Center manifolds and normal forms for a class of retarded functional differential equations with parameter associated with Fold-Hopf singularity

Applied Mathematics and Computation 181 (2006) 220–246 www.elsevier.com/locate/amc

Center manifolds and normal forms for a class of retarded functional differential equations with parameter associated with Fold-Hopf singularity R. Qesmi a b

a,*

, M. Ait Babram b, M.L. Hbid

a

De´partement de Mathe´matiques, Faculte´ des Sciences Semlalia, Universite´ Cadi Ayyad, B.P. S15, Marrakech, Morocco De´partement de Mathe´matiques, Faculte´ des Sciences et te´chniques Gue´liz, Universite´ Cadi Ayyad, Marrakech, Morocco

Abstract In this paper, we present explicit formulas for computing the coefficients of a center manifolds for the Fold-Hopf singularity in autonomous retarded functional differential equations. As consequence, normal forms associated with the flow on a center manifold up to an arbitrary order are derived. The explicit formulas have been implemented using the computer algebra system Maple. We apply our results to a delayed system in order to show the applicability of the methodology.  2006 Elsevier Inc. All rights reserved.

1. Introduction The dynamics at the onset of several instabilities in a physical system undergoing a bifurcation near an equilibrium point can often be reduced to a simple of ordinary differential equations by the application of both center manifolds and normal forms theories. References on the center manifolds and normal forms may be found, for example in [3,8]. Here the case of retarded functional differential equations (RFDEs) with parameters is considered following the computation method of center manifolds we developed in [12] for RFDEs associated with Hopf singularity without parameters. In the past few years, symbolic computations using computer languages such as Maple, Mathematica, and Macsysma have been introduced in computing center manifolds. However, it seems that even with a symbolic manipulator, the computation of center manifolds is still limited to lower-order approximation, since executing such a symbolic program usually quickly runs out of computer memory as the order of center manifolds increases. thus, computationally efficients methodologies and symbolic computer programs need to be developed, in particular, for computing higher-order center manifolds.

*

Corresponding author. E-mail address: [email protected] (R. Qesmi).

0096-3003/$ - see front matter  2006 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2006.01.030

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221

The center manifolds and normal forms of Fold-Hopf bifurcation has been discussed in [10,13] for ordinary differential equations. In [5,6], Faria and Magalhae`s have considered the computation of coefficients of normal forms for RFDEs of both Hopf and Bogdanov singularities. However, it is difficult to apply the method to compute the explicit expressions for these coefficients since it demands much more computation efforts for high-order normal forms. In [1,2], M. Ait Babram et al have considered RFDEs without parameters, and they derive an initial value problem (IVP) in finite dimension to compute the terms of center manifolds. However, the ingredients of the obtained (IVP) have not an explicit form, and the technicalities used to compute the solution of the (IVP) are theoretical and can not be programmed on machine. The attention of this paper will be focused on the development of methodology and software for computing the center manifolds of FoldHopf bifurcation for RFDEs with parameters. According to the structure of linearized equation of a retarded system evaluated at an equilibrium, the case considered in this paper corresponds to a pair of purely imaginary and a simple zero eigenvalues (Fold-Hopf), while the case studied in [12] for FDEs without parameters is only for one pair of purely imaginary eigenvalues (Hopf). As an important consequence, we obtain, for the general situation of bifurcation of Fold-Hopf for RFDEs, explicit formulas giving the coefficients of normal forms in terms of the coefficients of the original equation. For the notation background about the theory of RFDEs and all needed results in the remainder of this paper, we follow [9], as recalled in Section 2 of [12], but we use C n ¼ Cð½r; 0; Rn Þ; r P 0 since we need to work in realization spaces with different dimensions, depending on whether the parameters are incorporated or not incorporated in the realization space variables. The paper is organized as follows: A theorem of characterization of a center manifolds for RFDEs with parameters and its proof are given in Section 2, as consequence, a computational schemes of a center manifold and normal forms associated with Fold-Hopf are presented in the same section. Section 3 outlines the symbolic computation procedure using Maple, and an illustrative example is given in Section 4 to demonstrate the applicability of the obtained results. Conclusions are drawn in Section 5. The results of the examples including the center manifolds and normal forms obtained by executing the Maple programs are listed in the appendices. 2. Center manifolds for FDEs with parameters and main results In this section we present our main results concerning the computation of the terms of center manifolds and normal forms for the next class of RFDEs m X d xðtÞ ¼ Lj ðaÞxðt  rj Þ þ f ðxðtÞ; xðt  r1 Þ; . . . ; xðt  rm Þ; aÞ; dt j¼0

ð2:1Þ

where a 2 Rp , a # Lj(a) are a C1 functions with values in the space of square matrices of order n. f : Rnm  Rp ! Rn is assumed to be sufficiently smooth (f 2 C1) such that: f(0, . . . , 0, a) = 0 and Df(0, . . . , 0, a) = 0 for all a 2 Rp , r0 = 0 and rj > 0 for all 0 < j 6 m. If we denote r :¼ max16i6m{ri}, C n :¼ Cð½r; 0; Rn Þ the space of continuous functions from [r, 0] to Rn , and for / 2 Cn LðaÞ/ :¼ L0 ðaÞ/ð0Þ þ

m X

Lj ðaÞ/ðrj Þ; gð/; aÞ :¼ f ð/ð0Þ; /ðr1 Þ; . . . ; /ðrm Þ; aÞ

j¼1

then Eq. (2.1) reads as equation x_ ðtÞ ¼ LðaÞxt þ F ðxt ; aÞ.

ð2:2Þ

We denote L0 = L(0). In the sequel of this paper we assume that the following hypothesis is satisfied: (H) The linear equation x_ ðtÞ ¼ L0 xt has a pure imaginary pair (±ix) and a simple zero (k = 0) as characteristic values and no other characteristic values with zero real part.

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One way of considering center manifolds for a differential equations with parameters is to reduce the situation to the case of differential equations without parameters by considering the system x_ ðtÞ ¼ L0 xt þ ½LðaÞ  L0 xt þ F ðxt ; aÞ; _ ¼ 0. aðtÞ

ð2:3Þ ð2:4Þ

e :¼ C nþp , and the The solutions of this system are of the form ~xðtÞ :¼ ðxðtÞ; aðtÞÞ 2 Rnþp , the phase space is C system can be written ~x_ ðtÞ ¼ e L~xt þ Fe ð~xt Þ;

ð2:5Þ

where e Lðu; vÞ :¼ ðL0 u; 0Þ and Fe ðu; vÞ :¼ ð½Lðvð0ÞÞ  L0 u þ F ðu; vð0ÞÞ; 0Þ with u 2 Cn, v 2 Cp. It is now possible to apply to (2.5) the center manifold theory as introduced in [12]. e denote the infinitesimal generators associated with the equations x_ ðtÞ ¼ L0 xt , and x_ ðtÞ ¼ e Let A and A Lxt , _ ¼ 0, has a unique characteristic value k = 0, with multiplicity p, and respectively. The equation in Cp, aðtÞ the associated generalized eigenspace consists of the elements of Cp which are constant functions, and it is denoted here also by Rp . Let K :¼ rðAÞ \ iR and consider the decomposition Cn = Xc  Xs obtained as in Section 2 of [12]. In particular, we consider bases for Xc and X c denoted by U = (/1, . . . , /2), W = col(w1, . . . , w2), W e ¼ K f0g, X e s ¼ X s  R where R = e c ¼ X c  Rp ; X respectively, and satisfying (W, U) = I. We define K  e and the rows e e {v 2 Cp:v(0) = 0}, and consider for bases of X c and X c , respectively, the columns of the matrix U e of the matrix W, e ¼ U



U

0

0

Ip

 ;

e ¼ W



W

0

0

Ip

 ;

e  C e defined as in Section 2 of [12], and e Ui e ¼ I nþp , where hÆ , Æi is the bilinear form in C which satisfies h W; _e eB e with U¼U 2

0

0

0

3

6 0 e¼6 B 6 4 0

ix 0

0 0

0 0

7 7 7. 5

0

0

0

0p

ix

e ¼X e associated with K. e ec  X e s , where X e c is the invariant space of A Then we have the decomposition C In the sequel, we recall the definition of a local center manifold associated to Eq. (2.5). e s , the graph of ~h is said to be a local center manifold Definition 2.1 [9]. Given a C1 map ~ h from R3þp into X ~ ~ associated to Eq. (2.5) if and only if hð0Þ ¼ 0, Dhð0Þ ¼ 0, and there exists a neighborhood V of zero in R3þp such that, for each n 2 V, there exist d = d(n) > 0 and a function x defined on ]dr,d[ such that e þ~ x0 ¼ Un hðnÞ and x verifies Eq. (2.5) on ]d,d[ and satisfies the identity e xt ¼ UzðtÞ þ~ hðzðtÞÞ for t 2 ½0; d½; where z(t) is the unique solution of the ordinary differential equation ( d e e e zðtÞ ¼ BzðtÞ þ Wð0Þ Fe ð UzðtÞ þ~ hðzðtÞÞÞ; dt zð0Þ ¼ n;

n 2 R3þp .

ð2:6Þ

Remark 2.2. (i) If we write ~ hðn; aÞ ¼ ðhðn; aÞ; h0 ðn; aÞÞ for n 2 R3 and a 2 Rp , then Eq. (2.6) is equivalent to     BzðtÞ d zðtÞ ¼ þ dt a 0

Wð0Þ½ðLðað0Þ þ h0 ðzðtÞ; aÞð0ÞÞ  L0 ÞðUzðtÞ þ hðzðtÞ;aÞÞ þ F ðUzðtÞ þ hðzðtÞ;aÞ; að0Þ þ h0 ðzðtÞ;aÞð0ÞÞ 0

!

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223

2

3 ix 0 0 with B ¼ 4 0 ix 0 5 and zð0Þ ¼ n 2 R3 . Noting that h0(z(t), a)(0) = 0, because h0(z(t), a) 2 R, and drop0 0 0 ping the auxiliary equations introduced for handling the parameter, we get the reduced equation on the center manifold—represented as a graph over the n and a variables of h(n, a) for n and a sufficiently smalls—associated with the parametrized Eq. (2.1) as ( d zðtÞ ¼ BzðtÞ þ Wð0Þ½ðLðaÞ  L0 ÞðUzðtÞ þ hðzðtÞ; aÞÞ þ F ðUzðtÞ þ hðzðtÞ; aÞ; aÞ; dt zð0Þ ¼ n;

n 2 R3 .

(ii) It is noted that the invariance properties of center manifolds guarantee that any small solutions bifurcating from (0, 0, 0) must lie in any center manifold and thus we may follow the local evolution of bifurcating families of solutions in this suspended family of center manifolds, (see [13] for more details). 2.1. Characterization of a local center manifolds for the RFDEs In the following, we give an analytic characterization of a center manifold associated to Eq. (2.1). Theorem 2.3. Given a C1 map h from R3þp into Xs with h(0) = 0 and Dh(0) = 0, a necessary condition for the graph of h to be a local center manifold of Eq. (2.1) is that there exists a neighborhood V of zero in R3þp such that, for each (n, a) 2 V o ohðn; aÞ ohðn; aÞ ohðn; aÞ ðhðn; aÞÞðhÞ ¼ ix ðhÞWð0Þ½ðLðaÞ  Lð0ÞÞðUn þ hðn; aÞ ðhÞn1 þ ix ðhÞn2 þ oh on1 on2 on þ

ohðn; aÞ ðhÞWð0ÞF ðUn þ hðn; aÞ; aÞ þ UðhÞWð0Þ½ðLðaÞ  Lð0ÞÞðUn þ hðn; aÞ on

þ UðhÞWð0ÞF ðUn þ hðn; aÞ; aÞ o ðhðn; aÞÞð0Þ ¼ L0 hðn; aÞ þ ðLðaÞ  L0 ÞðUn þ hðn; aÞÞ þ F ðUn þ hðn; aÞ; aÞ. oh

ð2:7Þ ð2:8Þ

Proof. Let h be a graph of a local center manifold of Eq. (2.1), then from Definition 2.1 there exist a neighborhood V of zero in R3þp such that, for each (n, a) 2 V, there exist d > 0 such that the solution of (2.1) with initial data Un + h(n, a) exists on the interval ]dr,d[ and it is given by xt ¼ UzðtÞ þ hðzðtÞ; aÞ for t 2 d; d½; such that z(t) is solution of the equation ( d zðtÞ ¼ BzðtÞ þ Wð0Þ½ðLðaÞ  L0 ÞðUzðtÞ þ hðzðtÞ; aÞÞ þ F ðUzðtÞ þ hðzðtÞ; aÞ; aÞ; dt zð0Þ ¼ n; n 2 R3 ; 3 2 ix 0 0 with B ¼ 4 0 ix 0 5. 0 0 0 The variation of constants formula of Eq. (2.1) can be written as Z t xt ¼ T ðtÞ/ þ T ðt  sÞX 0 ½ðLðaÞ  L0 Þxs þ F ðxs ; aÞ ds; t P 0; 0

where (T(t))tP0 is the semi group solution of the linear equation x_ ðtÞ ¼ L0 xt . It follow from the decomposition of the phase space by K : Cn = Xc  Xs that the function h satisfies:

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hðzðtÞ; aÞ ¼ T ðtÞhðzðtÞ; aÞ þ

Z

t

T ðt  rÞX s0 F ðUzðrÞ þ hðzðrÞ; aÞ; aÞ dr. 0

Then 1 1 1 ½T ðtÞhðn; aÞ  hðn; aÞ ¼ ½hðzðtÞ; aÞ  hðzð0Þ; aÞ  t t t

Z

t

T ðt  rÞX s0 ½ðLðaÞ  L0 ÞðUzðrÞ þ hðzðrÞ; aÞÞ 0

þ F ðUzðrÞ þ hðzðrÞ; aÞ; aÞ dr; which implies from the fact that h and F are smooth and T(.) is a strongly continuous semi group on the Banach space Cn, that h(n) is in the domain of A, and   ohðn; aÞ Ahðn; aÞ ¼ fBn þ Wð0Þ½ðLðaÞ  L0 ÞðUn þ hðn; aÞÞ þ F ðUn þ hðn; aÞ; aÞg on ¼ X s0 ½ðLðaÞ  L0 ÞðUn þ hðn; aÞÞ þ F ðUn þ hðn; aÞ; aÞ Consequently, we have by evaluating the above equation at h 5 0 that   o ohðn; aÞ ðhðn; aÞÞðhÞ ¼ ðhÞ fBn þ Wð0Þ½ðLðaÞ  L0 ÞðUn þ hðn; aÞÞ þ F ðUn þ hðn; aÞ; aÞg oh on þ UðhÞWð0Þ½ðLðaÞ  L0 ÞðUn þ hðn; aÞÞ þ F ðUn þ hðn; aÞ; aÞ;

ð2:9Þ

which is the formula (2.7) of theorem. On the other hand, it result from the fact that the semi-flow t#xt = Uz(t) + h(z(t), a) exists on the open ]d, d[ that for h 2 ]d, 0] d d d ðUðhÞn þ hðn; aÞðhÞÞ ¼ x0 ðhÞ ¼ xðhÞ ¼ LðaÞðUzðhÞ þ hðzðhÞ; aÞÞ þ F ðUzðhÞ þ hðzðhÞ; aÞ; aÞ. dh dh dh Consequently, by the fact that

d Uð0Þn dh

¼ Uð0ÞBn ¼ L0 ðUnÞ, we obtain

o ðhðn; aÞÞð0Þ ¼ L0 hðnÞ þ ðLðaÞ  L0 ÞðUn þ hðn; aÞÞ þ F ðUn þ hðn; aÞ; aÞ; oh

ð2:10Þ

which achieve the proof of theorem. h 2.2. The computational scheme Let us recall that the function h which represent the center manifold for Eq. (2.1) has the same regularity as the nonlinearity F. From this fact and in view of the assumed smoothness on F, for all m 2 N, we can write m X hðn; aÞ ¼ hk ðn; aÞ þ vðn; aÞ for n 2 V ; k¼2

where hk is the homogeneous part of degree k and v (n, a) = o(j(n, a)jm). Let k 2 N, k P 2. The homogeneous parts of degree k of Eqs. (2.7) and (2.8) are respectively given by ohk ðn; aÞ ohk ðn; aÞ ohk ðn; aÞ ¼ ix n1 þ ix n2 þ Nk1 ðn; aÞ; oh on1 on2 o ðhk ðn; aÞÞð0Þ ¼ L0 hk ðn; aÞ þ Rk1 ðn; aÞ; oh where Nk1 ðn; aÞ ¼ UWð0ÞRk1 ðn; aÞ þ

k1 X ohkjþ1 ðn; aÞWð0ÞRj1 on j¼2

ð2:11Þ ð2:12Þ

R. Qesmi et al. / Applied Mathematics and Computation 181 (2006) 220–246

225

and Ri1 is the homogeneous part of degree i of R(n , a) = (L(a)  L0)(Un + h(n, a)) + F(Un + h(n, a), a). In particular, R1 is the homogeneous part of degree 2 of (L(a)  L0)Un + F(Un, a) which is independent from terms of a center manifolds. l If P n ¼ ðn1 ; n2 ; n3 Þ 2 R3 , q = (q1, q2, q3), a = (a1, . . . , ap), al ¼ al11 al22    app for l ¼ ðl1 ; . . . ; lp Þ 2 Np , and p jlj ¼ i¼1 li , then we can write X q q q hk ðn; aÞ ¼ hkðq;lÞ n11 n22 n33 al for some hkðq;lÞ 2 X s ; ðq;lÞ2Dk k1

N

X

ðn; aÞðhÞ ¼

q

q

q

3 l 1 2 N k1 ðq;lÞ n1 n2 n3 a

for some N kðq;lÞ 2 X s

ð2:13Þ

ðq;lÞ2Dk

and Rk1 ðn; aÞ ¼

X

q

q

q

n for some Rk1 ðq;lÞ 2 R ;

3 l 1 2 Rk1 ðq;lÞ n1 n2 n3 a

ð2:14Þ

ðq;lÞ2Dk

where Dk ¼ fðq; lÞ 2 N3  Np : jðq; lÞj ¼ kg. Theorem 2.4. Assume that (H) holds. Then the vector of the coefficients of the homogeneous part of degree k of a local center manifold associated with Eq. (2.1) is given in a unique way by the following recursive formulas: For k = 2, (q1, q2, q3, l) 2 D2: If (q1  q2  1)(q1  q2 + 1)(q1  q2) 5 0 then h2ðq1 ;q2 ;q3 ;lÞ ðhÞ ¼ eixðq1 q2 Þh h2ðq1 ;q2 ;q3 ;lÞ ð0Þ þ



1 ðeixh  eixðq1 q2 Þh Þ/1 ð0Þw1 ð0Þ ixðq1  q2  1Þ

1 ðeixh  eixðq1 q2 Þh Þ/2 ð0Þw2 ð0Þ ixðq1  q2 þ 1Þ  1 ð1  eixðq1 q2 Þh Þ/3 ð0Þw3 ð0Þ R1ðq1 ;q2 ;q3 ;lÞ . þ ixðq1  q2 Þ þ

ð2:15Þ

If q1  q2  1 = 0 then h2ðq1 ;q2 ;q3 ;lÞ ðhÞ

eixh h2ðq1 ;q2 ;q3 ;lÞ ð0Þ



þ heixh /1 ð0Þw1 ð0Þ þ  1 ixh þ ð1  e Þ/3 ð0Þw3 ð0Þ R1ðq1 ;q2 ;q3 ;lÞ . ix

¼

sinðxhÞ ixh ðe  eixh Þ/2 ð0Þw2 ð0Þ x ð2:16Þ

If q1  q2 + 1 = 0 then 

sinðxhÞ /1 ð0Þw1 ð0Þ þ ðheixh Þ/2 ð0Þw2 ð0Þ x  1 þ ð1  eixh Þ/3 ð0Þw3 ð0Þ R1ðq1 ;q2 ;q3 ;lÞ . ix

h2ðq1 ;q2 ;q3 ;lÞ ðhÞ ¼ eixh h2ðq1 ;q2 ;q3 ;lÞ ð0Þ þ

ð2:17Þ

If q1  q2 = 0 then h2ðq1 ;q2 ;q3 ;lÞ ðhÞ



1 ðeixh  1Þ/1 ð0Þw1 ð0Þ þ h/3 ð0Þw3 ð0Þ ix  1 ixh þ ðe  1Þ/2 ð0Þw2 ð0Þ R1ðq1 ;q2 ;q3 ;lÞ . ix

¼

h2ðq1 ;q2 ;q3 ;lÞ ð0Þ

þ

ð2:18Þ

For k > 2 and (q1, q2, q3, l) 2 Dk: hkðq1 ;q2 ;q3 ;lÞ ðhÞ ¼ eixðq1 q2 Þh hkðq1 ;q2 ;q3 ;lÞ ð0Þ þ

Z

h 0

eixðq1 q2 ÞðhsÞ N k1 ðq1 ;q2 ;q3 ;lÞ ðsÞ ds;

ð2:19Þ

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R. Qesmi et al. / Applied Mathematics and Computation 181 (2006) 220–246

where the vectors hkðq1 ;q2 ;q3 ;lÞ ð0Þ; ðq1 ; q2 ; q3 ; lÞ 2 Dk are given by solving the following systems: For (q1  q2  1) · (q1  q2 + 1)(q1  q2) 5 0 hkðq1 ;q2 ;lÞ ð0Þ ¼ ½Dðixðq1  q2 ÞÞ1 Ek1 ðq1 ;q2 ;lÞ

ð2:20Þ

for q1 = q2 + 1: M1

hkðq1 ;q2 ;q3 ;lÞ ð0Þ

! ¼

0

Ek1 ðq1 ;q2 ;q3 ;lÞ

! ð2:21Þ

vk1 ðq1 ;q2 ;q3 ;lÞ

for q1 = q2  1 M1

hkðq1 ;q2 ;q3 ;lÞ ð0Þ

! ¼

0

Ek1 ðq1 ;q2 ;q3 ;lÞ

! ð2:22Þ

vk1 ðq1 ;q2 ;q3 ;lÞ

and for q1 = q2 M2

hkðq1 ;q2 ;q3 ;lÞ ð0Þ 0

! ¼

Ek1 ðq1 ;q2 ;q3 ;lÞ

! ð2:23Þ

.

vk1 ðq1 ;q2 ;q3 ;lÞ

With M1 and M2 are the (n + 1) · (n + 1) matrices defined by ! DðxiÞ w> 1 ð0Þ M1 ¼ hw ; exi. i 0 1

and M2 ¼

! Dð0Þ w> 3 ð0Þ ; hw3 ; I Rn i 0

k1 and the second members Ek1 ðq1 ;q2 ;q3 ;lÞ and vðq1 ;q2 ;q3 ;lÞ are a vector given by means of the coefficients of the center manifolds already computed (see (2.27) and (2.29) in the proof below).

Proof. Let (q1, q2, q3, l) 2 Dk, from relation (2.11), we have ohkðq1 ;q2 ;q3 ;lÞ ðhÞ oh

¼ ixðq1  q2 Þhkðq1 ;q2 ;q3 ;lÞ ðhÞ þ N k1 ðq1 ;q2 ;q3 ;lÞ ðhÞ;

ð2:24Þ

or equivalently hkðq1 ;q2 ;q3 ;lÞ ðhÞ ¼ eixðq1 q2 Þh hkðq1 ;q2 ;q3 ;lÞ ð0Þ þ

Z 0

h

eixðq1 q2 ÞðhsÞ N k1 ðq1 ;q2 ;q3 ;lÞ ðsÞ ds

ð2:25Þ

However, for k = 2, we have that N1 ðn; aÞðhÞ ¼ UðhÞWð0ÞR1 ðn; aÞ ¼ ½eixh /1 ð0Þw1 ð0Þ þ eixh /2 ð0Þw2 ð0Þ þ /3 ð0Þw3 ð0ÞR1 ðn; aÞ; it follow that: Z Z h eixðq1 q2 ÞðhsÞ N 1ðq1 ;q2 ;q3 ;lÞ ðsÞ ds ¼ eixðq1 q2 Þh 0



Z

0

h

h

eixðq1 q2 1Þs ds/1 ð0Þw1 ð0Þþ

 eixðq1 q2 Þs ds/3 ð0Þw3 ð0ÞÞ R1 ðn; aÞ;

Z

h

eixðq1 q2 þ1Þs ds/2 ð0Þw2 ð0Þ

0

0

which implies, by discussing the cases whether q1  q2 2 {1, 0, 1} or not, the value of h2ðq1 ;q2 ;q3 ;lÞ ðhÞ given in theorem.

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227

The relation (2.12) is equivalent to ohkðq1 ;q2 ;q3 ;lÞ oh

ð0Þ ¼ L0 hkðq1 ;q2 ;q3 ;lÞ þ Rk1 ðq1 ;q2 ;q3 ;lÞ ;

which give by evaluating the relation (2.24) at h = 0 that k k1 L0 hkðq1 ;q2 ;q3 ;lÞ þ Rk1 ðq1 ;q2 ;q3 ;lÞ ¼ ixðq1  q2 Þhðq1 ;q2 ;q3 ;lÞ ð0Þ þ N ðq1 ;q2 ;q3 ;lÞ ð0Þ:

On the other hand, by taking the values of the linear operator L0 at the both hand sides of relation (2.25), we have Z .  0 k ixðq1 q2 Þ. k 0 ixðq1 q2 Þð.sÞ k1 L hðq1 ;q2 ;q3 ;lÞ ¼ L0 ðe Þhðq1 ;q2 ;q3 ;lÞ ð0Þ þ L e N ðq1 ;q2 ;q3 ;lÞ ðsÞ ds . 0

Consequently, we obtain from the expression of the characteristic equation that Dðixðq1  q2 ÞÞhkðq1 ;q2 ;q3 ;lÞ ð0Þ ¼ Ek1 ðq1 ;q2 ;q3 ;lÞ where Ek1 ðq1 ;q2 ;q3 ;lÞ

¼

Rk1 ðq1 ;q2 ;q3 ;lÞ

þL

0

Z . e

for ðq1 ; q2 ; lÞ 2 Dk ;

ixðq1 q2 Þð.sÞ

0

N k1 ðq1 ;q2 ;q3 ;lÞ ðsÞ ds



 N k1 ðq1 ;q2 ;q3 ;lÞ ð0Þ.

ð2:26Þ

ð2:27Þ

We will use also the fact that all center manifolds has rank in the subspace Xs, which implies in particular that hw1 ; hkðq1 ;q2 ;q3 ;lÞ ð.Þi ¼ 0 for all (q1, q2, q3, l) 2 Dk and k > 1. Consequently, we obtain from relation (2.25) that hw ; eixðq1 q2 Þ. ihk ð0Þ ¼ vk1 for ðq ; q ; q ; lÞ 2 D ; ð2:28Þ 1

ðq1 ;q2 ;q3 ;lÞ

1

2

3

k

is given by Z . ¼  w1 ; eixðq1 q2 Þð.sÞ N k1 ðsÞ ds . ðq1 ;q2 ;q3 ;lÞ

where the vector vk1 ðq1 ;q2 ;q3 ;lÞ

ðq1 ;q2 ;q3 ;lÞ

vk1 ðq1 ;q2 ;q3 ;lÞ 

0

We introduce the (n + 1) · (n + 1) matrices M1 and M2 defined by ! DðixÞ w> ð0Þ 1 M1 ¼ hw1 ; eix. i 0 and M2 ¼

Dð0Þ hw3 ; I Rn i

w> 3 ð0Þ 0

!

then it follows from the systems (2.26) and (2.28) that the vectors defined by: ! k h ð0Þ ðq1 ;q2 ;q3 ;lÞ ^ hkðq1 ;q2 ;q3 ;lÞ ð0Þ ¼ 0 satisfy M 1^ hkðq1 ;q2 ;q3 ;lÞ ð0Þ ¼

Ek1 ðq1 ;q2 ;q3 ;lÞ vk1 ðq1 ;q2 ;q3 ;lÞ

for q1 = q2 + 1, M 1^ hkðq1 ;q2 ;q3 ;lÞ ð0Þ for q1 = q2  1, and

¼

!

Ek1 ðq1 ;q2 ;q3 ;lÞ vk1 ðq1 ;q2 ;q3 ;lÞ

!

ð2:29Þ

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M 2^ hkðq1 ;q2 ;q3 ;lÞ ð0Þ for q1 = q2.

¼

Ek1 ðq1 ;q2 ;q3 ;lÞ

!

vk1 ðq1 ;q2 ;q3 ;lÞ

h

The uniqueness of hkðq1 ;q2 ;q3 ;lÞ ð0Þ is a consequence of the next lemma: Lemma. The (n + 1) · (n + 1) matrices M1 and M2 are invertibles. Proof. Let x 2 Rn , g 2 R such that   x M1 ¼ 0; g then (

DðixÞx þ gw> 1 ð0Þ ¼ 0; ix. hw ; e ix ¼ 0; 1

then from the fact that w1(0)D(ix) = 0, we have g = 0, and the above system becomes  DðixÞx ¼ 0; hw ; eix. ix ¼ 0. 1

From the fact that dim ker D(ix) = 1 and since we have D(ix)/1(0) = 0, then the first equation implies that x 2 span{/1(0)}, it follows that there exist c 2 R such that x = c/1(0), and from the second equation we have chw1,/1i = 0 which yields c = 0 and finally we obtain x = 0. Consequently M1 is invertible. By the same manner as before, we prove that M2 is invertible. Which completes the proof of theorem. h 2.3. Normal forms Having computed the center manifolds of Eq. (2.1), we obtain the associated reduced equation as follow: d zðtÞ ¼ BzðtÞ þ H ðzðtÞ; aÞ; dt 0 1 ix 0 0 where B ¼ @ 0 ix 0 A; z 2 R3 ; a 2 Rp and 0 0 0

ð2:30Þ

H ðz; aÞ :¼ Wð0Þ½ðLðaÞ  L0 ÞðUz þ hðz; aÞÞ þ F ðUz þ hðz; aÞ; aÞ. It is now easy to compute its normal forms up to a desired order. The basic idea of normal form theory consists of employing successive, near identity nonlinear transformations to eliminate the so-called non-resonant nonlinear terms, and the terms called resonant which cannot be eliminated are remained in normal forms. Assume that by a nonlinear transformation z ¼ u þ T ðu; aÞ;

u 2 R3 ;

a 2 Rp ;

ð2:31Þ

the above system is transformed to its normal form d uðtÞ ¼ BuðtÞ þ N ðuðtÞ; aÞ. dt

ð2:32Þ

Then if we replace (2.31) and (2.32) in Eq. (2.30) we obtain the following formula: DT ðu; aÞBu  BT ðu; aÞ ¼ H ðu þ T ðu; aÞ; aÞ  DT ðu; aÞN ðu; aÞ  N ðu; aÞ.

ð2:33Þ

In view of the regularity of h, we restrict our attention to the terms of degree lower than m for each m P 2.

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229

Let k X X

T ðu; aÞ ¼

q

q

q

T jðq;lÞ u11 u22 u33 al ;

N ðu; aÞ ¼

j¼2 ðq;lÞ2Dj

T j;1 ðq;lÞ

1

B j;2 C C T jðq;lÞ ¼ B @ T ðq;lÞ A; T j;3 ðq;lÞ

q

q

q

N jðq;lÞ u11 u22 u33 al ;

j¼2 ðq;lÞ2Dj

H ðu þ T ðu; aÞ; aÞ  DT ðu; aÞN ðu; aÞ ¼ 0

k X X

0

N j;1 ðq;lÞ

X

k X

1

q

j¼2 ðq;lÞ2Dj

B j;2 C C N jðq;lÞ ¼ B @ N ðq;lÞ A and N j;3 ðq;lÞ

q

q

C jðq;lÞ u11 u22 u33 al ; 0

C j;1 ðq;lÞ

1

B j;2 C C C jðq;lÞ ¼ B @ C ðq;lÞ A. C j;3 ðq;lÞ

Then from the fact that DT ðu; aÞBu ¼ ix

k X X oT ðu; aÞ oT ðu; aÞ q q q u1 þ ix u2 ¼ ixðq1  q2 ÞT jðq;lÞ u11 u22 u33 al ou1 ou2 j¼2 ðq;lÞ2D j

and

0

k P

P

q1 q2 q3 l T j;1 ðq;lÞ u1 u2 u3 a

B ix B j¼2 ðq;lÞ2Dj B B k BT ðu; aÞ ¼ B P P j;2 q1 q2 q3 l B ix T ðq;lÞ u1 u2 u3 a B j¼2 ðq;lÞ2Dj @ 0

1 C C C C C; C C A

the coefficients of the nonlinear transformation and the normal form, satisfy the following system for (q1, q2, q3, l) 2 Dj: j;1 j;1  ixðq1  q2 þ 1ÞT j;1 ðq1 ;q2 ;q3 ;lÞ ¼ C ðq1 ;q2 ;q3 ;lÞ  N ðq1 ;q2 ;q3 ;lÞ ; j;2 j;2  ixðq1  q2 ÞT j;2 ðq1 ;q2 ;q3 ;lÞ ¼ C ðq1 ;q2 ;q3 ;lÞ  N ðq1 ;q2 ;q3 ;lÞ ;

and j;3 j;3 ixðq1  q2  1ÞT j;3 ðq1 ;q2 ;q3 ;lÞ ¼ C ðq1 ;q2 ;q3 ;lÞ  N ðq1 ;q2 ;q3 ;lÞ .

Remark 2.5. Note that for a fixed k P 2, the coefficients C kðq1 ;q2 ;q3 ;lÞ ; ðq1 ; q2 ; q3 ; lÞ 2 Dk are given in terms of T jðq ;q ;q ;lÞ ; N jðq ;q ;q ;lÞ ; j 2 f2 . . . k  1g, because of T(0, 0) = DT(0, 0) = N(0, 0) = DN(0, 0) = 0. 1

2

3

1

2

3

Finally, the algorithm of computation of normal forms is the following: Being computed N jðq1 ;q2 ;q3 ;lÞ ; T jðq1 ;q2 ;q3 ;lÞ for all j 2 {2 . . . , k  1} and (q1, q2, q3, l) 2 Dj then N kðq1 ;q2 ;q3 ;lÞ ; T kðq1 ;q2 ;q3 ;lÞ are given by For (q1  q2 + 1)(q1  q2)(q1  q2  1) 5 0 then we can choose k;2 k;3 N k;1 ðq1 ;q2 ;q3 ;lÞ ¼ N ðq1 ;q2 ;q3 ;lÞ ¼ N ðq1 ;q2 ;q3 ;lÞ ¼ 0;

T k;1 ðq1 ;q2 ;q3 ;lÞ ¼

C k;1 ðq1 ;q2 ;q3 ;lÞ ixðq1  q2 þ 1Þ

;

T k;2 ðq1 ;q2 ;q3 ;lÞ ¼

C k;2 ðq1 ;q2 ;q3 ;lÞ ixðq1  q2 Þ

;

T k;3 ðq1 ;q2 ;q3 ;lÞ ¼

For q1  q2 + 1 = 0 then k;1 N k;1 ðq1 ;q2 ;q3 ;lÞ ¼ C ðq1 ;q2 ;q3 ;lÞ ;

T k;1 ðq1 ;q2 ;q3 ;lÞ ¼ 0;

N k;2 ðq1 ;q2 ;q3 ;lÞ ¼ 0;

T k;2 ðq1 ;q2 ;q3 ;lÞ ¼

C k;2 ðq1 ;q2 ;q3 ;lÞ ix

;

N k;3 ðq1 ;q2 ;q3 ;lÞ ¼ 0; T k;3 ðq1 ;q2 ;q3 ;lÞ ¼

C k;3 ðq1 ;q2 ;q3 ;lÞ 2ix

.

C k;3 ðq1 ;q2 ;q3 ;lÞ ixðq1  q2  1Þ

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For q1  q2 = 0 then N k;1 ðq1 ;q2 ;q3 ;lÞ ¼ 0; T k;1 ðq1 ;q2 ;q3 ;lÞ ¼

k;2 N k;2 ðq1 ;q2 ;q3 ;lÞ ¼ C ðq1 ;q2 ;q3 ;lÞ ;

C k;1 ðq1 ;q2 ;q3 ;lÞ ix

;

T k;2 ðq1 ;q2 ;q3 ;lÞ ¼ 0;

N k;3 ðq1 ;q2 ;q3 ;lÞ ¼ 0; T k;3 ðq1 ;q2 ;q3 ;lÞ ¼

C k;3 ðq1 ;q2 ;q3 ;lÞ ix

.

For q1  q2  1 = 0 then N k;1 ðq1 ;q2 ;q3 ;lÞ ¼ 0; T k;1 ðq1 ;q2 ;q3 ;lÞ ¼

N k;2 ðq1 ;q2 ;q3 ;lÞ ¼ 0;

C k;1 ðq1 ;q2 ;q3 ;lÞ 2ix

;

k;3 N k;3 ðq1 ;q2 ;q3 ;lÞ ¼ C ðq1 ;q2 ;q3 ;lÞ ;

T k;2 ðq1 ;q2 ;q3 ;lÞ ¼

C k;2 ðq1 ;q2 ;q3 ;lÞ ix

;

T k;3 ðq1 ;q2 ;q3 ;lÞ ¼ 0.

3. Outline of symbolic computer programs All the formulas presented in the previous section are given explicitly in terms of the coefficients of the original differential equations, and thus can be easily implemented on a symbolic computation system. The symbolic manipulation language Maple has been used to code these explicit formulas. In this section, we shall outline the computer programs. 3.1. Create the input file (a) Set and define the variables: n the dimension of the system. m the number of delays. r[l], l = 1, . . . , m the delays. omega the positive imaginary part of the purely imaginary eigenvalues. phi[i, j] (i = 1, 2, 3; j = 1, . . . , n) the jth component of the ith element of the basis U, of the generalized subspace associated with the critical eigenvalues. m0 the order of the center manifold to be computed. m1 the order of the normal forms to be computed. (b) Create the vector field of the original system: In order to create the vector field of the system, we must consider the parameters al, l = 1, . . . , p as a new dependent variables as follows: We put x[1, n + l] alpha[l],l = 1, . . . , p, where x[i, j] note the jth component of the dependent variables associated with the delay r[i]. (see the examples for more clarification). LP[l, i, j] the (i, j) th coefficient of the linear part of the vector field associated with the delay r[l] of the system. NL[i] the ith component of the nonlinearity of the vector field of the system. 3.2. Compute the center manifold and normal form (a) Compute the basis W. (b) Compute recursively the coefficients hiðq1 ;q2 ;q3 ;lÞ ; q1 þ q2 þ q3 þ l ¼ i; i ¼ 2; . . . ; j of a center manifolds: Pj1 • Compute the homogeneous part Hj of F ðUn þ i¼2 hi ðn; aÞ; aÞ. • Compute the coefficients of the nonlinear terms Nj1(n,a)(h) and Rj1(n, a). • Compute the initial data hjðq1 ;q2 ;q3 ;lÞ ð0Þ by formulas (2.20)–(2.23). • Compute the vector hjðq1 ;q2 ;q3 ;lÞ ðhÞ by formulas (2.15)–(2.19). • Construct the jth homogeneous part hj(n, a)(h) of a center manifolds.

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(c) Compute the normal forms N associated up to the desired order. (d) Write the reduced equation on center manifold and the normal form N in the output file.

4. Examples In this section, we shall apply the results presented in Section 2 and the Maple programs we developed to compute the center manifolds and normal forms for three examples. All the results including the center manifolds, normal forms obtained by executing the Maple programs on a PC (Pentium 2—400 MHz) are given in the appendices. We describe also in these appendices how to create the input file of each example. 4.1. Example 1 Consider the scalar RFDE x_ ðtÞ ¼ axðtÞ  bxðt  1Þ  cxðt  rÞ þ f ðxðtÞ; xðt  1Þ; xðt  rÞÞ; where f 2 C 1 ðR3 ; RÞ and f(0,0,0) = Df(0,0,0) = 0. and assume the characteristic equation k ¼ a  bek  cekr has only the eigenvalues ±ix, x > 0, and zero on the imaginary axis. This latter hypothesis is equivalent to a þ b þ c ¼ 0; b sinðxÞ þ c sinðxrÞ ¼ x; a þ b cosðxÞ þ c cosðxrÞ ¼ 0. If we suppose r, x such that sin((r  1)x) 5 0, then given r, x, these later equations uniquely determine the coefficients a0, b0, c0. If we introduce the small parameters a1, a2 and a3 by a1 ¼ a þ a0 ; a2 ¼ b þ b0 and a3 ¼ c þ c0 ; then the RFDE can be reads as x_ ðtÞ ¼ a0 xðtÞ  b0 xðt  1Þ  c0 xðt  rÞ þ a1 xðtÞ þ a2 xðt  1Þ þ a3 xðt  rÞ þ f ðxðtÞ; xðt  1Þ; xðt  rÞÞ. Here, we consider the case x :¼ p2 and r:¼2. which give a0 ¼  p4 ; b0 ¼ p2 and c0 ¼  p4. The quadratic term is considered to be x2 ðtÞ  xðtÞxðt  1Þ þ 2xðt  1Þxðt  2Þ. Then executing the Maple program developed in this paper yields the reduced equation on center manifold and the associated normal form up to third order in Appendix A. 4.2. Examples 2 Consider the following model for a network of two neurons with self-connection  u_ 1 ðtÞ ¼ u1 ðtÞ þ a11 f ðu1 ðt  sÞÞ þ a12 f ðu2 ðt  sÞÞ; u_ 2 ðtÞ ¼ u2 ðtÞ þ a21 f ðu1 ðt  sÞÞ þ a22 f ðu2 ðt  sÞÞ;

ð4:1Þ

where aij, i, j = 1, 2 are real constants, the delay s is positive and f 2 C 3 ðR; RÞ such that f(0) = 0. The particular case f(u) = tan gh, has been studied by many researchers (see [11,7]). The above mentioned authors investigated the linearized stability and delay induced oscillations. In [4], Faria studied the local bifurcation of system (4.1) under the assumptions f00 (0) = 0 and f000 (0) 5 0. Thus by deriving the normal form up to second

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order of the system. Here, we will consider the system (4.1) with the unique assumption f(0) = 0. We will derive the reduced equation on the center manifolds and the normal form associated to the equation. Scaling the time variable, t ! t/s and separating the linear from the nonlinear terms, (4.1) becomes _ ¼ sLut þ sF ðut Þ; uðtÞ

ð4:2Þ

where ut 2 C :¼ Cð½1; 0; R2 Þ, and L : C ! R2 ; F : C ! R2 are given by L/ ¼

/1 ð0Þ þ a11 f 0 ð0Þ/1 ð1Þ þ a12 f 0 ð0Þ/2 ð1Þ

!

/2 ð0Þ þ a21 f 0 ð0Þ/1 ð1Þ þ a22 f 0 ð0Þ/2 ð1Þ ! a11 gð/1 ð1ÞÞ þ a12 gð/2 ð1ÞÞ F ð/Þ ¼ ; a21 gð/1 ð1ÞÞ þ a22 gð/2 ð1ÞÞ

for / ¼ ð/1 ; /2 Þ and g(x) = f(x)  f 0 (0)x. The characteristic equation for the linearization of Eq. (4.2) at (0, 0) is 2

Dðk; sÞ ¼ ðk þ sÞ  sT ðk þ sÞek þ s2 De2k ¼ 0; where 1 T ¼ ða11 þ a22 Þf 0 ð0Þ; 2

2

D ¼ ða11 a22  a12 a21 Þf 0 ð0Þ .

The following result, which was proved in [4], gives the necessary and sufficient conditions for existence of Fold-Hopf singularity. pffiffiffiffiffiffiffiffiffiffiffiffi Theorem 4.1. Assume D > T2, D > 1 and let q ¼ D  1. Then for r > 0,s > 0, D(ir, s) = 0 if and only if there is an n 2 N , such that s = sn and r = rn, where sn,rn are defined by rn sn ¼ ; q

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T þ q D  T2 cos rn ¼ ; D

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D  T 2  qT sin rn ¼ D

and rn 2 ð2np; 2ðn þ 1ÞpÞ. At s ¼ sn ; n 2 N , the eigenvalues ±irn are simple. Moreover, k = 0 is a simple eigenvalue of the linearization of Eq. (4.2) at ð 0; 0 Þ if and only if D = 2T  1. The basis U can be chosen as U ¼ ð/1 ; /2 ; /3 Þ; /1 ðhÞ ¼ eirn h u; /2 ðhÞ ¼ /1 ðhÞ; /3 ðhÞ ¼ v   0 f ð0Þa12 c1 with u ¼ 1; f 0 ð0Þa , v ¼ 1; and cj :¼ f 0 (0)ajj  1, j = 1, 2. c2 12 For a fixed n 2 N , introduce the new parameter a1 by a1 = s  sn, then the RFDE (4.2) is written as _ ¼ sn Lut þ a1 Lut þ ða1 þ sn ÞF ðut Þ. uðtÞ pffiffiffi 3. The nonlinear terms are considered to be as ! 8/21 ð1Þ  6/22 ð1Þ þ 2/31 ð1Þ  32 /32 ð1Þ . 4/21 ð1Þ  2/22 ð1Þ þ /31 ð1Þ  12 /32 ð1Þ

pffiffi, q ¼ We choose D = 2, which give rn ¼ 5p , sn ¼ 35p 3 3

F ð/Þ ¼

Again we execute the Maple program developed in this paper to find the reduced equation on the center manifolds and the associated normal form, up to fourth-order, given in Appendix B.

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4.3. Example 3 Consider the 3-dimensional RFDE given by 8 2 2 2 > < x_ ðtÞ ¼ axðt  1Þ þ 2x ðt  1Þ þ 3zðtÞzðt  1Þ þ y ðt  1Þ þ 5z ðtÞ; _ ¼ yðt  1Þ þ x2 ðt  1Þ þ y 2 ðt  1Þ  yðtÞxðt  1Þ; yðtÞ > : z_ ðtÞ ¼ bzðtÞ þ x2 ðtÞ  y 2 ðtÞ þ 2z2 ðtÞ; where a, b are a real parameters. The characteristic equation of the linear part is given by ðk  bÞðk  aek Þðk  ek Þ ¼ 0. It has ±ix, x > 0, as purely imaginary eigenvalues and simple zero eigenvalue if and only if ak ¼  p2 þ 2kp and xk ¼ p2 þ 2kp. We choose as critical parameters a0 ¼  p2 and x0 ¼ p2. If we introduce the small parameters a1 and a2 by a1 ¼ a þ

p 2

and

a2 ¼ b;

then the RFDE can be reads as 8 p 2 2 2 > < x_ ðtÞ ¼  2 xðt  1Þ þ a1 xðt  1Þ þ 2x ðt  1Þ þ 3zðtÞzðt  1Þ þ y ðt  1Þ þ 5z ðtÞ; _ ¼ yðt  1Þ þ x2 ðt  1Þ þ y 2 ðt  1Þ  yðtÞxðt  1Þ; yðtÞ > : z_ ðtÞ ¼ a2 zðtÞ þ x2 ðtÞ  y 2 ðtÞ þ 2z2 ðtÞ. Executing the Maple program developed in this paper yields he results in Appendix C. 5. Conclusions Methodology and computer programs have been developed for computing explicit canter manifolds and normal forms for Fold-Hopf singularity. The calculations and formulas are given in a explicit iterative procedure, and thus are very easy to be implemented on a symbolic computation system. Symbolic computer programs written in Maple have been developed for automating the computations. It has been shown that by several examples that the method is computationally efficient and fast, particularly suitable for the computations of high-dimensional systems and higher-order center manifolds and normal forms. Appendix A The input data file of Example 1: n :¼ 1: m :¼ 3: r[1] :¼ 0: r[2] :¼ 1: r[3] :¼ 2: omega :¼ Pi/2: phi[1, 1] :¼ exp(I*omega*theta): phi[2, 1] :¼ exp(I*omega*theta): phi[3, 1] :¼ 1: LP[1, 1, 1] :¼ Pi/4: LP[2, 1, 1] :¼ Pi/2: LP[3, 1, 1] :¼ Pi/4: NL[1] :¼ x[1, 2]*x[1, 1]+x[1, 3]*x[2, 1]+x[1,4]*x[3, 1]+x[1, 1]**2x[1, 1]*x[2, 1]+2*x[2, 1]*x[3, 1]: m0 :¼ 2: The reduced equation on a center manifold up to the third-order is

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1 dn1=dt ¼  pn2  2ðð4l2p  8l2p2  4l2p4 þ 8l2p3 þ 8l1l3  12l12 2 þ 8l2l3  4l22  8l2l1 þ 4l32 þ 12l1pl2 þ 10l1p3 l2  2l1p4 l2  14l1p2 l2  4l1l3p3 þ 14l1l3p2  12l1l3p  2l2l3p3 þ 2l2l3p2  4l2l3p þ p5 l2 þ 8l12 p3  l12 p4  21l12 p2 þ 22l12 p þ 6pl22  p2 l22  4l32 p3 þ l32 p4 þ 7l32 p2  10l32 pÞn1Þ=ð2  2p þ p2 Þ3  ðð16l1p3 þ 8l1p4 þ 16l1p2 þ 8l3p þ 16l3p3  16l3p2  8l3p4  8l1p  16l2l3  16l22  16l2l1 þ 2l1p4 l3  2l2p4 l3  2p5 l1 þ 2p5 l3 þ 16l22 p3  3l22 p4 þ 32l1pl2 þ 16l1p3 l2  2l1p4 l2  40l1p2 l2  8l1l3p3 þ 24l1l3p2  24l1l3p þ 8l2l3p3  16l2l3p2 þ 32l2l3p  4l12 p3 þ l12 p4 þ 4l12 p þ 28pl22 1 3  28p2 l22 þ 12l32 p3  3l32 p4  24l32 p2 þ 20l32 pÞn2Þ=ð2  2p þ p2 Þ þ ððl1p4 þ l2p4 2 þ l3p4 þ 4p3  10l3p3  6l2p3  2l1p3 þ 10l2p2  8p2 þ 2l1p2 þ 18l3p2 þ 8l2p þ 8p 1  16l1p  32l2Þðl1 þ l2 þ l3Þn3Þ=ð2  2p þ p2 Þ2  ðð3l1p6 þ 3p6 l2 þ 3l3p6  12p5 l1 12  36p5 l3  24p5 l2 þ 72l2p4 þ 36l1p4 þ 84l3p4 þ 56l1p3  200l3p3 þ 40l2p3 þ 532l3p2  244l2p2  508l1p2  496l3p þ 480l2p þ 592l1p  496l2  528l1 þ 432l3Þn12 Þ=ð2  2p þ p2 Þ 2 þ ðð9p5  36p4  6l3p4  6l1p4 þ 6l2p4 þ 72p3 þ 62l1p3  14l3p3  40l2p3 þ 20l3p2 3 þ 116l2p2  72p2  116l1p2  24l3p þ 36p  68l2p þ 120l1p þ 56l3 1 3 þ 24l2  104l1Þn1n2Þ=ð2  2p þ p2 Þ  ðð48l3 þ 496l2 þ 144l1  96p4 þ 96p þ 192p3 12  192p2  1056l2p þ 1324l2p2 þ 24p5  488l1p3 þ 108l1p4 þ 1060l1p2 þ 496l3p þ 152l3p3  652l3p2 þ 60l3p4  784l1p þ 192l2p4  808l2p3  12p5 l1  36p5 l3  24p5 l2 þ 3p6 l2 1 3 þ 3l3p6 þ 3l1p6 Þn22 Þ=ð2  2p þ p2 Þ  ððl1p6 þ p6 l2 þ l3p6  20p5 l3  20p5 l1 þ 4p5 2 5 4 4 4  20p l2 þ 48l2p þ 40l1p  16p þ 104l3p4 þ 32p3  272l3p3 þ 80l1p3 þ 436l3p2  108l2p2  300l1p2  32p2 þ 16p þ 160l2p  448l3p þ 416l1p  192l1 1 þ 256l3Þn1n3Þ=ð2  2p þ p2 Þ3  ððl1p6 þ p6 l2 þ l3p6 þ 4p5 þ 16l1p4 2  48l2p4  16p4  48l3p4 þ 264l2p3  40l1p3 þ 200l3p3 þ 32p3  636l2p2  428l3p2  44l1p2  32p2 þ 160l1p þ 480l3p þ 800l2p þ 16p  224l1 3

 480l2  224l3Þn2n3Þ=ð2  2p þ p2 Þ þ 2ðð3l1p4 þ 3l2p4 þ 3l3p4  23l3p3 þ 4p3  19l2p3  15l1p3  8p2 þ 40l3p2 þ 32l2p2 þ 24l1p2 þ 8p  26l1p  2l2p  10l3p  8l3 2

 8l1  40l2Þn32 Þ=ð2  2p þ p2 Þ þ

1 ðp4  18p3 þ 30p2  56p þ 184Þn13 2 4 ð2  2p þ p2 Þ

þ

1 ð3p4 þ 6p3  126p2 þ 1264p  1368Þn12 n2 1 ð3p4  54p3 þ 18p2 þ 232p  1064Þn1n22 þ 2 2 12 12 ð2  2p þ p2 Þ ð2  2p þ p2 Þ

þ

1 ð3p4 þ 6p3 þ 138p2  736p þ 504Þn23 12 ð2  2p þ p2 Þ2



1 ð6p6  51p5 þ 129p4  148p3 þ 90p2 þ 68p  88Þn12 n3 3 ð2  2p þ p2 Þ3

3

R. Qesmi et al. / Applied Mathematics and Computation 181 (2006) 220–246



4 ð18p4  45p3 þ 76p2  98p þ 32Þn1n2n3 3 3 ð2  2p þ p2 Þ



1 ð6p6  51p5 þ 189p4  416p3 þ 474p2  356p þ 184Þn22 n3 3 3 ð2  2p þ p2 Þ

2 2

235

ðp6  20p5 þ 73p4  106p3 þ 92p2  48p þ 56Þn1n32 ð2  2p þ p2 Þ

3

ðp6  24p4 þ 118p3  310p2 þ 400p  256Þn2n32 ð2  2p þ p2 Þ

3

þ8

ð2p4  13p3 þ 22p2  10p  8Þn33 ð2  2p þ p2 Þ

2

;

1 dn2=dt ¼ pn1 þ 2ðð4l2p  8l2p2  4l2p4 þ 8l2p3 þ 8l1l3  12l12 þ 8l2l3 2  4l22  8l2l1 þ 4l32 þ 12l1pl2 þ 10l1p3 l2  2l1p4 l2  14l1p2 l2  4l1l3p3 þ 14l1l3p2  12l1l3p  2l2l3p3 þ 2l2l3p2  4l2l3p þ p5 l2 þ 8l12 p3  l12 p4  21l12 p2 3

þ 22l12 p þ 6pl22  p2 l22  4l32 p3 þ l32 p4 þ 7l32 p2  10l32 pÞð2 þ pÞn1Þ=ðð2  2p þ p2 Þ pÞ þ ðð16l1p3 þ 8l1p4 þ 16l1p2 þ 8l3p þ 16l3p3  16l3p2  8l3p4  8l1p  16l2l3  16l22  16l2l1 þ 2l1p4 l3  2l2p4 l3  2p5 l1 þ 2p5 l3 þ 16l22 p3  3l22 p4 þ 32l1pl2 þ 16l1p3 l2  2l1p4 l2  40l1p2 l2  8l1l3p3 þ 24l1l3p2  24l1l3p þ 8l2l3p3  16l2l3p2 þ 32l2l3p  4l12 p3 þ l12 p4 þ 4l12 p þ 28pl22  28p2 l22 þ 12l32 p3  3l32 p4  24l32 p2 þ 20l32 pÞ 1 3  ð2 þ pÞn2Þ=ðð2  2p þ p2 Þ pÞ  ððl1p4 þ l2p4 þ l3p4 þ 4p3  10l3p3  6l2p3  2l1p3 2 þ 10l2p2  8p2 þ 2l1p2 þ 18l3p2 þ 8l2p þ 8p  16l1p  32l2Þð2 þ pÞðl1 þ l2 1 2 þ l3Þn3Þ=ðpð2  2p þ p2 Þ Þ þ ðð3l1p6 þ 3p6 l2 þ 3l3p6  12p5 l1  36p5 l3  24p5 l2 12 þ 72l2p4 þ 36l1p4 þ 84l3p4 þ 56l1p3  200l3p3 þ 40l2p3 þ 532l3p2  244l2p2  508l1p2  496l3p þ 480l2p þ 592l1p  496l2  528l1 þ 432l3Þð2 þ pÞn12 Þ=ðð2  2p þ p2 Þ3 pÞ 2  ðð9p5  36p4  6l3p4  6l1p4 þ 6l2p4 þ 72p3 þ 62l1p3  14l3p3  40l2p3 þ 20l3p2 3 þ 116l2p2  72p2  116l1p2  24l3p þ 36p  68l2p þ 120l1p þ 56l3 þ 24l2  104l1Þ 1  ð2 þ pÞn1n2Þ=ðð2  2p þ p2 Þ3 pÞ þ ðð48l3 þ 496l2 þ 144l1  96p4 þ 96p þ 192p3  192p2 12  1056l2p þ 1324l2p2 þ 24p5  488l1p3 þ 108l1p4 þ 1060l1p2 þ 496l3p þ 152l3p3  652l3p2 þ 60l3p4  784l1p þ 192l2p4  808l2p3  12p5 l1  36p5 l3  24p5 l2 þ 3p6 l2 þ 3l3p6 1 þ 3l1p6 Þð2 þ pÞn22 Þ=ðpð2  2p þ p2 Þ3 Þ þ ððl1p6 þ p6 l2 þ l3p6  20p5 l3  20p5 l1 2 þ 4p5  20p5 l2 þ 48l2p4 þ 40l1p4  16p4 þ 104l3p4 þ 32p3  272l3p3 þ 80l1p3 þ 436l3p2  108l2p2  300l1p2  32p2 þ 16p þ 160l2p  448l3p þ 416l1p  192l1 þ 256l3Þð2 1 3 þ pÞn1n3Þ=ðð2  2p þ p2 Þ pÞ þ ððl1p6 þ p6 l2 þ l3p6 þ 4p5 þ 16l1p4  48l2p4  16p4 2  48l3p4 þ 264l2p3  40l1p3 þ 200l3p3 þ 32p3  636l2p2  428l3p2  44l1p2  32p2 3

þ 160l1p þ 480l3p þ 800l2p þ 16p  224l1  480l2  224l3Þð2 þ pÞn2n3Þ=ðð2  2p þ p2 Þ pÞ  2ðð3l1p4 þ 3l2p4 þ 3l3p4  23l3p3 þ 4p3  19l2p3  15l1p3  8p2 þ 40l3p2 þ 32l2p2 þ 24l1p2 þ 8p  26l1p  2l2p  10l3p  8l3  8l1  40l2Þ

236

R. Qesmi et al. / Applied Mathematics and Computation 181 (2006) 220–246 2

 ð2 þ pÞn32 Þ=ðpð2  2p þ p2 Þ Þ 

1 ðp4  18p3 þ 30p2  56p þ 184Þð2 þ pÞn13 2 4 pð2  2p þ p2 Þ



1 ð3p4 þ 6p3  126p2 þ 1264p  1368Þð2 þ pÞn12 n2 2 12 pð2  2p þ p2 Þ



1 ð3p4  54p3 þ 18p2 þ 232p  1064Þð2 þ pÞn1n22 2 12 pð2  2p þ p2 Þ



1 ð3p4 þ 6p3 þ 138p2  736p þ 504Þð2 þ pÞn23 2 12 pð2  2p þ p2 Þ

þ

1 ð6p6  51p5 þ 129p4  148p3 þ 90p2 þ 68p  88Þð2 þ pÞn12 n3 3 3 pð2  2p þ p2 Þ

þ

4 ð18p4  45p3 þ 76p2  98p þ 32Þð2 þ pÞn1n2n3 3 pð2  2p þ p2 Þ3

þ

1 ð6p6  51p5 þ 189p4  416p3 þ 474p2  356p þ 184Þð2 þ pÞn22 n3 3 pð2  2p þ p2 Þ3

þ2 þ2 8

ðp6  20p5 þ 73p4  106p3 þ 92p2  48p þ 56Þð2 þ pÞn1n32 ð2  2p þ p2 Þ3 p ðp6  24p4 þ 118p3  310p2 þ 400p  256Þð2 þ pÞn2n32 ð2  2p þ p2 Þ3 p ð2p4  13p3 þ 22p2  10p  8Þð2 þ pÞn33 pð2  2p þ p2 Þ2

dn3=dt ¼ ðð4l2p  8l2p2  4l2p4 þ 8l2p3 þ 8l1l3  12l12 þ 8l2l3  4l22  8l2l1 þ 4l32 þ 12l1pl2 þ 10l1p3 l2  2l1p4 l2  14l1p2 l2  4l1l3p3 þ 14l1l3p2  12l1l3p  2l2l3p3 þ 2l2l3p2  4l2l3p þ p5 l2 þ 8l12 p3  l12 p4  21l12 p2 þ 22l12 p 1 2 þ 6pl22  p2 l22  4l32 p3 þ l32 p4 þ 7l32 p2  10l32 pÞn1Þ=ðð2  2p þ p2 Þ pÞ  ðð16l1p3 2 4 2 3 2 4 þ 8l1p þ 16l1p þ 8l3p þ 16l3p  16l3p  8l3p  8l1p  16l2l3  16l22  16l2l1 þ 2l1p4 l3  2l2p4 l3  2p5 l1 þ 2p5 l3 þ 16l22 p3  3l22 p4 þ 32l1pl2 þ 16l1p3 l2  2l1p4 l2  40l1p2 l2  8l1l3p3 þ 24l1l3p2  24l1l3p þ 8l2l3p3  16l2l3p2 þ 32l2l3p  4l12 p3 þ l12 p4 þ 4l12 p þ 28pl22  28p2 l22 þ 12l32 p3  3l32 p4  24l32 p2 þ 20l32 pÞn2Þ=ðpð2  2p 1 þ p2 Þ2 Þ þ ðð8l2p  8l2p2 þ 4l1p3  8l1p2 þ 8l3p þ 4l3p3  8l3p2 þ 8l1p þ 4l2p3  32l2l3 4  32l22  32l2l1 þ 2l1p4 l3 þ 2l2p4 l3  6l22 p3 þ l22 p4  8l1pl2  8l1p3 l2 þ 2l1p4 l2 þ 12l1p2 l2  12l1l3p3 þ 20l1l3p2  16l1l3p  16l2l3p3 þ 28l2l3p2 þ 8l2l3p  2l12 p3 þ l12 p4 þ 2l12 p2  16l12 p þ 8pl22 þ 10p2 l22  10l32 p3 þ l32 p4 þ 18l32 p2 Þn3Þ=ðpð2  2p 1 ðð3l1p6 þ 3p6 l2 þ 3l3p6  12p5 l1  36p5 l3  24p5 l2 þ 72l2p4 þ 36l1p4 þ 84l3p4 24 þ 56l1p3  200l3p3 þ 40l2p3 þ 532l3p2  244l2p2  508l1p2  496l3p þ 480l2p þ 592l1p þ p2 ÞÞ 

1 2  496l2  528l1 þ 432l3Þn12 Þ=ðð2  2p þ p2 Þ pÞ þ ðð9p5  36p4  6l3p4  6l1p4 þ 6l2p4 3 þ 72p3 þ 62l1p3  14l3p3  40l2p3 þ 20l3p2 þ 116l2p2  72p2  116l1p2  24l3p þ 36p

R. Qesmi et al. / Applied Mathematics and Computation 181 (2006) 220–246

1 ðð48l3 þ 496l2 þ 144l1 24  96p4 þ 96p þ 192p3  192p2  1056l2p þ 1324l2p2 þ 24p5  488l1p3 þ 108l1p4 þ 1060l1p2  68l2p þ 120l1p þ 56l3 þ 24l2  104l1Þn1n2Þ=ðpð2  2p þ p2 Þ2 Þ

þ 496l3p þ 152l3p3  652l3p2 þ 60l3p4  784l1p þ 192l2p4  808l2p3  12p5 l1  36p5 l3 1  24p5 l2 þ 3p6 l2 þ 3l3p6 þ 3l1p6 Þn22 Þ=ðpð2  2p þ p2 Þ2 Þ  ððl1p6 þ p6 l2 þ l3p6  20p5 l3 4  20p5 l1 þ 4p5  20p5 l2 þ 48l2p4 þ 40l1p4  16p4 þ 104l3p4 þ 32p3  272l3p3 þ 80l1p3 þ 436l3p2  108l2p2  300l1p2  32p2 þ 16p þ 160l2p  448l3p þ 416l1p  192l1 1 þ 256l3Þn1n3Þ=ðpð2  2p þ p2 Þ2 Þ  ððl1p6 þ p6 l2 þ l3p6 þ 4p5 þ 16l1p4  48l2p4  16p4 4 4 3 3  48l3p þ 264l2p  40l1p þ 200l3p3 þ 32p3  636l2p2  428l3p2  44l1p2  32p2 þ 160l1p 2

þ 480l3p þ 800l2p þ 16p  224l1  480l2  224l3Þn2n3Þ=ðpð2  2p þ p2 Þ Þ þ ðð3l1p4 þ 3l2p4 þ 3l3p4  23l3p3 þ 4p3  19l2p3  15l1p3  8p2 þ 40l3p2 þ 32l2p2 þ 24l1p2 þ 8p  26l1p  2l2p  10l3p  8l3  8l1  40l2Þn32 Þ=ðpð2  2p þ p2 ÞÞ þ

1 ðp4  18p3 þ 30p2  56p þ 184Þn13 1 ð3p4 þ 6p3  126p2 þ 1264p  1368Þn12 n2 þ 8 24 pð2  2p þ p2 Þ pð2  2p þ p2 Þ

þ

1 ð3p4  54p3 þ 18p2 þ 232p  1064Þn1n22 24 pð2  2p þ p2 Þ

þ

1 ð3p4 þ 6p3 þ 138p2  736p þ 504Þn23 24 pð2  2p þ p2 Þ



1 ð6p6  51p5 þ 129p4  148p3 þ 90p2 þ 68p  88Þn12 n3 2 6 pð2  2p þ p2 Þ



2 ð18p4  45p3 þ 76p2  98p þ 32Þn1n2n3 2 3 pð2  2p þ p2 Þ



1 ð6p6  51p5 þ 189p4  416p3 þ 474p2  356p þ 184Þn22 n3 2 6 ð2  2p þ p2 Þ p

 

ðp6  20p5 þ 73p4  106p3 þ 92p2  48p þ 56Þn1n32 pð2  2p þ p2 Þ

2

ðp6  24p4 þ 118p3  310p2 þ 400p  256Þn2n32

þ4

pð2  2p þ p2 Þ

2

ð2p4  13p3 þ 22p2  10p  8Þn33 ð2  2p þ p2 Þp.

The normal form in polar coordinate: order = 3 3 3 5 dr=dt ¼ ððl1l3p5  l2l3p5 þ l12 p5  l22 p5  l32 p5 þ l2l1p5 þ 16l2l3 2 2 2 þ 16l22 þ 16l2l1  2l1p4 l3 þ 8l2p4 l3  29l22 p3 þ 11l22 p4  32l1pl2  22l1p3 l2 þ 44l1p2 l2 þ 6l1l3p3  24l1l3p2 þ 16l1l3p  18l2l3p3 þ 36l2l3p2  48l2l3p þ 25l12 p3  11l12 p4  20l12 p2 þ 8l12 p  32pl22 þ 36p2 l22  31l32 p3 þ 13l32 p4 þ 44l32 p2  24l32 pÞrrÞ=ðp%1Þ þ ðð112l3 þ 240l2 þ 112l1  520l2p þ 478l2p2 þ 84l1p3  38l1p4  42l1p2  360l3p  316l3p3 þ 446l3p2

237

238

R. Qesmi et al. / Applied Mathematics and Computation 181 (2006) 220–246

þ 142l3p4  88l1p þ 90l2p4  264l2p3  6p5 l1  38p5 l3  24p5 l2 9 9 9 þ p6 l2 þ l3p6 þ l1p6 ÞrrzÞ=ðp%1Þ 2 2 2 ð18p6  97p5 þ 272p4  638p3 þ 1068p2  1112p þ 512Þrrz2 þ p%1 ð 94 p4 þ 12 p3 þ 13p2  15p þ 3Þrr3 þ pð4  8p þ 8p2  4p3 þ p4 Þ %1 :¼ 8  24p þ 36p2  32p3 þ 18p4  6p5 þ p6 ; 1 1 3 1 dh=dt ¼ p þ ð16l1l3 þ l1l3p5  l2l3p5  l12 p5  l22 p5  l32 p5 2 2 2 2 2 2 2 5  3l2l1p þ 24l1  16l2l3 þ 8l2 þ 16l2l1  8l3  8l1p4 l3 þ 2l2p4 l3  15l22 p3 þ 8l22 p4  40l1pl2  54l1p3 l2 þ 22l1p4 l2 þ 56l1p2 l2 þ 34l1l3p3  52l1l3p2 þ 32l1l3p  2l2l3p3 þ 8l2l3p2 þ 8l2l3p  37l12 p3 þ 8l12 p4 þ 66l12 p2  56l12 p  24pl22 þ 22p2 l22 þ 3l32 p3  14l32 p2 þ 24l32 pÞ=ðp%1Þ þ ðð10l1p4 11 1 þ 24p5 l3  l3p6 þ l3p7  128l3  70l3p4 þ 232l3p þ 138l3p3 2 2 11 1  186l2p3 þ 10p5 l2  p6 l2 þ p7 l2  210l3p2 þ 294l1p2 þ 24p5 l1 2 2 11 1 6 2 4  l1p þ 294l2p þ 42l2p þ l1p7 þ 96l1  126l1p3  312l1p 2 2  200l2pÞzÞ=ðp%1Þ ð2p7  22p6 þ 89p5  134p4  6p3 þ 168p2  104p  112Þz2 p%1 1 5 9 4 21 3 56 2 ð 4 p þ 4 p  2 p þ 3 p  403 p þ 373Þrr2 %1 :¼ 8  24p þ 36p2  32p3 þ 18p4  6p5 þ p6 þ pð4  8p þ 8p2  4p3 þ p4 Þ  1 1 3 1 dz=dt ¼ 8l2l3  8l22  8l2l1 þ l1p4 l3 þ l2p4 l3  l22 p3 þ l22 p4 2 2 2 4 1  2l1pl2  2l1p3 l2 þ l1p4 l2 þ 3l1p2 l2  3l1l3p3 þ 5l1l3p2 2 1 1  4l1l3p  4l2l3p3 þ 7l2l3p2 þ 2l2l3p  l12 p3 þ l12 p4 2 4  1 2 2 5 5 1 9 2 2 2 2 2 2 3 2 4 2 þ l1 p  4l1 p þ 2pl2 þ p l2  l3 p þ l3 p þ l3 p z =ðpð2  2p þ p2 ÞÞ 2 2 2 4 2 þ

þ ðð3l1p4 þ 3l2p4 þ 3l3p4  23l3p3  19l2p3  15l1p3 þ 24l1p2 þ 40l3p2 þ 32l2p2  10l3p  2l2p  26l1p  8l1  8l3  1 1 1  40l2Þz2 Þ=ðpð2  2p þ p2 ÞÞ þ  l1p6  p6 l2  l3p6 þ p5 l2 8 8 8 3 5 1 5 11 þ p l3 þ p l1  3l1p4  l2p4  3l3p4 þ l3p3 þ 9l1p3 þ 16l2p3 2 2 2   5 45 23 2 2 2 þ l3p  l2p  l1p þ 4l1p þ 12l2p  8l3 þ 8l1 rr2 ðpð4  8p þ 8p2  4p3 þ p4 ÞÞ 2 2 2  6 17 5 53 4  p þ 2 p  2 p þ 47p3  47p2 þ 24p  8 rr2 z ð8p4  52p3 þ 88p2  40p  32Þz3 þ . þ pð4  8p þ 8p2  4p3 þ p4 Þ ð2  2p þ p2 Þp

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Appendix B The input data file of Example 2: n :¼ 2: m :¼ 2: omega :¼ 5*Pi/3: b11 :¼ 4:b12 :¼  3:b21 :¼ 2:b22 :¼  1: phi[1, 1] :¼ exp(I*omega*theta): phi[1, 2] :¼ (b22  1)*exp(I*omega*theta)/b21: phi[2, 1] :¼ exp(I*omega*theta): phi[2, 2] :¼ (b22  1)*exp(I*omega*theta)/b21: phi[3, 1] :¼ 1:phi[3, 2] :¼ (1  b11)/b12: r[1] :¼ 0: r[2] :¼ 1: LP[1, 1, 1]:¼ 1:LP[1, 1, 2] :¼ 0:LP[1, 2, 1] :¼ 0:LP[1, 2, 2]:¼  1:LP[2, 1, 1] :¼ 4:LP[2, 1, 2] :¼  3:LP[2, 2, 1] :¼ 2:LP[2, 2, 2] :¼ 1: NL[1] :¼ x[1, 3]*x[1, 1] + 4*x[1, 3]*x[2, 1] 3*x[1, 3]*x[2, 2] + 8*(x[1, 3] + 5*Pi/ (3*sqrt(3)))*(x[2, 1]**2)  6*(x[1, 3] + (5*Pi/(3*sqrt(3))))*(x[2, 2]**2) + 2*(x[1, 3] + (5*Pi/ (3*sqrt(3))))*(x[2, 1]**3)  3*(x[1, 3] + (5*Pi/(3*sqrt(3))))*(x[2, 2]**3)/2: NL[2] :¼ x[1, 3]*x[1, 2] + 2*x[1, 3]*x[2, 1]  x[1, 3]*x[2, 2] + 4*(x[1, 3] + (5*Pi/ (3*sqrt(3))))*(x[2, 1]**2)  2*(x[1, 3] + (5*Pi/(3*sqrt(3))))*(x[2, 2]**2) + (x[1, 3] + (5*Pi/ (3*sqrt(3))))*(x[2, 1]**3)  (x[1, 3] + (5*Pi/(3*sqrt(3))))*(x[2, 2]**3)/2: m0 :¼ 2: The reduced equation on a center manifold up to the third order is  pffiffiffi pffiffiffi 5 1  %2l1 15309l1 3 þ 1984500p3  1531250p6 3 dn1=dt ¼  pn2  3 8575 pffiffiffi pffiffiffi pffiffiffi pffiffiffi pffiffiffi 4 6 3 þ 953125l1p 3 198450p2 3  2756250p4 3 þ 591300l1p2 3 þ 2542500l1p

2723625l1p3  4021875l1p5  229635l1p þ 5512500p5 n1 =ð%13 p2 Þ pffiffiffi pffiffiffi pffiffiffi   2 3 %2l1 1701 3  24975p2 3 þ 17415p þ 52875p3  16875p4 3 þ 3125p5 n2 þ 3 1715 pffiffiffi pffiffiffi p%1 4  2 2 3 150 210p þ 189 þ 1325p 3  459p 3  2875p %2l1n1 þ 343 %33 p ffiffi ffi pffiffiffi pffiffiffi 4  237500p5 3  691875p4 þ 249375p3 3  90450p2  6615p 3 þ 5103 %2l1n1n2 þ 343 pffiffiffi pffiffiffi pffiffiffi 2  =ðp%13 Þ þ 134375p5 3  428125p4 þ 168000p3 3  71775p2  2835p 3 þ 3402 343 . pffiffiffi pffiffiffi 8    3%2l1n22 ðp%13 Þ  %2 918750p5 3  15309l1 þ 99225p2 þ 1378125p4 5145 pffiffiffi pffiffiffi pffiffiffi 330750p3 3 þ 765625p6 þ 136080l1 3p þ 1694250l1p3 3  859375l1p6  1174500l1p2 . pffiffiffi pffiffiffi 8   %2 15309l1 3 þ 992250p3 4027500l1p4 þ 1687500l1p5 3 n1n3 ðp%33 Þ  15435 pffiffiffi pffiffiffi pffiffiffi pffiffiffi pffiffiffi 765625p6 3  99225p2 3  1378125p4 3 þ 684450l1p2 3 þ 1946250l1p4 3 . pffiffiffi þ 359375l1p6 3  2662875l1p3  2278125l1p5  280665l1p þ 2756250p5 n2n3 ðp%13 Þ  pffiffiffi pffiffiffi pffiffiffi 5  %2p 621000p3 3  664200p2 þ 107730p 3  35721  990000p4 þ 200000p5 3 n13  252  25  .  pffiffiffi pffiffiffi pffiffiffi %2p 93240p2 3 %3 10000p4 þ 9000p3 3  14400p2 þ 3510p 3  1701  .252   pffiffiffi pffiffiffi pffiffiffi %1 10000p4  9000p3 3 þ 53622p þ 228600p3 þ 40000p5  94000p4 3  1701 3 n12 n2  pffiffiffi pffiffiffi 5  %2p 174600p2 þ 55890p 3 þ 18711  350000p4 þ14400p2  3510p 3 þ 1701  252

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.  pffiffiffi pffiffiffi pffiffiffi pffiffiffi þ237000p3 3 þ 200000p5 3 n1n22 %3 10000p4 þ 9000p3 3  14400p2 þ 3510p 3 pffiffiffi pffiffiffi pffiffiffi 5  %2p 3 63000p3 3 þ 70200p2 þ 2430p 3 þ 15309  210000p4 1701  756 pffiffiffi 3 . pffiffiffi pffiffiffi 5 þ 200000p 3 n2 ð%1ð10000p4  9000p3 3 þ 14400p2  3510p 3 þ 1701ÞÞ

ÞÞ

pffiffiffi pffiffiffi pffiffiffi 40 ðð34375p5 3 þ 10206  384750p2 þ 25515p 3  618750p4 þ 464625p3 3Þ%2n12 n3Þ=%33 3087 pffiffiffi pffiffiffi pffiffiffi 80 þ ðð42525p þ 322875p3 þ 3402 3 þ 6750p2 3  405000p4 3 3087 pffiffiffi pffiffiffi 40  þ 465625p5 Þ%2n1n2n3Þ=%13 þ 19845p þ 370125p3 þ 3402 3  38250p2 3 3087 . pffiffiffi p ffiffi ffi 1   %2 3529575p  53679375p3 338750p4 3 þ 321875p5 %2 3n22 n3 %13  9261 pffiffiffi pffiffiffi pffiffiffi pffiffiffi þ163296 3 þ 17593750p6 3 þ 10870200p2 3 þ 50490000p4 3 pffiffiffi pffiffiffi   1    77971875p5 n1n32 =%13  %2 5196875p5 3 þ 2543400p2  513135p 3 3087 . pffiffiffi 4 3 þ8550000p  3036375p 3 þ 4531250p6 þ 163296 n2n32 %13 pffiffiffi %1 :¼ 25p2  15p 3 þ 9 pffiffiffi pffiffiffi %2 :¼ 45p þ 9 3 þ 25p2 3 pffiffiffi %3 :¼ 9  25p2 þ 15p 3; pffiffiffi pffiffiffi pffiffiffi 5 2  15309l1 3 þ 1984500p3  1531250p6 3  198450p2 3 dn2=dt ¼ pn1  3 8575 pffiffiffi pffiffiffi pffiffiffi pffiffiffi  2756250p4 3 þ 591300l1p2 3 þ 2542500l1p4 3 þ 953125l1p6 3  2723625l1p3  4021875l1p5 . 229635l1p þ 5512500p5 l1n1 ðp2 %12 Þ pffiffiffi pffiffiffi pffiffiffi  52875p3  17415p þ 16875p4 3 þ 24975p2 3  3125p5 þ 1701 3 l12 n2 6  1715 p%22 pffiffiffi pffiffiffi   2 3 300 l1 210p þ 189 þ 1325p 3  459p 3  2875p4 n12  343 %22 pffiffiffi p ffiffi ffi pffiffiffi   8 l1 237500p5 3  691875p4 þ 249375p3 3  90450p2  6615p 3 þ 5103 n1n2 þ 343 %12 p pffiffiffi pffiffiffi p ffiffi ffi pffiffiffi  4 l1 3 134375p5 3  428125p4 þ 168000p3 3  71775p2  2835p 3 þ 3402 n22 þ 343 %12 p   p ffiffi ffi pffiffiffi pffiffiffi 16  918750p5 3 þ 15309l1  99225p2  1378125p4 þ 330750p3 3  765625p6  136080l1 3p 5145 . pffiffiffi pffiffiffi 1694250l1p3 3 þ 859375l1p6 þ 1174500l1p2 þ 4027500l1p4  1687500l1p5 3 n1n3 ð%12 pÞ pffiffiffi pffiffiffi pffiffiffi pffiffiffi pffiffiffi 16   15309l1 3 þ 992250p3  765625p6 3  99225p2 3  1378125p4 3 þ 684450l1p2 3 15435 pffiffiffi pffiffiffi þ 1946250l1p4 3 þ 359375l1p6 3  2662875l1p3  2278125l1p5  280665l1p . þ 2756250p5 n2n3 ð%12 pÞ pffiffiffi pffiffiffi pffiffiffi  5 621000p3 3  664200p2 þ 107730p 3  35721  990000p4 þ 200000p5 3 pn13 pffiffiffi pffiffiffi þ 126 10000p4 þ 9000p3 3  14400p2 þ 3510p 3  1701 þ

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pffiffiffi pffiffiffi pffiffiffi   25 93240p2 3 þ 1701 3  53622p  228600p3 þ 94000p4 3  40000p5 pn12 n2 pffiffiffi pffiffiffi  126 10000p4 þ 9000p3 3  14400p2 þ 3510p 3  1701 pffiffiffi pffiffiffi 5   174600p2 þ 55890p 3 þ 18711  350000p4 þ 237000p3 3 126 . pffiffiffi pffiffiffi pffiffiffi þ200000p5 3 pn1n22 10000p4  9000p3 3 þ 14400p2  3510p 3 þ 1701 pffiffiffi pffiffiffi pffiffiffi pffiffiffi  5 63000p3 3 þ 70200p2 þ 2430p 3 þ 15309  210000p4 þ 200000p5 3 p 3n23 pffiffiffi pffiffiffi  378 10000p4  9000p3 3 þ 14400p2  3510p 3 þ 1701 pffiffiffi pffiffiffi pffiffiffi  80 34375p5 3 þ 10206  384750p2 þ 25515p 3  618750p4 þ 464625p3 3 n12 n3  2 3087 %1p p ffiffi ffi ffiffiffi pffiffiffi   160 42525p  322875p3  3402 3  6750p2 3 þ 405000p4 3  465625p5 n1n2n3  3087 %22 pffiffiffi p ffiffi ffi pffiffiffi  pffiffiffi 80 19845p  370125p3  3402 3 þ 38250p2 3 þ 338750p4 3  321875p5 3n22 n3  3087 %22   pffiffiffi p ffiffiffi 2  3529575p  53679375p3 þ 163296 3 þ 17593750p6 3 9261 . pffiffiffi pffiffiffi þ10870200p2 3 þ 50490000p4 3  77971875p5 n1n32 %12 pffiffiffi pffiffiffi pffiffiffi 2  ðð  5196875p5 3 þ 2543400p2  513135p 3 þ 8550000p4  3036375p3 3 3087   þ4531250p6 þ 163296 n2n32 =%12 p ffiffi ffi %1 :¼ 25p2  15p 3 þ 9 pffiffiffi %2 :¼ 9  25p2 þ 15p 3;  pffiffiffi pffiffiffi pffiffiffi 1 dn3=dt ¼ l1 30618l1 3 þ 1984500p3  1531250p6 3  198450p2 3 1960 pffiffiffi pffiffiffi pffiffiffi pffiffiffi  2756250p4 3 þ 828225l1p2 3 þ 2171250l1p4 3 þ 1156250l1p6 3  2750625l1p3  pffiffiffi 1 3731250l1p5  433755l1p þ 5512500p5 n1=ð%12 p2 Þ þ l1 367500p4 3 þ 39690p 392 pffiffiffi pffiffiffi pffiffiffi þ551250p3  132300p2 3 þ 306250p5 þ 15309l1 3 þ 223425l1p2 3 þ 18750l1p5 pffiffiffi pffiffiffi 1  169695l1p  380250l1p3 þ 60000l1p4 3 n2=ðp%12 Þ  78121827l1 3 8820 pffiffiffi pffiffiffi þ8197969500p3 þ 144703125000p9  122232796875p6 3  148837500000p8 3 pffiffiffi pffiffiffi pffiffiffi  506345175p2 3  322776562500l1p8 3  22968750000p10 3 þ 228234375000l1p9 pffiffiffi pffiffiffi pffiffiffi þ 30156250000l1p10 3 þ 451967343750l1p7  21767484375p4 3 þ 34886149200l1p2 3 pffiffiffi pffiffiffi þ 254828328750l1p4 3 þ 60863906250l1p6 3  214691502375l1p3 . 485536781250l1p5  7861783860l1p þ 284651718750p7 þ 108502537500p5 n12 ðp%2%12 Þ pffiffiffi pffiffiffi 1   12055837500p5 3 þ 31627968750p7 3 þ 677055834l1  168781725p2 1470 pffiffiffi pffiffiffi  7656250000p10 þ 16078125000p9 3  7255828125p4 þ 910885500p3 3  953564062500l1p8 pffiffiffi pffiffiffi þ 2812500000l1p10 þ 168281250000l1 3p9 þ 867211875000l1 3p7  40744265625p6 pffiffiffi pffiffiffi  49612500000p8  2414612025l1 3p þ 38386679625l1p3 3  1426704890625l1p6 . pffiffiffi þ1951897500l1p2  333901591875l1p4 þ 502901662500l1p5 3 n1n2 ðp%2%12 Þ pffiffiffi pffiffiffi 1  þ 1328071059l1 3 þ 2732656500p3 þ 48234375000p9  40744265625p6 3 8820

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pffiffiffi pffiffiffi pffiffiffi pffiffiffi 49612500000p8 3  168781725p2 3  1755435937500l1p8 3  7656250000p10 3 pffiffiffi pffiffiffi þ 878484375000l1p9 þ 5781250000l1p10 3 þ 4998041718750l1p7  7255828125p4 3 pffiffiffi pffiffiffi pffiffiffi  7949963700l1p2 3  757538527500l1p4 3  2865511687500l1p6 3 þ 304777312875l1p3 þ 3187292793750l1p5  11867077530l1p þ 94883906250p7    pffiffiffi pffiffiffi 2  . 1  þ þ 36167512500p5 n22 1837500p5 3 þ 45927l1 p%3 9  25p2 þ 15p 3 294 pffiffiffi pffiffiffi pffiffiffi 2 4 3 6  198450p  2756250p þ 661500p 3  1531250p  371385l1 3p  3550500l1p3 3 . pffiffiffi þ 1875000l1p6 þ 2818800l1p2 þ 7571250l1p4  3078125l1p5 3 n1n3 ðp%12 Þ pffiffiffi pffiffiffi 5  þ 15309l1 3 þ 396900p3  306250p6 3 882 pffiffiffi pffiffiffi pffiffiffi pffiffiffi pffiffiffi  39690p2 3  551250p4 3 þ 114210l1p2 3 þ 42750l1p4 3 þ 212500l1p6 3  48600l1p3   5 pffiffiffi 5 pffiffiffi  466875l1p5  137781l1p þ 1102500p5 n2n3 =ðp%12 Þ þ p 3 þ l1  l1 3p n32 9 18 pffiffiffi pffiffiffi pffiffiffi  3 2 4 25 p 621000p 3  664200p þ 107730p 3  35721  990000p þ 200000p5 3 n13 þ %2 288 pffiffiffi pffiffiffi pffiffiffi   2 3 93240p þ 94000p4 3  40000p5 pn12 n2 3 þ 1701 3  53622p  228600p 125 þ %3 288 . pffiffiffi pffiffiffi pffiffiffi 25  2 þ 174600p þ 55890p 3 þ 18711  350000p4 þ 237000p3 3 þ 200000p5 3 pn1n22 ð%2Þ 288 pffiffiffi pffiffiffi pffiffiffi pffiffiffi 25 p 3 63000p3 3 þ 70200p2 þ 2430p 3 þ 15309  210000p4 þ 200000p5 3 n23  %3 864 p ffiffi ffi pffiffiffi 1   5677690190625p5 3 þ 5386021875000p7 3 þ 6666395904  345545562600p2 7056 pffiffiffi pffiffiffi þ 5740645365p 3  160625000000p10 þ 541968750000p9 3  5320406025000p4 pffiffiffi þ 1058529961125p3 3  11965985718750p6  4311759375000p8 n12 n3  .  pffiffiffi 2 1   %3 9  25p2 þ 15p 3 139753335015p  497418114375p3 10584 pffiffiffi pffiffiffi pffiffiffi þ 7499695392 3 þ 4813781250000p9  11250786093750p6 3  8970215625000p8 3 pffiffiffi pffiffiffi pffiffiffi þ 285018369300p2 3 þ 110000000000p10 3  1204444687500p4 3 þ 22947840000000p7   þ 9627328153125p5 n1n2n3 =ð%2%12 Þ pffiffiffi 1 pffiffiffi 3 87428244645p þ 731775940875p3 þ 7221928896 3 þ 3542906250000p9  21168 pffiffiffi pffiffiffi pffiffiffi pffiffiffi  11450488218750p6 3  7255696875000p8 3 þ 74530335600p2 3 þ 60625000000p10 3 pffiffiffi  2584885770000p4 3 þ 20482824375000p7 þ 12103053159375p5 n22 n3=ð%2%12 Þ pffiffiffi pffiffiffi pffiffiffi 25  þ p 950859  15196275p2 þ 3518750p5 3 þ 3405240p 3 þ 12078000p3 3 10584 pffiffiffi pffiffiffi  5  46875p5 3 þ 707400p2  830655p 3 15924375p4 n1n32 =%12 þ 3528   . p ffiffi ffi 25 2 5 pffiffiffi 2 2 4 3 6 %1 þ  p þ p 3 n33 7710000p þ 2885625p 3 þ 326592 þ 4531250p n2n3 18 36 pffiffiffi 2 %1 :¼ 25p  15p 3 þ 9 pffiffiffi pffiffiffi %2 :¼ 10000p4  9000p3 3 þ 14400p2  3510p 3 þ 1701 pffiffiffi pffiffiffi %3 :¼ 10000p4 þ 9000p3 3  14400p2 þ 3510p 3  1701.

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The normal form in polar coordinate: order = 3  pffiffiffi pffiffiffi 3 l12 1171875p5 3 þ 417150p2  59535p 3 þ 2013750p4 dr=dt ¼  17150 pffiffiffi 4 l1ð357210p  658125p3 3 þ 15309 þ 921875p6 rr=ðp2 %1Þ  3087 p ffiffi ffi p ffiffi ffi pffiffiffi p ffiffi ffi  4114800p3 þ 15309 3 þ 659375p6 3 þ 978480p2 3 þ 3195000p4 3 pffiffiffi  1  489888  36384375p5 3 þ 15957000p2  3948750p5 rrz=ðp%1Þ  6174 . pffiffiffi pffiffiffi  2202795p 3 þ 67590000p4  23965875p3 3 þ 26656250p6 rrz2 ð%1Þ pffiffiffi pffiffiffi 5  p 4640085p þ 64010250p3  168399 3  44250000p6 3  504 pffiffiffi pffiffiffi 13456125p2 3  62640000p4 3 þ 25000000p7 þ 117675000p5 rr3 . pffiffiffi pffiffiffi ð250000p6  375000p5 3 þ 855000p4  384750p3 3 þ 330075p2 pffiffiffi 57105p 3 þ 15309 pffiffiffi pffiffiffi %1 :¼ 625p4  750p3 3 þ 1125p2  270p 3 þ 81  pffiffiffi pffiffiffi 5 1 l12 6571125p3 þ 30618 3  535815p þ 1443825p2 3 dh=dt ¼ p  3 17150 . pffiffiffi pffiffiffi 4 l1ð5103 þ 5878125p4 3 þ 1953125p6 3  8803125p5 ðp2 %1Þ  1715 p ffiffi ffi pffiffiffi p ffiffi ffi þ 554850p2  59535p 3  833625p3 3 þ 2036250p4  871875p5 3 pffiffiffi  1  80031375p3  163296 3  2440935p þ 453125p6 z=ðp%1Þ  18522 . pffiffiffi pffiffiffi pffiffiffi 2 þ 14110200p 3 þ 75330000p4 3 þ 21593750p6 3  109171875p5 z2 ð%1Þ pffiffiffi pffiffiffi 5  p 443961  13743675p2 þ 1826145p 3 þ 21269250p3 3  504 pffiffiffi pffiffiffi  62280000p4 þ 36975000p5 3  37750000p6 þ 5000000p7 3 rr2 pffiffiffi pffiffiffi =ð250000p6  375000p5 3 þ 855000p4  384750p3 3 þ 330075p2 pffiffiffi 57105p 3 þ 15309 pffiffiffi pffiffiffi %1 :¼ 625p4  750p3 3 þ 1125p2  270p 3 þ 81   pffiffiffi pffiffiffi 5 pffiffiffi 1  52329375p4 3  2268000p2 3 dz=dt ¼  l1 3p þ l1 z2 þ 18 8820 pffiffiffi pffiffiffi þ 60187500p5  3240405p þ 413343 3 þ 1796875p6 3 þ 46433250p3 l1rr2 .  pffiffiffi pffiffiffi p 625p4 þ 750p3 3  1125p2 þ 270p 3  81   pffiffiffi pffiffiffi 25 5 pffiffiffi 1  6984375p5 3 þ 10117800p2  1652805p 3 þ  p 2 þ p 3 z3 þ 18 36 1176 pffiffiffi þ27216  8330000p4  4894125p3 3 þ 1843750p6 rr2 z . pffiffiffi pffiffiffi 625p4 þ 750p3 3  1125p2 þ 270p 3  81 .

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Appendix C The input data file of Example 3: n :¼ 3: m :¼ 2: omega :¼ Pi/2: phi[1, 1] :¼ exp(I*omega*theta): phi[1, 2] :¼ 0: phi[1, 3] :¼ 0: phi[2, 1] :¼ exp(I*omega*theta): phi[2, 2] :¼ 0: phi[2, 3] :¼ 0: phi[3, 1] :¼ 0: phi[3, 2] :¼ 0: phi[3, 3] :¼ 1: r[1] :¼ 0: r[2] :¼ 1: LP[1, 1, 1] :¼ 0:LP[1, 1, 2] :¼ 0:LP[1, 1, 3] :¼ 0:LP[1, 2, 1] :¼ 0:LP[1, 2, 2] :¼ 0:LP[1, 2, 3] :¼ 0:LP[1, 3, 1] :¼ 0: LP[1, 3,2] :¼ 0: LP[1, 3, 3] :¼ 0: LP[2, 1, 1] :¼  Pi/2:LP[2, 1, 2] :¼ 0:LP[2, 1, 3] :¼ 0:LP[2, 2, 1] :¼ 0:LP[2, 2, 2] :¼ 1:LP[2, 2, 3] :¼ 0:LP[2, 3, 1] :¼ 0:LP[2, 3, 2] :¼ 0: LP[2, 3, 3] :¼ 0: NL[1] :¼ x[1, 4]*x[2, 1] + 2*(x[2, 1]**2)  x[1, 3]*x[2, 1] + 3*x[1, 3]*x[2, 3] + (x[2, 2]**2) + 5*(x[1, 3]**2): NL[2] :¼ (x[2, 1]**2) + (x[2, 2]**2)  x[1, 2]*x[2, 1]: NL[3] :¼ x[1, 5]*x[1, 3] + (x[1, 1]**2)  (x[1, 2]**2) + 2*(x[1, 3]**2): m0 :¼ 2: m2 :¼ 3: The reduced equation on a center manifold up to the 3rd order is 1 ðp5 þ l1p4 þ 8p3 þ 8l1p2 þ 16p  16l1Þl1n1 dn1=dt ¼  pn2  4 ; 2 ð4 þ p2 Þ3 8 þ

pl12 ð4 þ p2 Þn2 ð4 þ p2 Þ3

8 ð15p5 þ 48l1p4 þ 120p3 þ 304l1p2 þ 240p  512l1Þn12 15 ð4 þ p2 Þ3

32 l1ð3p4  35p3 þ 24p2  20p þ 48Þn1n2 32 l1ð3p2  28Þn22 þ 15 15 ð4 þ p2 Þ2 ð4 þ p2 Þ3 ðp5 þ 2l1p4 þ 8p3 þ 16l1p2 þ 16p  32l1Þn1n3 þ4 ð4 þ p2 Þ3 pð4 þ p2 Þl1n2n3 þ 16 3 ð4 þ p2 Þ 

þ4 þ

ð8p3  3l2p3 þ 16l1p2 þ 32p  12l2p  64l1Þn32 ð4 þ p2 Þ

2

128 ð3p2  20p þ 12Þn12 n2 128 ð3p2  28Þn1n22  2 2 15 15 ð4 þ p2 Þ ð4 þ p2 Þ



64 ð9p2  4Þn13 15 ð4 þ p2 Þ2

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2 ð45p5 þ 192p4 þ 360p3 þ 1216p2 þ 720p  2048Þn12 n3 3 15 ð4 þ p2 Þ

þ

8 ð57p4  140p3 þ 456p2  80p þ 912Þn1n2n3 3 15 ð4 þ p2 Þ



2 ð45p3 þ 48p2 þ 180p  448Þn22 n3 ð65p4 þ 8p2  1040Þn1n32 4 2 3 15 ð4 þ p2 Þ ð4 þ p2 Þ

8

pð4 þ p2 Þn2n32 3

8

ð3p3 þ 8p2 þ 12p  32Þn33 2

ð4 þ p2 Þ ð4 þ p2 Þ 1 ðp5 þ l1p4 þ 8p3 þ 8l1p2 þ 16p  16l1Þl1n1 dn2=dt ¼ pn1  8 3 2 ð4 þ p2 Þ p l12 ð4 þ p2 Þn2  16 3 ð4 þ p2 Þ 16 ð15p5 þ 48l1p4 þ 120p3 þ 304l1p2 þ 240p  512l1Þn12 þ 3 15 pð4 þ p2 Þ 64 ð3p4  35p3 þ 24p2  20p þ 48Þl1n1n2 64 ð3p2  28Þl1n22 þ 3 2 15 15 pð4 þ p2 Þ pð4 þ p2 Þ 5 4 3 2 ðp þ 2l1p þ 8p þ 16l1p þ 16p  32l1Þn1n3 þ8 3 ð4 þ p2 Þ p l1ð4 þ p2 Þn2n3 þ 32 3 ð4 þ p2 Þ ð8p3  3l2p3 þ 16l1p2 þ 32p  12l2p  64l1Þn32 þ8 2 ð4 þ p2 Þ p 128 ð9p2  4Þn13 256 ð3p2  20p þ 12Þn12 n2 256 ð3p2  28Þn1n22  þ  2 2 15 pð4 þ p2 Þ2 15 15 pð4 þ p2 Þ pð4 þ p2 Þ 



4 ð45p5 þ 192p4 þ 360p3 þ 1216p2 þ 720p  2048Þn12 n3 3 15 ð4 þ p2 Þ p

þ

16 ð57p4  140p3 þ 456p2  80p þ 912Þn1n2n3 3 15 ð4 þ p2 Þ p



4 ð45p3 þ 48p2 þ 180p  448Þn22 n3 ð65p4 þ 8p2  1040Þn1n32  8 2 3 15 ð4 þ p2 Þ p pð4 þ p2 Þ

 16

ð4 þ p2 Þn2n32 3

 16

ð3p3 þ 8p2 þ 12p  32Þn33 2

ð4 þ p2 Þ ð4 þ p2 Þ p 2 ð4 þ p Þl1n1n2 dn3=dt ¼ l2n3  4 2 ð4 þ p2 Þ ðp5 þ 2l1p4  8p3  16l1p2  16p  32l1Þn22  þ 2n32 2 ð4 þ p2 Þ p 16 ð3p2  10p þ 12Þn12 n2 16 ð3p2  28Þn1n22 þ þ 15 pð4 þ p2 Þ 15 pð4 þ p2 Þ 3 2 8 ð9p  40p þ 36Þn2 ð4 þ p2 Þn1n2n3 ð16  8p2 þ p4 Þn22 n3 þ þ 4 þ 2 2 2 15 pð4 þ p2 Þ ð4 þ p2 Þ ð4 þ p2 Þ p þ 32

ðp2  4p þ 4Þn2n32 . pð4 þ p2 Þ

245

246

R. Qesmi et al. / Applied Mathematics and Computation 181 (2006) 220–246

The normal form in polar coordinate: order = 3 l12 ðp4 þ 12p2  32Þrr l1ðp4 þ 12p2  32Þrrz þ4 2 4 6 64 þ 48p þ 12p þ p 64 þ 48p2 þ 12p4 þ p6  88 3 32 2 32  3 4 2 2 rr  5 p þ 5 p  5 p þ 128 ð130p  24p þ 2112Þrrz 5 ; þ þ 2 4 6 2 4 64 þ 48p þ 12p þ p pð16 þ 8p þ p Þ

dr=dt ¼ 2

1 l12 ð3p2  4Þ l1ð3p2  4Þz þ 32 dh=dt ¼ p  16 2 4 6 2 pð64 þ 48p þ 12p þ p Þ pð64 þ 48p2 þ 12p4 þ p6 Þ  16 3 208 2 64  2 rr  5 p  15 p  5 p þ 1088 ð256p4  48p2 þ 4160Þz2 15 þ þ pð64 þ 48p2 þ 12p4 þ p6 Þ pð16 þ 8p2 þ p4 Þ l1ð16  8p2 þ p4 Þrr2 ð16  8p2 þ p4 Þrr2 z þ . dz=dt ¼  pð16 þ 8p2 þ p4 Þ pð16 þ 8p2 þ p4 Þ References [1] M. Ait Babram, O. Arino, M.L. Hbid, Approximation scheme of a system manifold for functional differential equations, J. Math. Anal. Appl. 213 (1997) 554–572. [2] M.Ait Babram, O. Arino, M.L. Hbid, Computational scheme of a center manifold for neutral functional differential equations, J. Math. Anal. Appl. 258 (2001) 396–414. [3] O. Diekman, S.A. Van Gils, S.M. Verduyn Lunel, H.O. Whalter, Delay equations. Functional-, complex-, and nonlinear analysis, Applied Mathematical Sciences, Vol. 110, Springer-Verlag, New York, 1995. [4] T. Faria, On a planar system modeling a neuron network with memory, J. Differen. Equat. 168 (2000) 129–149. [5] T. Faria, L.T. Magalha`es, Normal forms for retarded functional differential equations with parameters and applications to Hopf bifurcation, J. Differen. Equat. 122 (1995) 181–200. [6] T. Faria, L.T. Magalha`es, Normal forms for retarded functional differential equations with parameters and applications to Hopf bifurcation, J. Differen. Equat. 122 (1995) 181–200. [7] K. Gopalsamy, I. Leung, Delay induced periodicity in a neural netlet of excitation and inhibition, Physica D 89 (1996) 395–426. [8] J. Guckenheimer, P.J. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, New York, Heidelberg, Berlin, 1983. [9] J.K. Hale, S.M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993. [10] Yu.A. Kuznetsov, Elements of Applied Bifurcation Theory, Springer-Verlag, New York, 1994. [11] L. Olien, J. Be´lair, Bifurcations, stability, and monotonicity properties of a delayed neural network model, Physica D 102 (1997) 349– 363. [12] R. Qesmi, M. Aitbabram, M.L. Hbid, A Maple program for computing a terms of a center manifolds, and elements of bifurcations for a class of retarded functional differential equations with Hopf singularity, Appl. Math. Comput. 175 (2005) 42–78. [13] S. Wiggins, Introduction to applied nonlinear dynamical systems and chaos, Texts in Applied Mathematics, 2, Springer-Verlag, New York, 1990.