Limit cycle bifurcations in a class of piecewise smooth systems with a double homoclinic loop

Applied Mathematics and Computation 248 (2014) 235–245

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Limit cycle bifurcations in a class of piecewise smooth systems with a double homoclinic loop q Yuanyuan Liu a,⇑, Valery G. Romanovski a,b,c a

Department of Mathematics, Shanghai Normal University, Shanghai, PR China Center for Applied Mathematics and Theoretical Physics, University of Maribor, Maribor, Slovenia c Faculty of Natural Science and Mathematics, University of Maribor, Koroška 160, Maribor, Slovenia b

a r t i c l e

i n f o

Keywords: Hamiltonian system Double homoclinic loop Melnikov function Limit cycle

a b s t r a c t In this paper we consider a class of perturbed piecewise smooth systems. Applying the method of first order Melnikov function we give a lower bound for the maximal number of limit cycles bifurcated from a double homoclinic loop. As an application we construct a piecewise quadratic system with quartic perturbation, which has 11 limit cycles bifurcated from such loop. Ó 2014 Elsevier Inc. All rights reserved.

1. Introduction and preliminaries The bifurcation theory is an important part of the qualitative theory of differential systems. The study of bifurcations from singular points and periodic orbits is important for the analysis of many mathematical models, like, for instance, the predator–prey system, the Holling–Tanner model, the infection model (see [21,23,25] and the references therein), and many other models. The objects of the main interest in such models are usually isolated periodic orbits, since they describe auto-oscillating regimes of the system. It appears the main technique to study limit cycles is a perturbation of simple systems with annulus of periodic orbits, in particular, a perturbation of Hamiltonian systems. In the latter case an efficient tool used to estimate the number of limit cycles is the so-called Melnikov function, see e.g. [8,15] and references given there. The Melnikov function can be used to study the number of limit cycles bifurcated from a center, a homoclinic loop, a heteroclinic loop or an annulus consisting of a family of periodic orbit. For instance, the authors of [24] proved that generic planar quadratic Hamiltonian systems with the third degree polynomial perturbation can have eight small-amplitude limit cycles around a center. Roussarie [20] studied the following system



x_ ¼ Hy þ epðx; y; e; dÞ; y_ ¼ Hx þ eqðx; y; e; dÞ;

ð1:1Þ

where Hðx; yÞ; pðx; y; e; dÞ; qðx; y; e; dÞ are analytic functions, e P 0 is small and d 2 D  Rm is a vector parameter with D being a compact set. Under the assumption that the origin is a hyperbolic saddle, the author obtained the expansion of the Melnikov function Mðh; dÞ near the homoclinic loop L0 as follows 2

2

Mðh; dÞ ¼ c0 ðdÞ þ c1 ðdÞh ln jhj þ c2 ðdÞh þ c3 ðdÞh ln jhj þ Oðh Þ;

0 < h  1:

q The project was supported by National Natural Science Foundation of China (11271261), the Slovenian Research Agency and by a Curie International Research Staff Exchange Scheme Fellowship within the 7th European Community Framework Programme, FP7-PEOPLE-2012- IRSES-316338. ⇑ Corresponding author. E-mail addresses: [email protected] (Y. Liu), [email protected] (V.G. Romanovski).

http://dx.doi.org/10.1016/j.amc.2014.09.125 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.

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Y. Liu, V.G. Romanovski / Applied Mathematics and Computation 248 (2014) 235–245

Then the authors of [11] gave formulas for c1 and c2 , and recently in [10] the formula for the coefficient c3 has been obtained. The authors of [4] discussed bifurcations of periodic orbits of a class of planar systems with one switching line. They derived an expression of the first order Melnikov function and applied it to study the number of limit cycles bifurcated from the annulus. To match the theoretic development and applications to models of real life phenomena, many researchers paid a lot of attention to the area of piecewise smooth system, see monographs [1,5,14], articles [3,6,7,19] and references therein. It is  well-known that for smooth dynamical system (1.1) if Melnikov function Mðh; dÞ satisfies for some small jh j > 0 the condition 







Mðh ; dÞ ¼ M0 ðh ; dÞ ¼ M 00 ðh ; dÞ ¼    ¼ M k1 ðh ; dÞ ¼ 0;



M k ðh ; dÞ – 0;

then the corresponding system has at most k limit cycles near Lh for e > 0 sufficient small, and has at least one limit cycle if k is odd, see Theorem 6.1 in [15]. A similar version for non-smooth systems was given in the paper [18] by Liu and Han. A number of new results on the problem are obtained also in [2,12,16,17]. The authors of [16,17] discussed the piecewise smooth systems with the origin as a center and a degenerated saddle (the definition can be found in [12]), respectively. Differently, we treat the origin as a hyperbolic saddle in the unperturbed system of (1.1). In this paper, we consider a piecewise near-Hamiltonian system of the form (1.1), where e > 0 is small and d 2 D  Rm is a vector parameter, with D compact, and

( Hðx; yÞ ¼

Hþ ðx; yÞ;

x P 0;

H ðx; yÞ;

x < 0;

pðx; y; e; dÞ ¼

qðx; y; e; dÞ ¼





pþ ðx; y; e; dÞ;

x P 0;

p ðx; y; e; dÞ;

x < 0;

qþ ðx; y; e; dÞ;

x P 0;

q ðx; y; e; dÞ;

x < 0;

with H ; p ; q being analytical functions defined on R2 . We call the analytic systems

(

x_ ¼ Hþy þ epþ ðx; y; e; dÞ; y_ ¼ Hþx þ eqþ ðx; y; e; dÞ;



x_ ¼ Hy þ ep ðx; y; e; dÞ; y_ ¼ Hx þ eq ðx; y; e; dÞ;

ð1:2Þ

ð1:3Þ

the right subsystem and the left subsystem, respectively. For e ¼ 0, systems (1.1)–(1.3) become, respectively,



(

x_ ¼ Hy ; y_ ¼ Hx ; x_ ¼ Hþy ; y_ ¼ Hþx ;



x_ ¼ Hy ; y_ ¼ Hx :

ð1:4Þ

ð1:5Þ

ð1:6Þ

We will suppose that H(I) The origin is a hyperbolic saddle for both systems (1.5) and (1.6), and the equations H ðx; yÞ ¼ 0 for x P 0 define þ  two homoclinic loops L 0 with a critical point at the origin. See Fig. 1.1. Then L0 and L0 form a double homoclinic loop þS  L0 ¼ L0 L0 in system (1.4).    Further, denoting by L h the orbits fH ðx; yÞ ¼ h; x P 0; h 2 ða ; bÞ; a < 0 < bg we assume that for system (1.4).  H(II) There exist three families of periodic orbits fLþ jaþ < h < 0g; fL h h ja < h < 0g and fLh j0 < h < bg where S  þ  e Lh ¼ fLh j0 < h < bg fLh~ j0 < h < bg. Note that the closed orbits Lh ja
h H ðAðhÞÞ ¼ H ðBðhÞÞ ¼ e c and BA, c respectively. See Fig. 1.2. for the definition of closed orbit Lh and denote Lh jxP0 and Lh jx0 by AB

ð1:7Þ

Y. Liu, V.G. Romanovski / Applied Mathematics and Computation 248 (2014) 235–245

237

Fig. 1.1. The phase portraits of the right and left subsystems.

0

e < 0; Lþ ; L~ are closed orbits inside L0 . For 0 < h; h e < b; L ¼ Lþ S L~ is a periodic orbit Fig. 1.2. The phase portrait of system (1.4). For aþ < h < 0; a < h h h h h h under condition (1.7).

We recall that the unperturbed system of (1.1) has three families of closed orbits near the double homoclinic loop L0 , which will be broken after small perturbation. Then it is easy to direct that a limit cycle can exist near the invariant curve L0 if the relevant succession function has an odd-order zero solution. By the theory of Melnikov function method, we can study the lower bound of the maximal number of limit cycles. Moreover, by [9,18] for system (1.1) there exist three Melnikov functions defined as follows

M1 ðh; dÞ ¼

R

~ dÞ ¼ M2 ðh;

R

Mðh; dÞ

¼

Lþ h L ~

qþ dx  pþ dyje¼0 ;

h 2 ðaþ ; 0Þ;

e 2 ða ; 0Þ; q dx  p dyje¼0 ; h h  R i R Hy ðBÞ qþ dx  pþ dyje¼0 þ b q dx  p dyje¼0 ; Hþ ðBÞ b

h Hþ y ðAÞ H y ðAÞ

AB

y

BA

ð1:8Þ h 2 ð0; bÞ:

For simplicity we assume that the Hamiltonian functions H are analytic functions on R2 whose series expansions near the origin are given by

Hþ ðx; yÞ ¼ 2k ðy2  x2 Þ þ

X

þ

k > 0;



l > 0:

hij xi yj ;

iþjP3

H ðx; yÞ ¼ l2 ðy2  x2 Þ þ

X

hij xi yj ;

ð1:9Þ

iþjP3

Further, for C

x

functions, we can write

p ðx; y; 0; dÞ ¼

X

aij xi yj ;

iþjP0

q ðx; y; 0; dÞ ¼

X



bij xi yj

ð1:10Þ

iþjP0

near the origin. To use the method of Melnikov functions, we need to derive those expansions. From the relevant references, it is direct to present the expansions of functions M 1 ðh; dÞ and M 2 ðh; dÞ, which are listed in Section 2. For 0 < h  b we have the following theorem.

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Theorem 1.1. The Melnikov function Mðh; dÞ in (1.8) has the following form

Mðh; dÞ ¼

 X 1 i r 3i þ r 3iþ1 h2 þ r3iþ2 h ln h h ;

0 < h  b;

ð1:11Þ

iP0

where

ri

¼ r þi þ d0 r i ;

ri

¼ r þi þ d0 r i þ d1 r i3 ;

i ¼ 0; 1; 2; i ¼ 3; 4; 5

 with d0 ; d1 ; r þ i defined in (2.8) and r i in (2.16) in Section 2. Then,we can obtain a lower bound for the maximal number of limit cycles bifurcating from the double homoclinic loop L0 after small perturbation of family (1.2) and (1.3) is obtained.

Theorem

1.2. Suppose

the

assumptions

ri ðd0 Þ ¼ 0; i ¼ 0; 1; . . . ; k; r kþ1 ðd0 Þ –

(HI)

and

@ðr 0 ;r 1 ;...;r k Þ 0; rank @ðd j 1 ;d2 ;...;dm Þ d¼d0

(HII)

hold.

If

there

exist

d0 2 Rm and

16k4

satisfying

¼ k þ 1 with d ¼ ðd1 ; d2 ; . . . ; dm Þ 2 Rm and one of the following

conditions with respect to the value of k holds:    1. For k ¼ 1; r  0 ðd0 Þ ¼ 0; r 2 ðd0 Þr 3 ðd0 Þr 5 ðd0 Þ – 0.



  2. For k ¼ 2; r  j ðd0 Þ ¼ 0; j ¼ 0; 2; r 3 ðd0 Þr 5 ðd0 Þ – 0; rank

@ðr  ;r Þ  0 2 @ðd1 ;d2 ;...;dm Þ d¼d0

¼ 2.

 3. For k ¼ 3 or 4; r  j ðd0 Þ ¼ 0; j ¼ 0; 2; 3; r 5 ðd0 Þ – 0; rank

@ðr  ;r  ;r  Þ  0 2 3 @ðd1 ;d2 ;...;dm Þ

¼ 3.



d¼d0

 Then for some ðe; dÞ near ð0; d0 Þ system (1.1) has 3k þ 1  2 4k limit cycles near L0 . At last we apply the obtained theoretical results to a particular system and give an example of a quartic system with 11 limit cycles bifurcated from a double homoclinic loop. 2. The proof of Theorem 1.1 By the results of [13] the function M 1 ðh; dÞ can be expanded as

M1 ðh; dÞ ¼

X i ðcþ2i þ cþ2iþ1 h ln jhjÞh ;

0 < h  aþ ;

ð2:1Þ

iP0

where in particular

cþ0 cþ1 cþ2 cþ3

¼

H

Lþ 0

qþ dx  pþ dyje¼0 ;

aþ þbþ 01 10

¼ k ; H þ þ ¼ Lþ ðpþx þ qþy  aþ10  b01 Þje¼0 dt þ b cþ1 ; 0

  þ  þ  þ þ þ þ  þ þ  þ c þ ; ¼  2k12 3aþ30  b21 þ aþ12 þ 3b03  1k ð2b02 þ aþ11 Þ 3h03  h21 þ 2aþ20 þ b11 3h30  h12 þb 1 þ

for some constants b and bþ with

 h  i þ ¼ 1 3 5hþ 2 þ 5hþ 2 þ hþ 2  hþ 2 þ 2hþ hþ  2hþ hþ  1 3hþ  hþ þ 3hþ : b 30 03 21 12 30 12 21 03 40 22 04 2 k2 4k Performing the change of variables

x ¼ z;

y ¼ y;

t ! t

and taking into account (1.9) and (1.10) we see that system (1.6) can be written as

e y þ ee z_ ¼ H p ðz; y; e; dÞ;

e z þ ee q ðz; y; e; dÞ; y_ ¼  H

ð2:2Þ

where for ðz; yÞ near the origin

e yÞ ¼ H ðz; yÞ ¼ l ðy2  z2 Þ þ Hðz; 2 e p ðz; y; 0; dÞ ¼ p ðz; y; 0; dÞ ¼

X

X

e zi yj ; h ij

iþjP3

e a ij zi yj ;

iþjP0

e q ðz; y; 0; dÞ ¼ q ðz; y; 0; dÞ ¼

X

iþjP0

e b ij zi yj

ð2:3Þ

Y. Liu, V.G. Romanovski / Applied Mathematics and Computation 248 (2014) 235–245

239

with

e h ij ¼

(



hij ; i even; j P 0; e a ij ¼  hij ; i odd; j P 0;

(

aij ; aij ;

i even; j P 0; i odd; j P 0;

e b ij ¼

(



bij ; i odd; j P 0;  bij ; i even; j P 0:

e dÞ of system (2.2) has the following expansion e h; Thus, similar to (2.1), we obtain that the Melnikov function Mð

X e ln j hjÞ e h ei; ðe c 2i þ e c 2iþ1 h

e dÞ ¼ e h; Mð

e  a ; 0 < h

ð2:4Þ

iP0

where, in particular,

e c1

¼

ea 10 þeb 01

; n  h     io e e 03  h e 21 þ 2 e e 30  h e 12 e b 21 þ e b 03 þ l1 ð2 e b 02 þ e b 11 3 h a 30  e a 12 þ 3 e a 11 Þ 3 h a 20 þ e ¼ 2l1 2 3 e c1; þb

e c3

l

with

e ¼ 1 b 2



 3  e2 e2 þ h e2  h e2 þ 2h e 30 h e 12  2 h e 21 h e 03  1 ð3 h e 40  e e h h : 5 h 30 þ 5 h þ 3 Þ 22 04 03 21 12 4l 2

l

e dÞ relating the values of h; e d it is direct to obtain that For M 2 ð h;

e dÞ ¼ M2 ð h;

X e ln j hjÞ e h ei; ðc2i þ c2iþ1 h

e  a ; 0 < h

ð2:5Þ

iP0

where

c1

¼

a þb 01 10

l

1

c3

¼ 2l 2

n

;

o               c ; 3a30  b21 þ a12 þ 3b03  l1 ð2b02 þ a11 Þ 3h03  h21 þ 2a20 þ b11 3h30  h12 þb 1

with

 ¼ 1 b 2



l

i 1   2   2   2 3 h   2        5 h30 þ 5 h03 þ h21  h12 þ 2h30 h12  2h21 h03  ð3h40  h22 þ 3h04 Þ : 4l 2

Further, using the method in [10] and Lemma 2.6 in [22], we have

c0 ¼

I L 0

q dx  p dyje¼0 ;

c2 ¼

I L 0





ðpx þ qy  a10  b01 Þje¼0 dt þ b c1



with a constant b . Next, we will study the expansion of function Mðh; dÞ. From the result in [17], under the hypotheses H(I) and H(II), we have

Hþy ðAÞ Hy ðBÞ ¼ 1: Hy ðAÞ Hþy ðBÞ Then, function (1.8) becomes

Mðh; dÞ ¼ Iþ ðh; dÞ þ

Hþy ðAÞ  ~ I ðh; dÞ; Hy ðAÞ

ð2:6Þ

where

R Iþ ðh; dÞ ¼ b qþ dx  pþ dy; AB R I ð e h; dÞ ¼ b q dx  p dy: BA By [17,22] we have Hþ y ðAðhÞÞ H y ðAðhÞÞ

¼

X i di h ;

0 < h  b;

iP0

Iþ ðh; dÞ ¼

X 1 i ðr þ3i þ r þ3iþ1 h2 þ rþ3iþ2 h ln hÞh ; iP0

ð2:7Þ 0 < h  b;

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Y. Liu, V.G. Romanovski / Applied Mathematics and Computation 248 (2014) 235–245

where 2

d0 rþ0 rþ4

2

3ðhþ Þ 3ðhþ Þ 3hþ h 2hþ 2h d1 ¼  403l  4l033  4k03l203  l204 þ l04 ; k pffiffiffi 1 þ þ þ þ þ þ þ þ ¼ c0 ; r 1 ¼ 2 2k 2 a00 ; r 3 ¼ c2 þ d c1 ; r3iþ2 ¼ cþ2iþ1 ; i pffiffi pffiffiffih pffiffiffi 5 þ 7  þ 2 5 þ 3 þ  þ cþ ¼ 2 2 5k2 h03  2k2 h04 aþ00  4 3 2 k2 2aþ20 þ b11  aþ02  4 2k2 h03 aþ01 þ d 1

¼ lk > 0;

ð2:8Þ

þ being some constants. with d ; d e dÞ and obtain the following statement. Then, we study the expansion of I ð h; þ

e dÞ in (2.6) can be written as Lemma 2.1. The function I ð h;

h; dÞ ¼ I ð e

X 1 e ln e ei; h 2 þ r 3iþ2 h hÞ h ðr 3i þ r3iþ1 e

0 < h  b;

ð2:9Þ

iP0

where

pffiffiffi 1  r0 ¼ c0 ; r 1 ¼ 2 2l2 a00 ; r 3 ¼ c2 þ d c1 ; r 3iþ2 ¼ c2iþ1 ; i ¼ 0; 1; h i p ffiffi pffiffiffi pffiffiffi 5  5  7 3  2   c r4 ¼ 2 2 2l2 h04  5l2 ðh03 Þ a00 þ 4 3 2 l2 2a20 þ b11  a02 þ 4 2l2 h03 a01 þ d 1  are some constants. and d ; d 

0 e 0 e Proof. Let x0 < 0 and jx0 j small. Then the line x ¼ x0 can intersect the clockwise orbit L ~ at points A ð hÞ ¼ ðx0 ; a ð hÞÞ and h 0 e 0 0 0 e  0 d 0 0 0 Sd 0 0 d d d d B ð hÞ ¼ ðx0 ; b ð hÞÞ, and the orbit Lh is divided into three parts, BB ; B A and A A. We let L1 ¼ BB A A; L2 ¼ B A , which implies S 0 e 00 e  00 e 0 e L1 Lh~ ¼ L1 L2 . Correspondingly, in system (2.2), denote by A ð hÞ the point ðx0 ; a ð hÞÞ, by B ð hÞ the point ðx0 ; b ð hÞÞ and by e 00 00 S d d the curve BB A A (see Fig. 2.1).

By Green’s formula, it follows that

h; dÞ ¼ I ð e ¼

Z L ~

Z

q dx  p dyje¼0 ¼

h

dx þ q

L1

Z L2

I

Z I Z dx þ ! p dy q q dx  p dy þ ! p dy ¼ ! ! b AB b AB BA[ AB BA[ AB

Z e dÞ þ I2 ð e e dÞ; dx þ ! p dy I1 ð h; h; dÞ þ I3 ð h; q

ð2:10Þ

AB

where I2 is C x on R2 and

ðx; y; 0; dÞ ¼ q ðx; y; 0; dÞ  q ðx; 0; 0; dÞ þ q

Z 0

y

px ðx; v ; 0; dÞdy

 ðx; 0; 0; dÞ ¼ 0 and q y ¼ q with q qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi y þ px . pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P Using the method of [22], we let v ¼ sgnðyÞ H ð0; yÞ, which yields that for jyj small v ¼ y l2 þ jP1 he 0j yj is C x and has pffiffiffi 1 2 2 the inverse yðv Þ ¼ 2l v þ Oðv Þ. Note that for jv j small,

h; dÞ ¼ I3 ð e

Z

b0 ðe hÞ

hÞ a0 ðe

Z ph~ X X e iþ12 ; e ai h p ð0; yðv Þ; 0; dÞdyðv Þ ¼ pffiffi ai v i dv ¼ ~ h

iP0

e  b; 0
iP0

a2i a i ¼  2iþ1 where e and

e Fig. 2.1. The orbit L ~ in system (1.4) and the orbit L 1 in system (2.2)je¼0 h

ð2:11Þ

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Y. Liu, V.G. Romanovski / Applied Mathematics and Computation 248 (2014) 235–245

pffiffiffi 1 pffiffiffi 7 pffiffiffi 5  pffiffiffi 3  2  2l 2 a00 ; a2 ¼ 3 2l2 ½5ðh03 Þ  2lh04 a00  6 2l2 h03 a01 þ 2 2l2 a02 ; pffiffi pffiffiffi 1 pffiffiffi 7 pffiffiffi 5  3  2  e a 1 ¼ 2 2l2 ½5ðh03 Þ  2lh04 a00 þ 4 2l2 h03 a01  4 3 2 l2 a02 : a 0 ¼  2l2 a00 ; e

a0 ¼

e dÞ in (2.10). Denote eI 1 ð h; e dÞ ¼ R~ e We now consider I1 ð h; qðz; y; 0; dÞ, where L1

e q ðz; y; 0; dÞ  e q ðz; 0; 0; dÞ þ q¼e

Z

y

0

e p z ðz; y; 0; dÞ:

Using the derivation of Iþ ðh; dÞ in [22] we obtain that

X

eI 1 ð e h; dÞ ¼

e et ij Iij ð hÞ;

e  b; 0
iþjP0

e ¼ Iij ð hÞ

R u0 0

e ui ðu2 þ hÞ

jþ12

du;

h  b; 0
where u0 ðx0 Þ; et ij are constants such that

et 00 et 01 et 20

     i pffiffiffi 5 h e 30  h e 12 e a 10 þ e b 01 ; et 10 ¼ 4 2l2 2 h b 01 þ l 2 e b 11 ; a 10 þ e a 20 þ e ¼ l4 e        2 e 03  2l h e 04 e b 01  8l3 e h 03 2 e b 02 þ e a 11 ; a 10 þ e ¼ 4l4 5 h h      i  n   e 12 2 e e 40 h 30  h b 11  e h 21 e b 02 þ l 3 e b 21  4l3 e b 01 2l e h 22  3 h a 20 þ e a 11 þ 2 e a 30 þ e a 10 þ e ¼ 8l3 3 e    2  2 e 21  3ð h e 30 Þ2  h e 03 h e 21 þ e e 30  h e 12 h 12 h þ3 h

Then, it follows that

e dÞ ¼ I1 ð h;

X

e t ij Iij ð hÞ;

ð2:12Þ

iþjP0

where

t00 t01 t20

pffiffiffi 5           ¼  l4 a10 þ b01 ; t10 ¼ 4 2l2 2h30  h12 a10 þ b01 þ l 2a20 þ b11 ; h  i  2       ¼ 4l4 5 h03  2lh04 a10 þ b01  8l3 h03 2b02 þ a11 ;               

 ¼ 8l3 3h30  h12 2a20 þ b11  h21 a11 þ 2b02 þ l 3a30 þ b21 þ 4l3 a10 þ b01 2l h22  3h40 h io   2  2      2 : þ3 h21  3 h30  h03 h21 þ h12 h30  ðh12 Þ

e for h e near 0. Recall that Next, we study the expansion of function Iij ð hÞ

Z

Z

ur ðu2 þ hÞ

kþ12

kþ12

du ¼

urþ1 ðu2 þ hÞ 2k þ r þ 2

þ

ð2k þ 1Þh 2k þ r þ 2

3

1

ur ðu2 þ hÞ2 du ¼

ur1 ðu2 þ hÞ2 ðr  1Þh  rþ2 rþ2

Z

Z

ur ðu2 þ hÞ

k12

du;

1

ur2 ðu2 þ hÞ2 du:

Thus, we obtain

   3    e ¼ u2 þ h e 2/ e ej e Iij h j1 h þ aij h Ii0 j 0

ð2:13Þ

e and a being a constant. Further, with /j1 being a polynomial of degree j  1 in h ij

    ½2i

e þb e e hÞ ¼ u½i1 h Ii0 ð e i h I½iþ1 ½ i ;0 h 2

2

2

e and b is a constant. where u½i1 is a polynomial in h i 2 Note that

Z

u0

1

ðu2 þ hÞ2 du ¼

0

Z 0

u0

1

1 1i u0 2 h h h ðu þ hÞ2 þ ln u0 þ ðu2 þ hÞ2  ln h; 2 4 2

uðu2 þ hÞ2 du ¼

i 3 3 1 h 2 u0 þ h 2  h2 : 3

e Substituting (2.14) into (2.13), we present for small u0 and h

ð2:14Þ

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Y. Liu, V.G. Romanovski / Applied Mathematics and Computation 248 (2014) 235–245

(

e ¼ Iij ð hÞ

e þm h e i þjþ1 ; i P 0 odd; j P 0; Iij ð hÞ ij 2 e þm h e 2i þjþ1 ln h; e i P 0 even; j P 0 I ð hÞ ij

where Iij 2 C

mij ¼

ij

e and is a family of polynomials in h

x

8 iþ1 > < ð1Þ 2 ði1Þ!!ð2jþ1Þ!! ;

i odd;

i > : ð1Þ2þ1 ði1Þ!!ð2jþ1Þ!! ;

i even:

ð2jþiþ2Þ!!

2ð2jþiþ2Þ!!

e dÞ has the expansion (2.9) for h e > 0 small. Further, by (2.5) Inserting (2.11) and (2.12) into (2.10) yields that the function Ið h; and [10,22], the coefficients ri ði ¼ 0; 1; . . . ; 5Þ have the form as (2.9). h From the results of [17] we have

e hðhÞ ¼

X i ki h ;

ð2:15Þ

iP1

where

k1 ¼

l k

;

k2 ¼

1



k2



h04 þ

3l  2k2

þ

h03

2

l



k

þ

h04 

 3 þ  h03 h03 : 2k

h; dÞ in (2.9) Then, it follows that for I ð e 

  X  1 i e d ¼ I hðhÞ; r3i þ r3iþ1 h2 þ r 3iþ2 h ln h h I ðh; dÞ;

0 < h  b;

ð2:16Þ

iP0

where

r0 r4

r 1 ¼ r 1

¼ r0 ;

pffiffiffiffiffi k1 ;

3 2

r 3 ¼ r 3 k1 þ r 2 k1 ln k1 ;

2

1 kffiffiffi 2ffi ¼ rp þ r 4 k1 ;

2

r 2 ¼ r 2 k1 ;

r5 ¼ r 2 k2 þ r 5 k1 :

k1

Inserting (2.7) and (2.16) into (2.6) we can obtain Theorem 1.1. 3. The proof of Theorem 1.2 e dÞ in (2.5) can be written as By (2.15), the function M 2 ð h;

  X e ^c2i þ ^c2iþ1 h ln jhj hi M2 ðh; dÞ; d ¼ M2 hðhÞ;

0 < h  a ;

ð3:1Þ

iP0

where

^c0 ¼ c0 ;

^c1 ¼ c1 k1 ;

^c2 ¼ c1 k1 ln k1 þ c2 k2 ;

^c3 ¼ c3 k22 þ c1 k2 :

By Lemma 2.1, (2.16) and (3.1), it follows that

^c0 ¼ r 0 ;

^c1 ¼ r 2 ;

^c2 ¼ r 3  d r 2 ;

^c3 ¼ r5



with d a constant. Taking into account that

cþ0 ¼ r þ0 ;

cþ1 ¼ r þ2 ;

þ

cþ2 ¼ r þ3  d r þ2 ;

cþ3 ¼ r þ5

for coefficients of M 1 ðh; dÞ and Iþ ðh; dÞ by (2.1), (1.11) and (3.1), we have the following lemma. Lemma 3.1. The expansions of functions M 1 ; M 2 and M can be written as

Mðh; dÞ

1

3

2

2

¼ r 0 þ r 1 h2 þ r 2 h ln h þ r3 h þ r 4 h2 þ r5 h ln h þ Oðh Þ; 0 < h  b;  þ 2 2 M1 ðh; dÞ ¼ r þ0 þ r þ2 h ln jhj þ rþ3  d r þ2 h þ r þ5 h ln jhj þ Oðh Þ; 0 < h  aþ ;   2 2 M2 ðh; dÞ ¼ r 0 þ r 2 h ln jhj þ r3  d r 2 h þ r 5 h ln jhj þ Oðh Þ; 0 < h  a ; 

where d are constants and

ri

¼ r þi þ d0 r i ;

ri

¼ r þi þ d0 r i þ d1 r i3 ;

d0 ; d1 ; r þ i

i ¼ 0; 1; 2; i ¼ 3; 4; 5

with defined in (2.7) and r  i in (2.16). We now can prove Theorem 1.2 and only consider the case of k ¼ 4 (the other cases are treated similarly). For d0 > 0 in Theorem 1.2, let d0 2 Rm be such that

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Y. Liu, V.G. Romanovski / Applied Mathematics and Computation 248 (2014) 235–245

r5 ðd0 Þ > 0; r i ðd0 Þ ¼ 0; i ¼ 0; 1; . . . ; 4; r5 ðd0 Þ > 0; r j ðd0 Þ ¼ 0; j ¼ 0; 2; 3; rank

@ðr 0 ; r 1 ; . . . ; r 4 Þ ¼5 @ðd1 ; d2 ; . . . ; dm Þ

and by the third condition

rank

@ðrþ0 ; r þ2 ; rþ3 Þ @ðr0 ; r 2 ; r3 Þ ¼ rank ¼ 3: @ðd1 ; d2 ; . . . ; dm Þ @ðd1 ; d2 ; . . . ; dm Þ

Since Lh near L0 with jhj small, we have

for 0 < h  1; for 0 < h  1;

2

h ln jhj > 0; h ln h < 0;

h ln jhj < 0; 2

h ln h < 0:

Thus, if we take the above coefficients satisfying

r1 ¼ r þj ¼ r j ¼ 0; j ¼ 0; 2; 3;

0 < r 4  1;

which yields that r 0 ¼ r 2 ¼ r3 ¼ 0, then function M has a positive zero h1 and M 1 and M 2 have no negative zeros. Further let  r1 ¼ rþ i ¼ r i ¼ 0; i ¼ 0; 2 and

0 < r þ3  1; 0 < r 3  1; 0 < r 3  r 4  1; which yields r0 ¼ r2 ¼ 0, so that M 1 and M 2 have two more negative zeros h2 and h3 and at the same time M has one more  positive zero h4 . Let r 1 ¼ r þ 0 ¼ r 0 ¼ 0 and

0 < r þ2  r þ3  1; 0 < r2  r3  1; 0 < r2  r 3  r 4  1; which implies r0 ¼ 0. Then, we obtain that M 1 and M 2 have two negative zeros h5 and h6 and M has a positive zero h7 . Fur ther, we let rþ 0 ¼ r 0 ¼ r 0 ¼ 0 and

0 < r þ2  r þ3  1; 0 < r2  r3  1; 0 < r1  r 2  r 3  r 4  1: which yields that M has one more positive zero þ þ   0 < rþ 0 < r 0  r 2  r 3  1 and 0  r 2  r 3  1;

h8

and

M1

and

M2

have

no

zeros.

Finally,

let

0 < r 0  r 1  r2  r 3  r 4  1: Then, the function M has a positive zero h9 and M 1 and M 2 have two zeros h10 and h11 . Hence, it is direct to see that Mðh; dÞ has 5 simple positive zeros, and M1 ðh; dÞ has 3 simple negative zeros and M 2 ðh; dÞ has 3 negative zeros near h ¼ 0. Thus, there is a system of the form (1.1) with 11 limit cycles near L0 . The proof of Theorem 1.2 is ended.

Fig. 4.1. The phase portrait of the unperturbed system of (4.1).

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Y. Liu, V.G. Romanovski / Applied Mathematics and Computation 248 (2014) 235–245

4. A quartic systems with 11 limit cycles Consider the system

(

8   3 _ > < x ¼ y þ eða0 þ a3 x Þ; 3 X x < 0:  i 2 > : y_ ¼ x þ x þ e bi x y;

x_ ¼ y þ eaþ2 x2 ; x P 0; þ þ þ y_ ¼ x  x2 þ eyðb0 þ b1 x þ b3 x3 Þ;

ð4:1Þ

i¼0

Clearly, in this case

Hþ ðx; yÞ ¼

1 2 1 ðy  x2 Þ þ x3 ; 2 3

H ðx; yÞ ¼

1 2 1 ðy  x2 Þ  x3 ; 2 3

h. The phase portrait of system (4.1)je¼0 is shown in Fig. 4.1. and k ¼ l ¼ 1 and h ¼ e By (1.4), Lemmas 2.1, 3.1 and Theorem 1.1, we have þ

rþ0

þ

þ

þ

þ

þ

þ

¼ 65 b0 þ 36 b þ 432 b ; r þ1 ¼ 0; r þ2 ¼ b0 ; r þ3 ¼ 12aþ2 þ 6b1 þ 36 b  db0 ; 35 1 385 3 5 3 pffiffi  pffiffi þ þ þ þ 4 2 5 þ ¼  3 2a2 þ b1  8 9 2 b0 ; r þ5 ¼ aþ2 þ 12 b0 þ 12 b1 ; pffiffiffi      ¼  65 b0 þ 36 b  36 b þ 432 b ; r1 ¼ 2 2a0 ; r 2 ¼ b0 ; 35 1 35 2 385 3 pffiffi pffiffi         5  ¼ 6b1  6b2 þ 36 b þ eb0 ; r 4 ¼ 4 3 2 b1 þ 8 9 2 b0 ; r 5 ¼  32 a3 þ 12 b0  12 b1  12 b2 ; 5 3

rþ4 r0 r3

r i ¼ r þi þ r i ;

i ¼ 0; 1; . . . ; 5:

Then, it follows that

0 A ¼@

B

@ðrþ ;rþ ;r þ Þ 0 2 5 þ þ þ ;b ;b ;b aþ 2 0 1 3

ð

36 35

B ¼@ 0

1

0

12 d 6 0 6 36  5 35      @ðr ;r 1 ;r 2 ;r 3 ;r 4 Þ B ¼ @ða ;a0 ;;b 0 þ    ¼ @ 1 ;b ;b ;b Þ 0

C

Þ

6 5

0

3

0

1

2

3

e

0 ;r 1 ;r 2 ;r3 ;r 4 Þ ¼ @ aþ ;bþ @ðr ð 0 ;bþ1 ;bþ3 ;a0 ;b0 ;b1 ;b2 ;b3 Þ 02 6 36 0 5 35 B B 0 0 0 B B 1 0 ¼B 0 B B 12 d 6 @ pffiffi pffiffi pffiffi  832  892  432

6

432 385

0 0

432 385

1

C 0 A;

36 5 36 35

0 6

432 385

1

C 0 A;

36 5

0  65 pffiffiffi 2 2 0 0 1

36 5

0

0

0

e

pffiffi 8 2 9

36 35

36 35

432 385

0 0

0 0

0 0

6 6 pffiffi 4 2 0 3

36 5

0

1 C C C C C; C C A

which yields that rank A ¼ rank B ¼ 3; rank C ¼ 5.   þ þ   þ þ 55 þ   For aþ 2 > 0; a3 < 0; b0 ¼ a0 ¼ b3 ¼ 0; b1 ¼ 20a2 ; b3 ¼  3 a2 ; b1 ¼ b2 ¼ 22a2 we obtain that

ri ¼ 0; i ¼ 0; 1; 2; 3; 4; r j ¼ 0; j ¼ 0; 2; 3 and r 5 > 0; r 5 > 0: By Theorem 1.2 we conclude that system (4.1) has 11 limit cycles near the double homoclinic loop L0 ¼ Lþ 0  limit cycles inside Lþ 0 , 3 limit cycles inside L0 and 5 limit cycles outside L0 .

S

L 0 , with 3

Acknowledgment The authors would like to give thanks to Prof. Maoan Han for his helpful discussions during the preparation of the paper and the referees for valuable comments and suggestions which help to improve the paper. 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] X. Chen, W. Zhang, Isochronicity of centers in a switching Bautin system, J. Differ. Equ. 252 (2012) 2877–2899. [3] B. Coll, A. Gasull, R. Prohens, Degenerate Hopf bifurcations in discontinuous planar systems, J. Math. Anal. Appl. 253 (2001) 671–690. [4] Z. Du, Y. Li, Bifurcation of periodic orbits with multiple crossings in a class of planar Filippov systems, Math. Comput. Model. 55 (2012) 1072–1082. [5] A.F. Filippov, Differential Equations with Discontinuous Righthand Sides, Kluwer Academic, Netherlands, 1988. [6] E. Freire, E. Ponce, J. Ros, The focus-center-limit cycle bifurcation in symmetric 3D piecewise linear systems, SIAM J. Appl. Math. 65 (2005) 1933–1951. [7] A. Gasull, J. Torregrosa, Center-focus problem for discontinuous planar differential equations, Int. J. Bifur. Chaos Appl. Sci. Eng. 13 (2003) 1755–1765. [8] M. Han, Asymptotic expansions of Melnikov function and limit cycle bifurcations, Int. J. Bifur. Chaos. 12 (2012) 1–30.

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