Computation of focus quantities ofthree-dimensional polynomial systems

Valery G. Romanovski, Douglas S. Shafer

(1.Center for Applied Mathematics and Theoretical Physics,University of Maribor, Krekova 2, SI-2000 Maribor, Slovenia;2.Faculty of Natural Science and Mathematics, University of Maribor,Koro?ka cesta 160, SI-2000 Maribor, Slovenia;3.Mathematics Department, University of North Carolina at Charlotte, Charlotte,North Carolina 28223, USA)

1 Introduction and Background

Suppose an analytic system of differential equations

(1)

Although much of the theoretical content of this paper applies to systems innwe will limit our considerations to three-dimensional systems (1) since already in this situation the computational burden in practical problems is enormous,lying at or beyond the limit of what is feasible.Moreover the general problems in this context of which we are aware are in3.

SupposeUis an open neighborhood of the origin in3,f:U→3is a real analytic mapping,and that df(0) has one non-zero and two pure imaginary eigenvalues.By an invertible linear change of coordinates and a possibly negative rescaling of time the system of differential equationscan be written in the form

(2)

whereλis a positive real number.We will let X denote the corresponding vector field

(3)

on a neighborhood of the origin.

For system (2),for everyr∈there exists in a sufficiently small neighborhood of the origin aCrinvariant manifoldWc,the local center manifold,that is tangent to the (u,v)-plane at the origin and which contains all the recurrent behavior of system (2) in a neighborhood of the origin in3([1,§4.1],[2,§2],[3]).It is not necessarily unique,but the local flows near the origin on any twoCr+1center manifolds areCrconjugate in a neighborhood of the origin[4].This fact justifies our abuse of language in speaking below of a center on "the" center manifold.The following theorem,which originates from the work of Lyapunov[5],is proved in [6,§13].Analyticity ofWcis a consequence of the analyticity of the distinguished normalizing transformation that brings the system to its quasi-normal form.Uniqueness follows from the same fact (as well as from a general theorem,Theorem 3.2 in [3]).

Theorem1(Lyapunov Center Theorem). For system (2) with corresponding vector field (3),the origin is a center for X|Wcif and only if X admits a real analytic local first integral of the formΦ(u,v,w)=u2+v2+… in a neighborhood of the origin in3.Moreover when there exists a center the local center manifoldWcis unique and is analytic.

By Theorem 1 existence of a center of X|Wcis equivalent to existence of a first integral for X,so we can restrict our efforts to investigation of conditions for existence of an integralΦ,which can be assumed to have no constant term,hence must have the formΦ(u,v,w)=u2+v2+….

In this paper we investigate the situation in which family (2) has the form

(4)

whereS1,S2,andS3are fixed finite subsets of03,every element of which satisfiesp+q+r≥2.Introducing the complex variablex=u+iv,the first two equations in (4) are equivalent to a single equationXis a polynomial function.Adjoining to this equation its complex conjugate,replacingeverywhere byy,regardingyas an independent complex variable,and replacingwbyzsimply as a notational convenience we obtain the complexification of family (4),

(5)

In this contribution we find and exploit the structure inherent in the focus quantities and in the coefficients ofΨthat allows an increase in the range of what is computationally feasible.Our algorithm increases the speed of computations considerably.Our result shows that the structure of the focus quantities for the three dimensional family of interest is similar to the structure of the focus quantities of two-dimensional systems and thus can be used for the analysis of limit cycle bifurcations in the framework of the approach suggested in [7].Additionally,although it is outside the main thrust of this study we also extend,in Theorem 2,a result of [8] on the structure of the set in parameter space corresponding to systems having a center on the center manifold at the origin.

2 The Focus Quantities and the Center Variety

In this section we investigate the existence of a first integralΨfor a system in family (5) by a straightforward approach,that is,by computing the coefficients of ZΨand equating them to zero.The special form of the coefficients ofΨand ZΨis more easily expressed and the discussion is much simplified by expressing Z andΨin the forms

(6)

and

(7)

whereS?-1×0×0is a set oftriples,all satisfying 1≤p+q+r≤N-1, andT?0×0×-1is a set ofmtriples,all satisfying 1≤P+Q+R≤N-1,for someN≥2.For the same reason we write

(8)

(although the first nonzero terms are of order three) with a similar shift in the first two subscripts.The family (6) is not completely general,but contains every family that arises as the complexification of a real family (2).

With the notation just introduced ZΨis

(9)

We seek an expression for the coefficientgk1,k2,k3ofxk1+1yk2+1zk3.The seven terms in (9) that arise from the product of a monomial times another monomial or a sum are clear.The power onzin the product of the sums in the first line of (9) isn+r=k3.Fix the value ofras some values,0≤s≤min{k3,N}.The product is

where we have allowedj=0 in the first sum with the obvious understanding that the term is not present whenj=0.

For any pair (j,k) satisfyingj+k≥3-k3+s,the powers onxandyin the product arek1+1 andk2+1,respectively,if and only ifp=k1-j+1 andq=k2-k+1,so the coefficient sought is

-ijvj-1,k-1,k3-sak1-j+1,k2-k +1,s,

although it is not present if (k1-j+1,k2-k+1,s)?S.

Sincej+k=(k1-p+1)+(k2-q+1)=k1+k2+2-(p+q),the largest relevant value ofj+koccurs whenp+qis minimal.Becausep+q+s≥1,p+q≥1-s;because (p,q)∈-1×0,p+q≥-1.Thus min{p+q}=max{1-s,-1},hencej+k≤k1+k2+min{s+1,3}.Thus forr=sthe contribution to the coefficient ofzk3is

so that summing over all admissiblerwe obtain

(10)

where the prime on the sums signifies that the corresponding term is not present if (k1-j+1,k2-k+1,s)?S.

By practically identical reasoning the contribution to the coefficientgk1,k2,k3ofxk1+1yk2+1zk3made by the product of the sums in the second line of (9) is

(11)

where the prime on the sums now signifies that the corresponding term is not present if (k2-k+1,k1-j+1,s)?S(a reversal of the entries in the first two positions from the case for the product of sums in the first line of (9)).

The power onzin the product of the sums in the third line of (9) isn+R=k3.Fix the value ofRas some values,where in this case -1≤s≤min{k3,N-1}.The product is

For any pair (j,k) satisfyingj+k≥3-k3+s,the powers onxandyin the product arek1+1 andk2+1,respectively,if and only ifP=k1-j+1 andQ=k2-k+1,so the coefficient sought is

-(k3-s)vj-1,k-1,k3-sck1-j+1,k2-k +1,s,

although it is not present if (k1-j+1,k2-k+1,s)?T.

Sincej+k=(k1-P+1)+(k2-Q+1)=k1+k2+2-(P+Q),the largest relevant value ofj+koccurs whenP+Qis minimal.BecauseP+Q+s≥1,P+Q≥1-s;because (P,Q)∈0×0,P+Q≥0.Thus min{P+Q}=max{1-s,0},hencej+k≤k1+k2+min{s+1,2}.Thus forR=sthe contribution to the coefficient ofzk3is

so that summing over all admissibleRwe obtain

(12)

where the double prime on the sums signifies that the corresponding term is not present if (k1-j+1,k2-k+1,s)?T.

Combining (10),(11),and (12) with the five terms in the product (9) arising from monomials and that do not cancel out gives us that the coefficientgk1,k2,k3ofxk1+1yk2+1zk3is

-iak1,k2,k3+i(k1+1)vk1,k2,k3+ibk1,k2,k3-i(k2+1)vk1,k2,k3-λk3vk1,k2,k3

(13)

where the primes and double primes on the sums have the same meaning as just above.

The maximum of the sum of the subscripts onvαβγin the sums isk1+k2+k3-1.Thus except when (k1,k2,k3)=(K,K,0) forK∈,the equationgk1k2k3=0 can be solved uniquely forvk1k2k3in terms of the known quantitiesvαβγwithα+β+γ

(14)

where we have incorporated the terms -iaKK0andibKK0from the first two lines in (13) into the sums using the assignmentsv110=1 andvαβγ=0 forα+β+γ=2 but (α,β,γ)≠(1,1,0) from the second expression in (7).

The focus quantitiesg000=0 andg110are uniquely determined,but the remaining ones depend on the choices made forvKK0,K∈,K≥1.Once such an assignment is madeΨis determined and satisfies

ZΨ(x,y,z)=g110(xy)2+g220(xy)3+….

(15)

We will need an expression for the coefficientvk1k2k3ofxk1+1yk2 +1zk3inΨ.To simplify the expression forvk1k2k3,we note that if the first two inner sums in (13) are started atj+k=2-k3+sthen the additional terms picked up are precisely the two "loose" terms -iak1,k2,k3andibk1,k2,k3.For suppose thats=k3-t,where 0≤t≤k3.Thenj+k=2+(k3-t)-k3=2-t≥0 so in factt∈{0,1,2}.

Fort=0,s=k3so the sum now starts atj+k=2+s-k3=2.Thus additional terms arise for (j,k)∈{(2,0),(1,1),(0,2)} for whichvj-1,k-1,k3-s=vj-1,k-1,k3-tis respectivelyv1,-1,0=0,v000=1,andv-1,1,0=0,so in the first sum the only additional term isak1k2k3and in the second sum the only additional term isbk1k2k3.

Fort=1,s=k3-1 so the sum now starts atj+k=2+s-k3=1.Thus additional terms arise for (j,k)∈{(1,0),(0,1)} for whichvj-1,k-1,k3-s=vj-1,k-1,1is respectivelyv0,-1,1=0 andv-1,0,1=0,so no terms are added.

Fort=2,s=k3-2 so the sum now starts atj+k=2+s-k3=0.Thus additional terms arise for (j,k)=(0,0) for whichvj-1,k-1,k3-s=vj-1,k-1,2isv-1,-1,2=0,so no terms are added.

From (13) we have then that when -λk3+(k1-k2)i≠0,

(16)

where the primes on the first two sums indicate thatak1-j+1,k2-k +1,sis to be replaced by 0 if (k1-j+1,k2-k+1,s)?Sandbk1-j+1,k2-k +1,sis to be replaced by 0 if (k2-k+1,k1-j+1,s)?S,and the double prime on the third sum indicates that the corresponding term does not appear if(k1-j+1,k2-k+1,s)?T.Like the focus quantities,these coefficients are polynomials in the coefficients of the nonlinear terms in (6).

RemarkFormulas similar to(14) and (16) are obtained also in [9,10].In fact,these formulas are generalizations of ones obtained for the two-dimensional case in [11-13].

The vanishing of all the focus quantities is obviously sufficient for existence of a formal first integral for system (6).In [8] it was shown to be necessary.As a corollary to this it was deduced that for family (4),for each fixed choice ofλthere exists a varietyVC(λ) in the space of admissible coefficients of the polynomials such that the origin is a center for X |Wcif and only if the coefficients of the components of X lie inVC.Further analysis yields the following result.

Theorem2Consider a family (4) on3.Let (λ,A,B,C) denote a parameter string and letEdenote the set of admissible parameters,the subset ofMfor someMthat corresponds toλ≠0.There exists a varietyVCinMsuch that the system (4) with parameter string (λ,A,B,C)∈Ehas a center on the local center manifold at the origin in3if and only if (λ,A,B,C) lies inE∩VC.

ProofThe functionsvk1k2k3fork1+k2+k3=0 are constants.Thus by (16) whenλis allowed to assume complex values fork1+k2+k3=1 each functionvk1k2k3fails to exist only at four values,each of the formλ=ri,r∈. Arguing recursively on the value ofk1+k2+k3we see that eachvk1k2k3,if it is notvkk0=0,fails to exist only at a finite number of values,each of the formλ=ri,r∈.

This discussion and equation (14) show thatgkk0fails to exist only at a finite number of valuesri,r∈,and that when a common denominator of all the fractions is taken andgkk0is simplified it has the form

dk(λ)=(r1λ+is1)…(rhλ+ish)

forrj∈andsj∈for allj.

Z=∪{(ri,a,b,c) :riis a root ofdkfor somek}

a point ofVIcorresponds to a system on3in family (6) for which there exists a formal first integral of the form (7).The setVIis not closed,hence does not form a variety inN.

On the other hand,when we restrict to the case that (6) is the complexification of a real family (4),λmust be real and toVIthere corresponds a varietyVCin the spaceMof the original coefficients (λ,A,B,C) of the original real family.Points inE∩VCcorrespond to exactly those elements of the original family (4) for which there is a center at the origin in the center manifold.

3 The Structure of the Focus Quantities and a Computational Algorithm

Suppose the nonlinearities in (2),hence in (6),are the full set of homogeneous quadratic polynomials,so that in (6)

S={(1,0,0),(0,1,0),(0,0,1),(-1,2,0),(-1,1,1),(-1,0,2)}

and

T={(2,0,-1),(1,1,-1),(1,0,0),(0,2,-1),(0,1,0),(0,0,1)}.

Easy computations show that forΨgiven by (7) and ZΨexpressed by (8) we have that,for example,

and,definingG110=[λ(-2+λi) (λ-2i]-1for simplicity,

g110=G110(-4a001c1,1,-1+4b001c1,1,-1+4λa010a100-4λb010b100+2λib1,-1,1c0,2,-1+2λia-1,1,1c2,0,-1+

λ2b1,-1,1c0,2,-1-λ2a001c1,1,-1+λ2b001c1,1,-1-λ2a-1,1,1c2,0,-1+λ3a010a100-λ3b010b100).

For every monomial in any of these expressions,if the exponent on each factor is multiplied times the subscript on that factor,regarded as an ordered triple,and the resulting triples are added for that monomial,the result is the subscriptαβγonvαβγorgαβγ.

To prove that this is true in general,we write the indexing setsSandTas

S={(p1,q1,r1),…,(p,q,r)}

and

T={(P1,Q1,R1),…,(Pm,Qm,Rm)}

ap1 q1 r1,…,ap q r,bq p r,…bq1 p1 r1,cP1 Q1 R1,…,cPm Qm Rm

so that a single monomial in the polynomial ring with these coefficients as indeterminates is

This monomial will be denoted

[ν]=[ν1,…,ν]

for short.The ring of polynomials with these coefficients as indeterminates and coefficients in a fieldk(typicallyor) will be denotedk[a,b,c].For an elementfofk[a,b,c],Supp(f) is the collection of exponent stringsν∈0for which the coefficient of [ν] infis nonzero.

Define a mappingL:0→3by

L(ν1,…,ν) =ν1(p1,q1,r1)+…+ν(p,q,r) +ν(q,p,r)+…+ν(q1,p1,r1) =

ν(P1,Q1,R1)+…+ν(Pm,Qm,Rm)

(p1ν1+…+pν+qν+…+q1ν+P1ν+…+Pmν,

q1ν1+…+qν+pν+…+p1ν+Q1ν+…+Qmν,

r1ν1+…+rν+rν+…+r1ν+R1ν+…+Rmν).

(17)

Definition3For (j,k,n)∈××,a polynomialf=∑ν∈Supp(f)f(ν)[ν] in[a,b,c] is a (j,k,n)-polynomial if,for everyν∈Supp(f),L(ν)=(j,k,n) for the mappingLdefined by (17).

The structure in the focus quantitiesgkk0as well as in the coefficientsvα,β,γofΨare given in the following theorem.

Theorem4Let a family (6) be given.There exists a formal power seriesΨ(x,y,z) of the form (7) and functionsgkk0,k≥1,such that

1.equation (15) holds;

2.for every fixedλ∈+,for every triple (j,k,n)∈-1×-1×0,j+k+n≥1,vjkn∈[a,b,c] andvjknis a (j,k,n)-polynomial;

3.for everyk∈,vkk0=0;and

4.for every fixedλ∈+,for everyk∈,igkk0∈[a,b,c] andgkk0is a (k,k,0)-polynomial.

ProofDisplay (8) and the discussion following it show that if in (7) the coefficientvk1 k2 k3ofxk1+1yk2+1zk3is defined as zero ifk1-k2=k3=0 and inductively onk1+k2+k3by (16) otherwise then equation (15) holds withgKK0given by(14). It is clear that for any fixed value ofλ∈+,vjknandigkk0lie in[a,b,c].Thus we need only show,again for fixedλ∈+,thatvjknis a (j,k,n)-polynomial and thatgkk0is a (k,k,0)-polynomial.

The proof thatvjknis a (j,k,n)-polynomial is by mathematical induction onj+k+n.

Basis step.Ifj+k+n=0 but (j,k,n)≠(0,0,0) thenvjkn≡0 so Supp(vjkn)=? and the result holds vacuously.Sincev000≡1=1[(0,…,0)],Supp(v000) is the singleton set containing only (0,…,0),on whichLevaluates to (0,0,0).

Inductive step.The idea of the proof is the following.

i. A monomialapqr,bqpr,orcPQRis [μ] for someμ=(0,…,1,…,0) andL(μ) is (p,q,r) in the first case,(q,p,r) in the second case,and (P,Q,R) in the third.

ii. In each sum in (16) the inductive hypothesis applies tovj-1,k-1,k3-s.

ii. Since every monomial [ν] invα,β,γis an (α,β,γ)-polynomial and every monomial invα,β,γapqr,vα,β,γbqpr,andvα,β,γcPQRis [ν][μ]=[ν+μ],by linearity ofL,for every monomial appearing in any of the sums

L(ν+μ) =L(ν)+L(μ) =

(j-1,k-1,k3-s)+(k1-j+1,k2-k+1,s)=(k1,k2,k3).

In detail,supposevα,β,γis an (α,β,γ)-polynomial wheneverα+β+γ≤dand fix (k1,k2,k3) withk1+k2+k3=d+1.Consider any summand

vj-1,k-1,k3-sak1-j+1,k2-k +1,s

in the first sum in (16), which is present if and only if (k1-j+1,k2-k+1,s)∈S,in which case it is (pw,qw,rw) for somew∈{1,…,}.Thenak1-j+1 ,k2-k+1,s=[μ] forL(μ)=(pw,qw,rw)=(k1-j+1,k2-k+1,s).Then

Since

(j-1)+(k-1)+(k3-s)=(j+k)+k3-s-2≤

(k1+k2+min{s+1,3})+k3-s-2≤k1+k2+k3-1=d,

by the inductive hypothesisL(ν)=(j-1,k-1,s) and by additivity ofL

L(ν+μ)=L(ν)+L(μ)=(j-1,k-1,s)+(k1-j+1,k2-k+1,s)=(k1,k2,k3).

For a summand in the second sum the proof is the same except that now when the monomialbk1-j+1,k2-k +1,sis present (k1-j+1,k2-k+1,s) is (qw,pw,rw) for somew∈{1,…,} so thatbk1-j+1 ,k2-k+1,s=[μ] forL(μ)=(qw,pw,rw),which is again (k1-j+1,k2-k+1,s) and the proof continues as before.

The proof for any term in the third sum involves a similar small change in detail.

The proof thatgkk0is a (k,k,0)-polynomial is along exactly the same lines.

The structure in the coefficientsvα,β,γofΨand in the focus quantitiesgkk0may be exploited to compute them efficiently.To do so we define a mappingV:0→recursively with respect to |ν|=ν1+…+νas follows:

V(0,…,0)=1;

(18)

forν≠(0,…,0)

V(ν)=0 ifL1(ν)-L2(ν)=L3(ν)=0,

(19)

and otherwise

(20)

where

Lemma5Ifν∈0is such that eitherL1(ν)<-1,L2(ν)<-1, orL3(ν)<0,thenV(ν)=0.

ProofThe proof is by induction on |ν|.

a.if 1≤w≤:L(ν)=(pw,qw,rw)∈-1×0×0so the result holds vacuously;

Inductive step.Suppose the lemma holds for allνwith |ν|≤dand letνbe such that |ν|=d+1 and eitherL1(ν)<-1,L2(ν)<-1,orL3(ν)<0 butL3(ν)(L1(ν)-L2(ν))≠0.

a. ifμarises in the first sum,

L1(μ)=L1(ν1,…,νj-1,…ν)=L1(ν)-pj<-1;

b. ifμarises in the second sum,

L1(μ)=L1(ν1,…,νj-1,…ν)=L1(ν)-q<-2;

c. ifμarises in the third sum,

L1(μ)=L1(ν1,…,νj-1,…ν)=L1(ν)-Pj-2<-2.

The arguments ifL2(ν)<-2 orL3(ν)<-1 are practically identical.

The proof for the casesL2(ν)=-2 andL3(ν)=-1 are analogous.

Theorem6For a family (6) letΨbe the formal series of the form (7),let {gkk0:k∈} be the functions given by (16) and(14) which satisfy the conditions of Theorem 4,and letVbe the mapping defined by (18),(19),and (20).Then

1. forν∈Supp(vk1 k2 k3) the coefficientvk1 k2 k3(ν)of [ν] isV(ν);

2. forν∈Supp(gk k 0) the coefficientgk k 0(ν)of [ν] is

ProofThe proof of part (1) is by induction onk1+k2+k3.

Basis step.Fork1+k2+k3=0,Supp(vk1k2k3)=? except forv000and

yieldsv000(0,…,0)=1=V(0,…,0).

Inductive step.Suppose statement (1) holds for allvk1k2k3for whichk1+k2+k3≤d.Let (k1,k2,k3) be such thatk1+k2+k3=d+1 but (k1-k2)k3≠0 and fix anyν∈0for whichL(ν)=(k1,k2,k3) (sincevk1k2k3is a (k1,k2,k3)-polynomial).We will find the contribution of each sum in (16) to

Consider the first sum in (16).If for any triple (j,k,s) the corresponding summand actually appears,then the subscript onak1-j+1,k2-k +1,smust actually be (pw,qw,rw) for some indexw∈{1,…,},and conversely for any suchwat most one triple (j,k,s) is possible:

j=k1-pw+1,k=k2-qw+1,s=rw.

(21)

Thus the first sum in (16) can be expressed as (omitting the factor -i)

(22)

where the prime indicates that the corresponding summand appears only if

k1-pw≥-1,k2-qw≥-1, andk3-rw≥0.

(23)

(μ1,…,μ)=(ν1,…,νw-1,…,ν),

and it must be the case that

νw-1≥0 and (ν1,…,νw-1,…,ν)∈Supp(vk1-pw,k2-qw,k3-rw).

(24)

(25)

where the prime on the sum indicates that the corresponding term is not present if any one of the conditions in (23) and (24) fails.

ν1+…+ (νw-1) +…+ν=|ν|-1=(k1+k2+k3) -1=(d+1)-1=d

the induction hypothesis applies,and

so the same substitution in (25) leaves the sum unchanged.

Ifw∈{1,…,} is such thatνw≥1 but at least one condition in (23) fails,the corresponding summand in (25) is absent.Then because

L(ν1,…,νw-1,…,ν)=(k1-pw,k2-qw,k3-rw)

by Lemma 5V(μ1,…,νw-1,…,ν)=0 so the same replacement may be made in (25) without changing the sum.

In summary,the first sum in (16) has the same value as the first sum in (20).

The argument for the second sum in (16) is analogous.For convenience temporarily letW=2-w+1.Whenbk1-j+1,k2-k +1,sis present its subscript is(qW,pW,rW).Equalities (21) are replaced by

j=k1-qW+1,k=k2-pW+1,s=rW

and (22) is replaced by

where the prime now indicates that the corresponding summand appears only if

k1-qW≥-1,k2-pW≥-1, andk3-rW≥0.

(26)

Condition (24) is replaced by

νw-1≥0 and (ν1,…,νw-1,…,ν)∈Supp(vk1-qW,k2-pW,k3-rW)

(27)

where the prime on the sum indicates that the corresponding term is not present if any one of the conditions in (26) and (27) fails.

Repeating the rest of the argument for the first sum almost verbatim we find that the second sum in (16) has the same value as the second sum in (20).

The argument for the third sum in (16) is analogous.For convenience temporarily letW=w-2.When the coefficientck1-j+1,k2-k +1,sis present its subscript is (PW,QW,RW).Equalities (21) are replaced by

j=k1-PW+1,k=k2-QW+1,s=RW

and (22) is replaced by

where the double prime indicates that the corresponding summand appears only if

k1-PW≥-1,k2-QW≥-1, andk3-RW≥0.

(28)

Condition (24) is replaced by

νw-1≥0 and (ν1,…,νw-1,…,ν)∈Supp(vk1-PW,k2-QW,k3-RW)

(29)

where the double prime on the sum indicates that the corresponding term is not present if any one of the conditions in (28) and (29) fails.

Repeating the rest of the argument for the first sum almost verbatim we find that the third sum in (16) has the same value as the third sum in (20),concluding the proof of statement (1).

The same kind of argument gives statement (2) of the theorem.

Mathematica code for an algorithm for computation of the focus quantities based on Theorems 4 and 6,as it applies to system (30) below,is given in the Appendix.It is clear how to modify it so as to apply it to any system of the form (5),and to obtain from its output the focus quantities expressed in terms of the coefficients of the system (4),of which it is the complexification.

4 Efficiency of computations

As is well-know studies on the center problem even for two dimensional systems often involve very laborious computations (see,e.g.[14,15]).To check the efficiency of our algorithm we have performed the computation of the first five focus quantities for the system

(30)

Whenλ=1 this is the system considered in [8].

The first quantity of the system computed practically immediately is

(31)

Since the other quantities are so long we do not present them here but simply note that computations based on(14) and (16) on the one hand and those based on the structurein the focus quantities,Theorems 4 and 6,on the other agree and have the following related timings.Computing in Mathematica on a desktop computer using an algorithm based on formulas(14) and (16),which the reader can easily reproduce,the computation ofg220throughg550took 2,5,119,and 1388 seconds of CPU time,respectively.Computation ofg220throughg550on the same machine but using the Mathematica code presented in the Appendix,based on Theorems 4 and 6,required 2,5,54,and 642 seconds of CPU time,respectively.

The computational test shows that the advanced algorithm based on Theorem 6 is more efficient than the one based on formulas(14) and (16),which is to be expected,since in the former case the computations involve only arithmetic operations with complex numbers while in the latter case one must perform polynomial computations.

Appendix

We used the following Mathematica code based on the formulas of Theorem 6 to compute the focus quantities of system (30).In the code "m1" stands for "-1" and "la" forλ.

l1 [nu1_,nu2_,nu3_,nu4_,nu5_,nu6_,nu7_,nu8_,nu9_] :=

0 nu1+0 nu2-1 nu3+1 nu4+0 nu5+1 nu6+1 nu7+1 nu8+

0 nu9;

l2 [nu1_,nu2_,nu3_,nu4_,nu5_,nu6_,nu7_,nu8_,nu9_] :=

1 nu1+0 nu2+1 nu3-1 nu4+0 nu5+0 nu6+1 nu7+0 nu8+

1 nu9;

l3 [nu1_,nu2_,nu3_,nu4_,nu5_,nu6_,nu7_,nu8_,nu9_] :=

0 nu1+1 nu2+1 nu3+1 nu4+1 nu5+0 nu6-1 nu7+0 nu8+

0 nu9;

gg[k_]:=gg[k]=Module[{fq},fq=0;

Do[If [ (l1[k1,k2,k3,k4,k5,k6,k7,k8,k9]== k &&

l2[k1,k2,k3,k4,k5,k6,k7,k8,k9]== k &&

l3[k1,k2,k3,k4,k5,k6,k7,k8,k9]== 0),

{g[ k1,k2,k3,k4,k5,k6,k7,k8,k9]=

Together[

(-I ((l1[k1-1,k2,k3,k4,k5,k6,k7,k8,k9]+1)

v[k1-1,k2,k3,k4,k5,k6,k7,k8,k9] +

(l1[k1,k2-1,k3,k4,k5,k6,k7,k8,k9]+1)

v[k1,k2-1,k3,k4,k5,k6,k7,k8,k9] +

( l1[k1,k2,k3-1,k4,k5,k6,k7,k8,k9]+1)

v[k1,k2,k3-1,k4,k5,k6,k7,k8,k9]) +

I ((l2[k1,k2,k3,k4-1,k5,k6,k7,k8,k9]+1)

v[k1,k2,k3,k4-1,k5,k6,k7,k8,k9] +

(l2[k1,k2,k3,k4,k5-1,k6,k7,k8,k9]+1)

v[k1,k2,k3,k4,k5-1,k6,k7,k8,k9] +

(l2[k1,k2,k3,k4,k5,k6-1,k7,k8,k9]+1)

v[k1,k2,k3,k4,k5,k6-1,k7,k8,k9]) -

(l3[k1,k2,k3,k4,k5,k6,k7-1,k8,k9]

v[k1,k2,k3,k4,k5,k6,k7-1,k8,k9] +

l3[k1,k2,k3,k4,k5,k6,k7,k8-1,k9]

v[k1,k2,k3,k4,k5,k6,k7,k8-1,k9] +

l3[k1,k2,k3,k4,k5,k6,k7,k8,k9-1]

v[k1,k2,k3,k4,k5,k6,k7,k8,k9-1] ))],

fq=fq+g[ k1,k2,k3,k4,k5,k6,k7,k8,k9]

a010^k1 a001^k2 am111^k3 b1m11^k4 b001^k5

b100^k6 c11m1^k7 c100^k8 c010^k9 ],

{k1,0,k},{k2,0,k},{k3,0,k},{k4,0,k},

{k5,0,k},{k6,0,k},{k7,0,k},{k8,0,k},

{k9,0,k}];

Factor[fq]]

v[0,0,0,0,0,0,0,0,0]=1;

v[k1_,k2_,k3_,k4_,k5_,k6_,k7_,k8_,k9_] :=

0 /; (( l1[k1,k2,k3,k4,k5,k6,k7,k8,k9]==

l2[k1,k2,k3,k4,k5,k6,k7,k8,k9] &&

l3[k1,k2,k3,k4,k5,k6,k7,k8,k9]== 0) || k1<0 ||

k2<0 || k3<0 || k4<0 || k5<0 || k6<0 || k7<0 ||

k8<0 || k9<0);

v[k1_,k2_,k3_,k4_,k5_,k6_,k7_,k8_,k9_] :=

v[k1,k2,k3,k4,k5,k6,k7,k8,k9]=

Together[1/( la l3[k1,k2,k3,k4,k5,k6,k7,k8,

k9]-(l1[k1,k2,k3,k4,k5,k6,k7,k8,k9] -

l2[k1,k2,k3,k4,k5,k6,k7,k8,k9] ) I )

(-I ((l1[k1-1,k2,k3,k4,k5,k6,k7,k8,k9]+1 )

v[k1-1,k2,k3,k4,k5,k6,k7,k8,k9] +

(l1[k1,k2-1,k3,k4,k5,k6,k7,k8,k9]+1 )

v[k1,k2-1,k3,k4,k5,k6,k7,k8,k9] +

(l1[k1,k2,k3-1,k4,k5,k6,k7,k8,k9]+1 )

v[k1,k2,k3-1,k4,k5,k6,k7,k8,k9]) +

I ( ( l2[k1,k2,k3,k4-1,k5,k6,k7,k8,k9]+1 )

v[k1,k2,k3,k4-1,k5,k6,k7,k8,k9] +

(l2[k1,k2,k3,k4,k5-1,k6,k7,k8,k9]+1)

v[k1,k2,k3,k4,k5-1,k6,k7,k8,k9] +

(l2[k1,k2,k3,k4,k5,k6-1,k7,k8,k9]+1)

v[k1,k2,k3,k4,k5,k6-1,k7,k8,k9]) -

(l3[k1,k2,k3,k4,k5,k6,k7-1,k8,k9]

v[k1,k2,k3,k4,k5,k6,k7-1,k8,k9] +

l3[k1,k2,k3,k4,k5,k6,k7,k8-1,k9]

v[k1,k2,k3,k4,k5,k6,k7,k8-1,k9] +

l3[k1,k2,k3,k4,k5,k6,k7,k8,k9-1]

v[k1,k2,k3,k4,k5,k6,k7,k8,k9-1] ))]

gg[1] // Factor

The output of the last command is (I (a001-b001) c11m1)/la,which is the first focus quantity (31).

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