P 2;
aij ¼ 0;
j P 1;
aþij ¼ bij ¼ 0:
Fig. 3.1. A disk U.
62
Y. Xiong, M. Han / Applied Mathematics and Computation 242 (2014) 47–64
Then, in view of (1.6) and Lemmas 2.1, 2.2, one can derive that 3 l X d þ ð0Þ MðhÞ ¼ b0;2iþ1 a00i 2h þ 12c2 i¼0
!i
3 l1 X d þ ð1Þ T 00 ðhÞ þ b1;2iþ1 a10i 2h þ 12c2 i¼0
!i
l l1 X pffiffiffi X þ þ T 10 ðhÞ þ h b0;2iþ1 /þ0i ðhÞ þ b1;2iþ1 /þ1i ðhÞ i¼0
!
i¼0
2lþ1 X pffiffiffi iþ1 qi0 ð hÞ : þ i¼0
From (3.12), we further have 3
l l1 l X X MðhÞ X T 00 ðhÞ T 10 ðhÞ u d þ þ þ ð0Þ ð1Þ pffiffiffi ¼ b0;2iþ1 a00i ui qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ b1;2iþ1 a10i ui qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ b0;2iþ1 /þ0i 2 3 3 2 24c u d u d h i¼0 i¼0 i¼0 24c2
2
3 l1 X u d þ b1;2iþ1 /þ1i þ 2 24c2 i¼0
2
! þ
l X
g2i a2i;0
i¼0
!
24c2
3
u d 2 24c2
!i þ
l X
g2iþ1 a2iþ1;0
i¼0
3
u d 2 24c2
!i sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 u d : 2 24c2
ð3:33Þ
Note that for 0 < juj 1
2 !12 !12 !k12 3 3 3 3 X pffiffiffi pffiffiffi 1 d d ð2k 1Þ!! d k qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 2 u ¼ 24 þ ð1Þ u k 5; ð2kÞ!! 12c2 12c2 12c2 u d3 kP1 24c2 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 !12 !12 !12 !kþ12 3 3 3 3 3 3 X ð1Þk ð2k 3Þ!! u d 1 d 1 4 d 1 d d ¼ pffiffiffi þ uþ u k 5: ¼ pffiffiffi u 2 24c2 2 ð2kÞ!! 12c2 12c2 12c2 12c2 2 2 kP2 Thus, in the light of Lemma 3.6 and the above, (3.33) can be rewritten as for 0 < u 1
X 5 1 5 MðhÞ pffiffiffi ¼ c0 þ ~c0 juj6 þ ci þ ci juj6 þ ~ci juj6 juji ; h iP1
ð3:34Þ
where
ci ¼
l X g2j j¼i
2
j
3
C ij
d 12c2
!ji
þ þ þ þ þ þ a2j;0 þ O b01 ; b03 ; . . . ; b0;2lþ1 ; b11 ; b13 ; . . . ; b1;2l1 ; a10 ; a30 ; . . . ; a2lþ1 ;
!12 3 d þ þ þ þ þ þ þ O b01 ; b03 ; . . . ; b0;2lþ1 ; b11 ; b13 ; . . . ; b1;2l1 ; a10 ; a30 ; . . . ; a2l1 ; p ffiffiffi lþ1 2 12c 2 2 !12 3 d ð0Þ þ ~c0 ¼ d0 a000 b01 ; 24c2 !12 3 d þ þ þ ð0Þ þ ~ci ¼ d0 a00i b0;2iþ1 þ O b01 ; b03 ; . . . ; b0;2i1 ; i ¼ 1; 2; . . . ; l; 2 24c !12 3 d ð1Þ þ c1 ¼ d1 a100 b11 ; 24c2 !12 3 d þ þ þ þ ð1Þ ci ¼ d1 a10;i1 b1;2i1 þ O b11 ; b13 ; . . . ; b1;2i3 ; i ¼ 2; 3; . . . ; l: 24c2 clþ1 ¼
g2lþ1 a2lþ1;0
Similar to (3.34), one has for 0 < u 1 5
MðhÞ ¼ v 0 þ v~ 0 juj6 þ
X
v i v i juj
1 6
X 5 ~ juj56 þ juj16 þ d ~ juj56 juji ; þ v~ i juj6 ui ¼ d0 þ d di þ d 0 i i
iP1
iP1
where
v~ i ¼
r0 ~
i ¼ 0; 1; . . . ; l;
v i ¼
r1
i ¼ 0; 1; . . . ; l 1;
vi
i ¼ 0; 1; . . . ; l;
d0
d1 ¼ ci ;
ci ; ci ;
i P 0; ¼ ð1Þi1 v ; di ¼ ð1Þ v i ; d i i i
~ ¼ ð1Þi v~ : d i i
ð3:35Þ
Y. Xiong, M. Han / Applied Mathematics and Computation 242 (2014) 47–64
63
Let
þ þ þ þ þ þ þ d ¼ aþ00 ; b01 ; aþ02 ; b11 ; b03 ; aþ04 ; b13 ; b05 ; . . . ; aþ0;2l ; b1;2l1 ; b0;2lþ1 ; a2lþ1;0 : Then, by (3.35), one can obtain that ðlþ1Þðlþ2Þ @ðc0 ; ~c0 ; c1 ; c1 ; ~c1 ; . . . ; cl ; cl ; ~cl ; clþ1 Þ @ð~c0 ; ~c1 ; . . . ; ~cl ; c1 ; c2 ; . . . ; cl ; c0 ; c1 ; . . . ; cl ; clþ1 Þ ¼ ð1Þ 2 þl det @d @d
D11 0 , detðDÞ; ¼ det D21 D22
det
where D21 is a ðl þ 2Þ ð2l þ 1Þ matrix, and
0
D11 ¼
B B B 1 ! 2 B B 3 B d B B 2 24c B B B B @ 0
D22
B B B B B ¼B B B B @
ð0Þ
d0 a000 ..
. ð0Þ
d0 a00l ð1Þ
d1 a100 ..
. ð1Þ
d1 a10;l1
1
g0 g2 2
..
1
0
. g2l 2l
g2lþ1
0
2lþ2
d3 24c 2
C C C C C C C; C C C C C A
C C C C C C: C C C 12 A
Thus,
detðDÞ ¼
2 Y
3
detðDii Þ ¼
i¼1
d 24c2
!l1
g2lþ1 2
lþ2
ðr0 Þlþ1 ðr1 Þl
l l1 Y g að0Þ Y i
i¼0
00i
2i
að1Þ 10i – 0;
i¼0
which means that c0 ; ~c0 ; c1 ; c1 ; ~c1 ; . . . ; cl ; cl ; ~cl ; clþ1 can be taken as free parameters. Summarizing the above, we obtain
8 X 5 1 5 > 6 ci þ ci juj6 þ ~ci juj6 juji ; 0 < u 1; > > c0 þ ~c0 juj þ < MðhÞ iP1 pffiffiffi ¼ X 5 1 5 > 6 h > ~ ci þ ci j rd11 jjuj6 ~ci j rd00 jjuj6 ð1Þi juji ; 0 < u 1; > : c0 þ c0 juj þ
ð3:36Þ
iP1
where c0 ; ~c0 ; c1 ; c1 ; ~c1 ; . . . ; cl ; cl ; ~cl ; clþ1 are independent with each other. Thus, we choose them satisfying
0 < c0 ~c0 ð1Þc1 ð1Þ2 c1 ð1Þ~c1 ð1Þl cl ð1Þlþ1 cl ð1Þl ~cl ð1Þlþ1 clþ1 1; pffiffi in (3.36) has already been changed l þ 1 þ l þ 1 þ l ¼ n þ l þ 1 and 2l þ 1 ¼ n times for 0 < u 1 such that the sign of MðhÞ h and 0 < u 1 respectively. Thus, MðhÞ can have 2n þ nþ1 zeros for 0 < juj 1 : n þ nþ1 zeros are positive and n zeros 2 2 are negative. Similarly, it is easy to prove that MðhÞ can have 2n þ nþ1 zeros for 0 < juj 1; n þ nþ1 of which are negative and n of 2 2 which are positive. This ends the proof. h
Acknowledgment The authors would like to thank the reviewers for their valuable comments and suggestions, which improve the presentation of the paper a lot. References [1] M. di Bernardo, C.J. Budd, A.R. Champneys, P. Kowalczyk, Piecewise Smooth Dynamical Systems, Theory and Applications, Springer-Verlag, London, 2008. [2] M.S. Branicky, Stability of switched and hybrid systems, in: Proceedings of the 33rd IEEE Conference on Decision Control, Lake Buena Vista, FL, 14–16, 1994, pp. 3498–3503. [3] A.F. Filipov, Differential Equations with Discontionuous Righthand Sides, Kluwer Academic, Netherlands, 1988.
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