Echo waves and coexisting phenomena in coupled brusselators

Chaos, Solitons and Fractals 13 (2002) 621±632

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Echo waves and coexisting phenomena in coupled brusselators Tianshou Zhou *, Suochun Zhang Institute of Applied Mathematics, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing 100080, People's Republic of China Accepted 3 October 2000

Abstract In this paper we study the echo wave and coexisting phenomena in linearly coupled brusselators and put forward a new method to prove the existence of the echo wave. We ®nd that the existence of the echo wave is determined by that of the periodic solution of the associate oscillator. In addition, we also ®nd that the stable steady state may coexist with the echo wave and that the echo wave may coexist with the in-phase wave under suitable conditions, and specify the coexisting regimes on parameters. Ó 2001 Elsevier Science Ltd. All rights reserved.

1. Introduction Lefever, Nicolis and Prigogine [1,2] pointed out that the single brusselator …1 ‡ B†x ‡ x2 y;

x_ ˆ F1 …x; y†  A y_ ˆ G1 …x; y†  Bx

…1†

x2 y;

is one of the simplest but most fundamental models which displays biological and chemical oscillations, where A > 0; B > 0 and the derivatives are with respect to time t. The coupled brusselators are derived from a single oscillator (1) through a linear di€erence form and usually described by the following ordinary di€erential equations x_ 1 ˆ F1 …x1 ; y1 † ‡ D11 …x2 y_ 1 ˆ G1 …x1 ; y1 † ‡ D21 …x2

x1 † ‡ D12 …y2

y1 †;

x1 † ‡ D22 …y2

y1 †;

x_ 2 ˆ F1 …x2 ; y2 † ‡ D11 …x1 y_ 2 ˆ G1 …x2 ; y2 † ‡ D21 …x1

x2 † ‡ D12 …y1

y2 †;

x2 † ‡ D22 …y1

y2 †:

We let  D



D11 D21

D12 D22



be a nonzero coupling matrix.

*

Corresponding author. E-mail address: [email protected] (T. Zhou).

0960-0779/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 0 - 0 7 7 9 ( 0 0 ) 0 0 2 2 2 - 8

…2a† …2b†

622

T. Zhou, S. Zhang / Chaos, Solitons and Fractals 13 (2002) 621±632

For the sake of convenience, we begin by introducing several de®nitions. De®nition 1 (Single oscillator and associate oscillator). System (1) is called the single oscillator of the system ((2a), (2b)), while system x_ ˆ F1 …x; y† 2D11 x 2D12 y; y_ ˆ G1 …x; y† 2D21 x 2D22 y

…3†

is called the associate oscillator corresponding to system (1). De®nition 2 (In-phase wave and echo wave). If the system (2a) and (2b) has a solution of the form: x1 ˆ x2 ˆ /…t†; y1 ˆ y2 ˆ w…t† (where /…t† and w…t† are periodic functions with the same period T ), the solution is called an in-phase wave; if x1 ˆ /…t†; y1 ˆ w…t†; x2 ˆ /…t a† and y2 ˆ w…t a† (where a > 0 is a constant) is a periodic solution of system (2a, 2b), we call it an echo wave. In other words, the in-phase wave consists of two cells (one is system (2a) and the other system (2b)) which move around the same trajectory with the same phase, while the echo wave also consists of two cells moving in the same periodic orbit but with a ®xed nonzero phase di€erence. There have been many authors, such as Krinsky [3], Tyson [4], Aronson et al. [5] and Bar-Eli [6], who have investigated in-phase and echo waves (in spite of some di€erences in terminology). Some of them ®rst determined the approximate parameter regimes where echo waves or in-phase waves exist by approximate estimates and then tested them through numerical simulations. For instance, Krinsky [3] and Tyson [4] just took this line and studied the echo wave in coupled Hodgkin±Huxley equations and in coupled oregonators, respectively. Other authors, however, such as Aronson [5], investigated in-phase waves and echo waves by the perturbation method and assumed that the coupling coecients or matrix are weak. In this paper we put forward a new method (or concept) pertaining to in-phase waves and echo waves, which may be perspectively applied to other analogous systems. The main results we obtain are that we mathematically prove the existence of the echo wave in coupled brusselators and give the conditions that the stable steady state of system (2a, 2b) coexists with the echo wave and that the echo wave coexists with the inphase wave. Besides, we point out that the in-phase wave is made up of the periodic solution of the single oscillator, while the existence of the echo wave is determined by that of the associate oscillator. 2. The matching condition In this section we shall prove that if the system comprising (2a, 2b) has an echo wave with a minimum positive period T , then the phase di€erence between two cells in the echo wave must be T =2. In fact, let x1 ˆ /…t†; y1 ˆ w…t†; x2 ˆ /…t a† and y2 ˆ w…t a† be the echo wave of system (2a, 2b); then it follows from system (2a, 2b) that _ /…t _ w…t

a† ˆ F1 …/…t

a†; w…t

a†† ‡ D11 …/…t

2a†

/…t

a†† ‡ D12 …w…t

2a†

w…t

a††;

a† ˆ G1 …/…t

a†; w…t

a†† ‡ D21 …/…t

2a†

/…t

a†† ‡ D22 …w…t

2a†

w…t

a††;

_ /…t _ w…t

a† ˆ F1 …/…t

a†; w…t

a†† ‡ D11 …/…t†

/…t

a†† ‡ D12 …w…t†

w…t

a††;

a† ˆ G1 …/…t

a†; w…t

a†† ‡ D21 …/…t†

/…t

a†† ‡ D22 …w…t†

w…t

a††:

…4†

By subtraction of the third from the ®rst and the fourth from the second in the above equalities, respectively, one obtains D11 …/…t

2a†

/…t†† ‡ D12 …w…t

2a†

w…t†† ˆ 0;

D21 …/…t

2a†

/…t†† ‡ D22 …w…t

2a†

w…t†† ˆ 0:

…5†

 6ˆ 0 and det …D†  ˆ 0 and discuss in detail the second case Once we distinguish between the two cases det …D† with the solution …/…t†; w…t†; /…t a†; w…t a†† of system (2a, 2b), we can reduce /…t 2a† ˆ /…t† or w…t 2a† ˆ w…t†. Since T is the minimum period of /…t† or w…t†, a ˆ T =2…mod T †, or a ˆ 0.

T. Zhou, S. Zhang / Chaos, Solitons and Fractals 13 (2002) 621±632

623

Here we make a point: through the following discussions, one will see that the existence of the echo wave is determined by that of the periodic solution of the associate oscillator. However, in order to study the associate oscillator more conveniently (including its existence and uniqueness of limit cycle, see Section 3) and specify the coexisting regimes in parameters more brie¯y (see Section 5), we assume that the coupling is the case: D11 ˆ D > 0; D12 ˆ D21 ˆ D22 ˆ 0. For other coupling cases there is no intrinsically distinct dealing method. 3. Existence and uniqueness of limit cycle of associate oscillator (3) Note that system (3) has a unique positive steady state, denoted by P2‡ ˆ …A=…1 ‡ 2D†; B…1 ‡ 2D†=A†. Below, we shall prove that system (3) has a unique stable limit cycle in R2‡ ˆ f…x; y†jx > 0; y > 0g. Through some simple operations, one easily sees that the matrix of linearized equations of system (3) at P2‡ is 0  2 1 A B 1 2D 1‡2D B C M1 ˆ @   2 A; A B 1‡2D whose eigenpolynomial is "  2 A 2 k ‡ 1 ‡ 2D ‡ 1 ‡ 2D

# B k‡

A2 ˆ 0: 1 ‡ 2D

…6†

Following the procedure of proof in [1,2], one can prove that there exists a limit cycle in system (3) at 2 least when B > 1 ‡ 2D ‡ …A=…1 ‡ 2D†† . Now we prove that it is unique. For this, by introducing trans2 formations n ˆ x…1 ‡ 2D†=A 1, g ˆ x ‡ y A=…1 ‡ 2D† B…1 ‡ 2D†=A and s ˆ …1 ‡ n† At=…1 ‡ 2D†, we change system (4) into the canonical Lienard equations as follows: dn ˆ g F …n†; ds dg ˆ g…n†; ds where

…7†

"

A F …n† ˆ n3 1 ‡ 2D , …1 ‡ n†



B…1 ‡ 2D† A

 2A n2 1 ‡ 2D

B…1 ‡ 2D† A

A 1 ‡ 2D



A 1 ‡ 2D

2 ! # n

2

2

and g…n† ˆ n=…1 ‡ n† . Note that x > 0 implies n > 1. A series of facts below is easily veri®ed: "  2 #  2 A 1 ‡ 2D A 0 < 0 when B > 1 ‡ 2D ‡ B 1 2D ; F …0† ˆ 1 ‡ 2D A 1 ‡ 2D ng…n† > 0 for n 6ˆ 0 and n > 1; Z 1 G…n† ˆ g…n† dn ! ‡1 when n ! 0

d dn



F 0 …n† g…n†



" A 2…1 ‡ 2D† ‡ ‡ B ˆ 1 ‡ 2D An2

when n 6ˆ 0 and n >

1:

1 ‡ 0; 

A 1 ‡ 2D

#

2 1

2D

1 ‡ 2D A…1 ‡ n†

2

>0

624

T. Zhou, S. Zhang / Chaos, Solitons and Fractals 13 (2002) 621±632

According to a lemma in [8], one obtains that the multiplicity of the limit cycle of system (3) is less than one for B > 1 ‡ 2D ‡ …A=…1 ‡ 2D††2 . Combining it with the results mentioned above, one knows that the limit cycle of system (3) is existent, stable and unique. In addition, analogous to the proof in [7], one obtains that system (3) has no closed orbit, and singular point P2‡ is globally asymptotically stable when 2 B 6 1 ‡ 2D ‡ …A=…1 ‡ 2D†† .

4. Echo wave 4.1. Uniqueness of positive steady state of system (2a, 2b) Let S ‡ ˆ …P1‡ ; P1‡ †  …A; B=A; A; B=A†. Obviously, S ‡ is a positive steady state of system (2a, 2b). In order to prove that it is unique, one must ®rst assume that …x1 ; y1 ; x2 ; y2 † is a positive steady state of system (2a, 2b), and then it must satisfy F1 …x1 ; y1 † ‡ D…x2 F1 …x2 ; y2 † ‡ D…x1

x1 † ˆ 0; x2 † ˆ 0;

…8†

G1 …x1 ; y1 † ˆ G1 …x2 ; y2 † ˆ 0; x1 > 0; y1 > 0; x2 > 0; y2 > 0:

If x2 > x1 , then F1 …x1 ; y1 † < 0 and F1 …x2 ; y2 † > 0 from Eqs. (8). From F1 …x1 ; y1 † < 0 and G1 …x1 ; y1 † ˆ 0, one can deduce that x1 > A. However, x2 < A from F1 …x2 ; y2 † < 0 and G1 …x2 ; y2 † ˆ 0 is impossible because of the assumption x1 < x2 . For the same reason, x2 < x1 is also impossible. In view of this, the positive steady state of system (2a, 2b) is unique. 4.2. Existence of echo wave of system (2a, 2b) To prove the existence of the echo wave of system (2a, 2b), one may make use of the basic fact given below. For general coupled oscillators x_ 1 ˆ f …x1 ; x2 † ‡ D…x3 ‡ x1 †; x_ 2 ˆ g…x1 ; x2 †; x_ 3 ˆ f …x3 ; x4 † ‡ D…x1 ‡ x3 †; x_ 4 ˆ g…x3 ; x4 †; where functions f …x; y† and g…x; y† have the nature f … x; y† ˆ spectively, the single oscillator corresponding to (9) is x_ ˆ f …x; y†; y_ ˆ g…x; y†:

…9†

f …x; y† and g… x; y† ˆ

g…x; y†, re-

…10†

If system (10) has a unique periodic solution (or limit cycle), denoted by x ˆ /…t† and y ˆ w…t† with period T , then according to the known assumption, …/…t†; w…t†† and … /…t†; w…t†† are all periodic solutions of (10), and these two periodic solutions must be in the same periodic orbit with a ®xed nonzero phase difference. Through Fourier analyses, one may obtain that /…t† and w…t† have the following Fourier expansions, respectively,     ‡1 ‡1 X X 2p…2k 1†t 2p…2k 1†t /…t† ˆ i and w…t† ˆ i ; …11† a2k 1 exp b2k 1 exp T T 1 1 which have the nature /…t† ˆ /…t ‡ T =2† and w…t† ˆ w…t ‡ T =2†. Thus, it can be seen that system (9) has an echo wave of the form …/…t†; w…t†; /…t ‡ T =2†; w…t ‡ T =2††.

T. Zhou, S. Zhang / Chaos, Solitons and Fractals 13 (2002) 621±632

625

However, if f …x; y† or g…x; y† does not have such a symmetric nature about origin, one may still obtain the existence of the echo wave of system (9) by reducing it to a normal form in which all even terms are removed through a series of transformations. Below, we carry out this line concretely for system (2a, 2b). Let g1 ˆ x1 A; g2 ˆ y1 B=A; g3 ˆ x2 A and g4 ˆ y2 B=A. Then system (2a, 2b) is changed to g_ 1 ˆ …B g_ 2 ˆ

A2 g2

Bg1

g_ 3 ˆ …B g_ 4 ˆ

1†g1 ‡ A2 g2 ‡ 2Ag1 g2 ‡ 2Ag1 g2

B 2 g A 1

1†g3 ‡ A2 g4 ‡ 2Ag3 g4 ‡

Bg3

A2 g4

2Ag3 g4

B 2 g A 3

B 2 g ‡ g2 g21 ‡ D…g3 A 1

g1 †;

g2 g21 ; B 2 g ‡ g4 g23 ‡ D…g1 A 3

…12†

g3 †;

g4 g23 :

Denote M2 by matrix   B 1 A2 M2 ˆ ; B A2 whose eigenpolynomial k2 ‡ …1 ‡ A2

B†k ‡ A2 ˆ 0

has two roots, denoted by k ˆ a  ib, where a ˆ …B 2

1

…13† q 2 2 A2 †=2 and b ˆ ‰…1 ‡ A† BŠ…B …1 A† †=2.

2

Below, we assume …1 A† < B < …1 ‡ A† , so that b > 0. The eigenvector corresponding to eigenvalue k‡ is denoted by v‡ . Let P1 ˆ P2 ˆ …Re …v‡ †; Im …v‡ ††, block matrix 0 1 0 1 n1 g1   B g2 C B n2 C P1 0 C B C Pˆ and g ˆ P n; where g ˆ B @ g3 A and n ˆ @ n3 A: 0 P2 g4 n4 Then Eqs. (12) become 0 1 0 10 1 a b 0 0 n1 n_1 B n_ C B b a 0 0 CB n2 C D B 2Cˆ B CB C @ n_ A @ 0 0 a b A@ n3 A ‡ b ‰a…n3 3 0 0 b a n4 n_4 0 …0† 1 0 …0† 1 V31 …n1 ; n2 † V21 …n1 ; n2 † B …0† C B …0† C B V …n ; n † C B V …n ; n † C ‡ B 22…0† 1 2 C ‡ B 32…0† 1 2 C: @ V23 …n3 ; n4 † A @ V33 …n3 ; n4 † A …0† …0† V34 …n3 ; n4 † V24 …n3 ; n4 † …0†

0

n1 † ‡ b…n4

1 b B a‡1 C C n2 †ŠB @ b A a 1 …14†

…0†

Expressions of v‡ ; p; V2j …n; g† and V3j …n; g† …j ˆ 1; 2† are given in Appendix A. Referring to Appendix B, one can obtain the normal form of (12) or (2a, 2b) (given below) through successive transformations of the form n ˆ x ‡ U…k† …x†; 0

1 0 x_ 1 B x_ 2 C B B CˆB @ x_ 3 A @ x_ 4

a b 0 0

b a 0 0

0 0 a b

10 1 0 x1 B x2 C D 0C CB C ‡ ‰a…x3 b A@ x 3 A b a x4

0

x1 † ‡ b…x4

0

1

…15†

…k† 1 V2k‡1;1 b B C …k† C B a ‡ 1 C X B V2k‡1;2 B C; B C x2 †Š@ ‡ …k† C b A kP1 B @ V2k‡1;3 A …k† a 1 V 2k‡1;4

…16†

626

T. Zhou, S. Zhang / Chaos, Solitons and Fractals 13 (2002) 621±632

where 0

1 /…k† x1 B /…k† C B 2 C U…k† …x† ˆ B x…k† C; @ /x3 A

0

1 x1 B x2 C C xˆB @ x3 A ; x4

/…k† x4

…k†

…k† /…k† xj ˆ /xj …x1 ; x2 ; x3 ; x4 † …j ˆ 1; 2; 3; 4† are homogeneous polynomials of degree k, and V2k‡1;j ˆ …k†

V2k‡1;j …x1 ; x2 ; x3 ; x4 † …j ˆ 1; 2; 3; 4† are also homogeneous polynomials of degree 2k prove that …k†

…1† V2k‡1;j …x1 ; x2 ; x3 ; x4 † ˆ …k†

…k†

V2k‡1;j …x1 ; x2 ; x3 ; x4 †

1. Appendix C will

for 1 6 j 6 4;

…k†

…2† V2k‡1;j …x1 ; x2 ; x1 ; x2 † ˆ V2k‡1;j 2 …x1 ; x2 ; x1 ; x2 † for j ˆ 3; 4: …1†

To prove it, we specially give expressions of V3;j …x1 ; x2 ; x3 ; x4 † …j ˆ 1; 2†. In order to ®nd out the relationship between Eqs. (16) and the following single oscillator x_ ˆ …B y_ ˆ

Bx

1

2D†x ‡ A2 y 2 ‡ 2Axy ‡ A2 y 2

2Axy

B 2 x A

B 2 x ‡ yx2 ; A

…17†

yx2 ;

one needs to consider the following equations 0 1   X a…1 2D†x ‡ b…1 2D†y x_ A‡ ˆ @ 2Da…a‡1† b2 y_ x ‡ ‰a ‡ 2D…a ‡ 1†Šy b kP1

! …k† V2k‡1;1 …x; y; x; y† : …k† V2k‡1;2 …x; y; x; y†

…18†

Since only linear terms are added by the coupling and since these are unchanged by transformations like the form (15), single oscillator (17) may be changed into a normal form in which all even terms disappear (the procedure for obtaining the normal form in the two-dimensional case is given in [9]). Eqs. (18) may be viewed as a normal form of system (3) or (17) (but there is a little di€erence between them because a normal form of a system is usually a canonical form at the neighborhood of a singular point of the system and is not unique). Since system (3) has a unique limit cycle under B > 1 ‡ 2D ‡ 2 …A=…1 ‡ 2D†† , system (18) still has a unique limit cycle under the same condition. It follows from Eqs. (18) that, if …/…t†; w…t†† is a periodic solution of Eqs. (16) with period T , so is … /…t†; w…t††. Analogous to the discussions in Section 1, /…t† and w…t† should satisfy     T T / t‡ ˆ /…t† and w t ‡ ˆ w…t†: …19† 2 2 Therefore, system (16) has echo wave …/…t†; w…t†; /…t ‡ T =2†; w…t ‡ T =2††, and this implies that system (2a, 2b) has also an echo wave under conditions: max……1 A†2 ; 1 ‡ 2D ‡ …A=…1 ‡ 2D††2 † < B < …1 ‡ A†2 (where condition …1 A†2 < B < …1 ‡ A†2 is to make the normal form procedure feasible, while condition 1 ‡ 2D ‡ …A=…1 ‡ 2D††2 < B guarantees the existence of the limit cycle). 5. Coexisting phenomena By examining the conditions of stability of the positive steady state of system (2a, 2b) and those of existence of the limit cycle of the associate oscillator, one easily ®nds that the stable positive steady state may coexist with the echo wave in coupled brusselators. This is a very interesting phenomenon. Tyson [4] ever conjectured that this case is also there in coupled oregonators.

T. Zhou, S. Zhang / Chaos, Solitons and Fractals 13 (2002) 621±632

627

Below, we further specify the conditions. Since S ‡ ˆ …A; B=A; A; B=A† is a unique positive steady state of system (2a, 2b), its matrix of the linearized equations at S ‡ is 0 B M ˆB @

B

1 D B D 0

A2 A2 0 0

B

D 0 1 D B

1 0 0 C C; A2 A A2

the roots of whose eigenpolynomial are q B 1 A2  …B 1 A2 †2 4A2 k1;2 ˆ ; 2 q 2 2 B 1 A2 2D  …B 1 A2 2D† 4A2 …1 ‡ 2D† k3;4 ˆ : 2

…20†

From system (20), S ‡ is stable when B < 1 ‡ A2 and unstable when B > 1 ‡ A2 . However, one knows that 2 2 2 system (2a, 2b) has an echo wave when max……1 A† ; 1 ‡ 2D ‡ …A=…1 ‡ 2D†† † < B < …1 ‡ A† from the above. Therefore, there may be a common regime between these parameter conditions. Here, the phenomenon of coexisting stable positive steady state and echo wave occurs. Concretely speaking, if 2 2 max……1 A† ; 1 ‡ 2D ‡ …A=…1 ‡ 2D†† < B < 1 ‡ A2 , then the stable steady state coexists with the echo wave. But the condition implies that parameters A and D must satisfy extra conditions. In fact, let x ˆ 1 ‡ 2D; then D > 0 implies x > 1. De®ne H …x†  x3 …1 ‡ A2 †x2 ‡ A2 . From H 0 …x† ˆ 0, one obtains x ˆ 0 or x ˆ …2=3†…1 ‡ A2 †. Since H …0† ˆ A2 > 0, H ……2=3†…1 ‡ A2 †† ˆ …4=27†…1 ‡ A2 †3 ‡ A2 < 0, H …1† ˆ 0; limx!1 H …x† ˆ 1. There must be a root, denoted by x0 , between p xˆ  0 and x ˆ 1 or between x ˆ 1 and x ˆ ‡1. Since x > 1 is needed, …2=3†…1 ‡ A2 † > 1 gives A > 1= 2. At that moment, for 0 < x < x0 ; H …x† < 0, which gives 0 < D < …x0 1†=2 , where x0 is the positive root greater than one of equation x3 …1 ‡ A2 †x2 ‡ A2 ˆ 0. To sum p up, one obtains the coexisting conditions of the stable positive steady state with the echo wave: A > 1= 2; 0 < D < …x0 1†=2 and max…1 ‡ 2D ‡ …A=…1 ‡ 2D††2 ; …1 A†2 † < B < 1 ‡ A2 . In addition, it is easy for one to prove the existence of the in-phase wave of system (2a, 2b). In fact, Lefever et al. [1,2] proved that the single oscillator (1) has a stable limit cycle when B > 1 ‡ A2 . Furthermore, Qin and Zeng [7] proved that the limit cycle is unique and globally asymptotically stable for B > 1 ‡ A2 and that the unique singular point P1‡ ˆ …A; B=A† of system (1) is globally asymptotically stable, and hence system (1) has no closed orbit when B 6 1 ‡ A2 . Let x ˆ /…t† and y ˆ w…t† be the periodic solution of system (1). Obviously, x1 ˆ x2 ˆ /…t† and y1 ˆ y2 ˆ w…t† is a periodic solution of system (2a, 2b) for any D > 0. This shows that when B > 1 ‡ A2 there always exists an in-phase wave for all values D > 0 in the coupled brusselators. Therefore, one may similarly give the coexisting conditions of the echo wave with the in-phase wave: max…1 ‡ 2D ‡ …A=…1 ‡ 2D††2 ; 1 ‡ A2 † < B < …1 ‡ A†2 .

6. Conclusion Through the discussion above, one sees that there is always an in-phase wave for any D > 0 when 2 2 2 B > 1 ‡ A2 and that there is an echo wave when max……1 A† ; 1 ‡ 2D ‡ …A=…1 ‡ 2D†† † < B < …1 ‡ A† . Through numerical simulations, one ®nds that the in-phase wave and the echo wave seem stable and unique in their existing regimes. We conjecture that this is correct (of course, it needs to be proved). Now admitting the conclusion, one accordingly concludes that the in-phase wave of system (2a, 2b) is determined by the single oscillator and the existence of the echo wave by that of the periodic solution of its associate oscillator. Thus, one practically ®nds a mechanism through which one easily obtains in-phase waves and echo waves in coupled brusselators. For other linearly coupled models [3±6], one may still obtain the same conclusions as in coupled brusselators. Detailed discussions will be given in another paper.

628

T. Zhou, S. Zhang / Chaos, Solitons and Fractals 13 (2002) 621±632

Acknowledgements The paper is partly supported by the National Natural Scienti®c Foundation (19671089). …0†

…0†

Appendix A. Expressions of v‡ , P, V2k and V3k …k ˆ 1; 2; 3; 4†  v‡ ˆ

 a ib ; a ‡ 1 ‡ ib

0

a Ba ‡ 1 P ˆB @ 0 0

b b 0 0

0 0 a 1‡a

1 0 0 C C; bA b

…0†

4a…1 ‡ a†An21 4b…1 ‡ 2a†An1 n2 ‡ 4b2 An22 ;    2a aB 2 …0† …1 ‡ a†A ‡ 2a…1 ‡ a† V22 …n1 ; n2 † ˆ n1 ‡ 2 4a…1 ‡ a†A b 2A

V21 …n1 ; n2 † ˆ

…0†

…0†

…0†

…0†

…0†

…0†

  aB n1 n2 ‡ b …4a ‡ 2†A A

 B 2 n; A 2

V23 …n3 ; n4 † ˆ V21 …n1 ; n2 †; V24 …n3 ; n4 † ˆ V22 …n3 ; n4 †; V33 …n3 ; n4 † ˆ V31 …n1 ; n2 † ˆ 0; …0†

V32 …n1 ; n2 † ˆ …0†

1 2 ‰a …1 ‡ a†n31 ‡ ab…3a ‡ 1†n21 n2 ‡ b2 …3a ‡ 1†n1 n22 ‡ b3 n32 Š; b …0†

V34 …n3 ; n4 † ˆ V32 …n3 ; n4 †:

Appendix B In this appendix we give the general procedure for reducing the coupled system to normal form. Let …0† …0† …0† X_ ˆ V1 …X † ‡ V2 …X † ‡ V3 …X † ‡


;

…B:1†

where 0

…0†

V1

1 ax1 ‡ bx2 B bx1 ‡ ax2 C C ˆB @ ax3 ‡ bx4 A bx3 ‡ ax4

…0†

and Vk …X † are k-order homogeneous polynomials …k ˆ 2; 3; . . .†. Consider the transformation X 0 ˆ X ‡ U…k† …X †

…B:2†

with inverse X ˆ X0

U…k† …X 0 † ‡ O…2k

1†;

…B:3†

where U…k† …X † is a homogeneous polynomial of degree k, and O…2k 1† is a formal sum of some homogeneous polynomials of degree more than 2k 1. By substituting (B.3) into (B.1) and leaving out the primes, one obtains …0† X_ ˆ V1 …X †

…0†

…0†

L‰U…k† …X †Š ‡ V2 …X † ‡


‡ Vk …X † ‡


;

…B:4†

T. Zhou, S. Zhang / Chaos, Solitons and Fractals 13 (2002) 621±632

629

where …0†

…0†

L‰U…k† …X †Š ˆ rV1 …X †U…k† …X †

rU…k† …X †V1 …X †

…B:5†

(r is the derivative operator). To remove all the terms of degree k one must solve …0†

L‰U…k† …X †Š ˆ Vk …X †:

…B:6†

If L is invertible, (B.6) will give …0†

U…k† …X † ˆ L 1 ‰Vk …X †Š:

…B:7†

The next step is to ®nd out under what condition L is invertible. Firstly, L may be viewed as a ®nitedimensional matrix, and it is possible to delete all the terms of degree k if and only if det L 6ˆ 0. Secondly, let us assume the form of U…k† to be 0 …k† 1 Ux1 …x1 ; x2 ; x3 ; x4 † B U…k† …x ; x ; x ; x † C B 2 1 2 3 4 C U…k† …X † ˆ B x…k† C; @ Ux3 …x1 ; x2 ; x3 ; x4 † A U…k† x4 …x1 ; x2 ; x3 ; x4 † then 0

a B b …k† L‰U …x1 ; x2 ; x3 ; x4 †Š ˆ B @ 0 0

b a 0 0

0 0 a b

10

1 /…k† x1 0 B /…k† C 0C C CB B x2 C b A@ /…k† A x3 …k† a / x4

0

…k† 1

oUx

B ox1 B oU…k† B x2 B ox1 B oU…k† B x3 B ox1 @ …k† oUx

4

ox1

…k† 1

…k† 1

…k† 1

oUx

oUx

oUx

ox2 …k† oUx

ox3 …k† oUx

ox4 …k† oUx

ox2 …k† oUx

ox3 …k† oUx

ox2 …k† oUx

ox3 …k† oUx

ox2

ox3

2

3

4

2

3

4

1

C0 C 2 C B ox4 CB …k† C@ oUx C 3 ox4 C A …k† oUx

1 ax1 ‡ bx2 bx1 ‡ ax2 C C: ax3 ‡ bx4 A bx3 ‡ ax4

4

ox4

…B:8† For the sake of convenience, let z1 ˆ x1 ‡ ix2 and z2 ˆ x3 ‡ ix4 (where i ˆ as follows: 0 1 0 1 U…k† a ib 0 0 0 B z1 C C B 0 CB U…k† a ‡ ib 0 0 …k† C B z1 C L‰U …z1 ; z1 ; z2 ; z2 †Š ˆ B …k† B C @ 0 A 0 a ib 0 @ U z2 A …k† 0 0 0 a ‡ ib Uz2 1 0 …k†

oUz

B oz11 B …k† B oUz1 B oz 1 B …k† B oUz2 B B oz1 @ oU…k† z2

…k†

…k†

oUz1 oz1 …k† oUz

oUz1 oz2 …k† oUz

oz1 …k† oUz2

oz2 …k† oUz2

1

oz1 …k† oUz 2

oz1

oz1

1

oz2 …k† oUz 2

oz2

p 1). Then (B.8) is rewritten

…k†

oUz1 oz2 …k† oUz

C0 1 C …a ib†z1 C 1 CB z1 C oz2 CB …a ‡ ib† C: …k† C@ A oUz2 C …a ib†z2 oz2 C …a ‡ ib†z2 …k† oU A

…B:9†

z2

oz2

Denote 0

…k;l†

g1;‡

1 zl1zk1 l B 0 C C ˆB @ 0 A; 0

0

…k;l†

g1;

1 0 B zl zk l C 1 1 C ˆB @ 0 A; 0

…k;l†

g2;‡

1 0 B 0 C C ˆB @ zl zk l A; 2 2 0

0

0

…k;l†

g2;

1 0 B 0 C C ˆB @ 0 A; zl2zk2 l

…B:10†

630

T. Zhou, S. Zhang / Chaos, Solitons and Fractals 13 (2002) 621±632 …k;l†

where l ˆ 0; 1; 2; . . . ; k. It is easy for one to see that all gj; …j ˆ 1; 2† are linear independent eigenvectors of …k;l† L, and the eigenvalues corresponding to them, denoted by kj; , in turn satisfy …k;l†

…k;l† …k;l†

L‰gj; Š ˆ kj; gj; ;

…B:11†

where …k;l†

kj; ˆ …2

k†a

ib…k

2l  1†:

…B:12†

At the Hopf bifurcation a ˆ 0 and b 6ˆ 0, and it reduces k ˆ 2l  1. This means that L is invertible if and only if k is even. By applying the coordinate transformations successively, one ®nally obtains the normal form of system (B.1) with all even nonlinear terms removed, i.e., Eqs. (16). Appendix C In this appendix we give the concrete expressions of U…2† …x† in the above-mentioned normal form procedure and prove that …k†

V2k‡1;j …x1 ; x2 ; x3 ; x4 † ˆ …k†

…k†

V2k‡1;j …x1 ; x2 ; x3 ; x4 † …j ˆ 1; 2; 3; 4†;

…k†

V2k‡1;j …x1 ; x2 ; x1 ; x2 † ˆ V2k‡1;j 2 …x1 ; x2 ; x1 ; x2 †

…C:1†

…j ˆ 3; 4†:

…C:2†

Here, we omit writing other expressions of U…k† …x†, because they are complicated. Let us assume the form of U…2† to be 0 …2† 1 0 …2† 1 Ux1 …x1 ; x2 ; x3 ; x4 † U1 B …2† C B U…2† …x ; x ; x ; x † C BU C B 2 1 2 3 4 C U…2† …X † ˆ B 2…2† C ˆ B x…2† …C:3† C: @ U3 A @ Ux3 …x1 ; x2 ; x3 ; x4 † A …2† …x1 ; x2 ; x3 ; x4 † Ux…2† U4 4 To remove the second-order terms, one must make 0 …2† …2† …2† oU1 oU1 oU1 0 10 …2† 1 ox ox ox 1 2 3 B …2† U1 a b 0 0 …2† …2† B oU2 oU2 oU2 …2† C B B b a 0 0 CB ox1 ox2 ox3 B CB U 2 C C B oU …2† …2† …2† @ 0 0 a b AB oU3 oU3 @ U3…2† A B 3 B ox1 ox2 ox3 @ …2† 0 0 b a …2† …2† …2† U4 oU oU oU 4

ox1

4

4

ox2

ox3

…2†

oU1 ox4 …2† oU2 ox4 …2† oU3 ox4 …2† oU4 ox4

1

0 C C CB CB C@ C A

1 ax1 ‡ bx2 bx1 ‡ ax2 C C ax3 ‡ bx4 A bx3 ‡ ax4

1 …0† …2† …2† …2† …2† …x ; x † D‰a…/ / † ‡ b…/ / †Š V 1 2 3 1 4 2 C B …0†21 B V …x ; x † ‡ D…1‡a† ‰a…/…2† /…2† † ‡ b…/…2† /…2† †Š C C B 22 1 2 3 1 4 2 b ˆB C …2† …2† …2† C B V21…0† …x3 ; x4 † ‡ D‰a…/…2† / † ‡ b…/ / †Š 3 1 4 2 A @ …0† …2† …2† …2† …2† D…a‡1† V22 …x3 ; x4 † ‰a…/3 /1 † ‡ b…/4 /2 †Š b 0

…C:4†

…2†

In order to ®nd out the expressions of Uk …k ˆ 1; 2; 3; 4†, generally one can assume …2†

…j†

…j†

…j†

…j†

…j†

…j†

…j†

…j†

Uj …x1 ; x2 ; x3 ; x4 † ˆ a1 x21 ‡ a2 x22 ‡ a3 x23 ‡ a4 x24 ‡ 2a5 x1 x2 ‡ 2a6 x1 x3 ‡ 2a7 x1 x4 ‡ 2a8 x2 x3 …j†

…j†

‡ 2a9 x2 x4 ‡ 2a10 x3 x4 :

…C:5† …j†

By substitution of (C.5) into (C.4), one can obtain a group of equations for ak (1 6 j 6 10; k ˆ 1; 2; 3; 4) …0† …0† …2† and then solve them. However, since V21 …x; y† and V22 …x; y† determine the form of /k …k ˆ 1; 2†, one can assume that

T. Zhou, S. Zhang / Chaos, Solitons and Fractals 13 (2002) 621±632 …2†

…1†

…1†

…1†

…1†

…1†

631

…1†

/1 …x1 ; x2 ; x3 ; x4 † ˆ a1 x21 ‡ a2 x22 ‡ a3 x23 ‡ a4 x24 ‡ 2a5 x1 x2 ‡ 2a6 x3 x4 : …2†

…2†

…2†

…C:6†

…2†

…2†

Because of the symmetry between U1 and U3 and between U2 and U4 , one may let U3 …x1 ; x2 ; x3 ; x4 † ˆ …2† …2† …2† U1 …x3 ; x4 ; x1 ; x2 † and U4 …x1 ; x2 ; x3 ; x4 † ˆ U2 …x3 ; x4 ; x1 ; x2 † and thus greatly decrease the number of equations. At this moment, the following equalities are obtained by balancing the coecients of various powers x1 or x2 from Eqs. (C.4): …1†

…1†

…1†

…1†

…1†

D†a1 ‡ Dba3 ˆ

…1†

D†a2 ‡ Dba4 ˆ 4b2 A;

a…1 ‡ D†a1 ‡ Daa3 ‡ 2ba5 ‡ b…1 a…1 ‡ D†a2 ‡ Daa4

…2†

…2†

…2†

…1†

a…1 ‡ D†a3 ‡ 2ba6 ‡ Dba1 ‡ b…1

…1†

a…1 ‡ D†a4

Daa1 Daa2

…1†

…1†

2ba5 ‡ b…1

…2†

…1†

…1†

ba1 ‡ ba2 …1†

…1†

…2†

D†a3 ˆ 0;

…1†

…2†

D†a4 ˆ 0;

2ba6 ‡ Dba2 ‡ b…1

…1†

…1†

aDa5 ‡ Daa6 ‡ b…1

…1†

…1†

…1†

4a…1 ‡ a†A;

…2† …2†

…2†

…2†

D†a5 ‡ bDa6 ˆ

…2†

2b…2a ‡ 1†A;

…2†

aa6 ‡ bDa5 ‡ b…1 D†a6 ˆ 0; ba3 ‡ aDa5 ‡ ba4   Da…1 ‡ a† Da…1 ‡ a† …1† …1† …2† …2† …2† a3 ‡ ‰D…1 ‡ a† aŠa1 D…1 ‡ a†a4 2ba5 b a1 b b   a aB 2…1 ‡ a†A ‡ 4a…1 ‡ a† ˆ ; b A   Da…1 ‡ a† Da…1 ‡ a† …1† …1† …2† …2† …2† a4 ‡ ‰D…1 ‡ a† aŠa2 D…1 ‡ a†a4 2ba5 b a2 b b   B ˆ b 2…1 ‡ a†A ; A   Da…1 ‡ a† …1† Da…1 ‡ a† …1† …2† …2† …2† a1 ‡ D…1 ‡ a†a1 ‰D…1 ‡ a† aŠa3 ‡ 2ba6 ˆ 0; b a3 b b   Da…1 ‡ a† …1† Da…1 ‡ a† …1† …2† …2† …2† a2 ‡ D…1 ‡ a†a2 ‡ ‰D…1 ‡ a† aŠa4 2ba6 ˆ 0; b a4 b b   Da…1 ‡ a† Da…1 ‡ a† …1† aB …1† …2† …2† …2† a6 ‡ ba2 ‡ ‰D…1 ‡ a† aŠa5 ; D…1 ‡ a†a6 ˆ 4a…1 ‡ a†A b a5 b b A   Da…1 ‡ a† …1† Da…1 ‡ a† …1† …2† …2† …2† …2† a5 ‡ ba3 ‡ ba4 D…1 ‡ a†a5 ‡ ‰D…1 ‡ a† aŠa6 ˆ 0: b a5 b b …j†

By solving them one obtains the expressions of ak (j ˆ 1; 2; 1 6 k 6 6). Because they are complicated, we omit writing them. In the procedure of removing the higher-order even terms, similar treatment is followed. In order to prove (C.1) and (C.2), one has to examine the normal form procedure in Appendix A carefully. For instance, after removing the second-order terms, all the third-order terms are, respectively, …1†

…2†

…2†

…2†

…2†

4a…1 ‡ a†Ax1 /1 4b…1 ‡ 2a†A…x1 /2 ‡ x2 /1 † ‡ 4b2 Ax2 /2 ;     a aB aB …1† …2† …2† …2† 2…1 ‡ a†A ‡ 4a…1 ‡ a† V3;2 …x1 ; x2 ; x3 ; x4 † ˆ x1 /1 ‡ 2 4a…1 ‡ a†A …x1 /2 ‡ x2 /1 † b A A   B …2† …0† ‡ b 2…1 ‡ 2a†A x2 /2 ‡ V3;2 …x1 ; x2 †; A

V3;1 …x1 ; x2 ; x3 ; x4 † ˆ

…1†

…1†

V3;3 …x1 ; x2 ; x3 ; x4 † ˆ V3;1 …x3 ; x4 ; x1 ; x2 †

and

…1†

…1†

V3;4 …x1 ; x2 ; x3 ; x4 † ˆ V3;2 …x3 ; x4 ; x1 ; x2 †;

and they obviously satisfy (C.1) and (C.2). For the higher-order case, the same conclusion holds.

632

T. Zhou, S. Zhang / Chaos, Solitons and Fractals 13 (2002) 621±632

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