Conditions for a centre and the bifurcation of limit cycles in a class of cubic systems
Conditions for a centre and the bifurcation of limit cycles in a class of cubic systems
N G Lloyd and J M Pearson Department of Mathematics, The University College of Wales, Aberystwyth, UK
1.
Introduction
For a class of cubic systems = P(x,y), y = Q(x,y)
(1.1)
in which P is linear, we consider the closely related problems of the number of limit cycles which can bifurcate out of the origin and the conditions under which the origin is a centre. in [8] the Russian mathematician Kuktes gave conditions which were said to be necessary and sufficient for the origin to be centre for systems of the form
= Xx+y, ~ =x+XY+alx2+a2xy+a3y2+a4x3+a5x2y+a6xy2+a7Y 3. (1.2) Our interest in these particular systems was stimulated by the work of Jin and Wang as reported in [7]. They describe computations suggesting that the conditions proposed by Kukles are In [3] we proved that this is indeed so, and we also incomplete (we give details in Section 3).
gave a full description of the bifurcation of limit cycles from the origin in the case when a 7 = 0. The computations for the full system (1.2) present quite severe technical problems, and we shall describe this work elsewhere. There the possibility a 2 = 0 is excluded, and our purpose The in this paper is to deal with this special case. We give necessary and sufficient conditions for the origin to be a centre, and prove that up to six limit cycles can bifurcate from the origin. of the technique recently developed by Colin Christopher, conditions for a centre are not covered by those given by Kukles, and are obtained by means described in [4], exploiting the consequences of the existence of invariant algebraic curves. This approach also yields other conditions for a centre when a 2 ~ 0; these are different from the Kukles conditions and those given here, and are described in [4].
23t This investigation is part of our group's continuing programme of research on the limit cycles of polynomial systems. We haw.~ concentrated on socalled small amplitude limit cycles, that is, limit cycles which bifurcate out of a cdticat point under perturbation of the coefficients arising in the equations themselves, and much of our recent work has been on cubic systems.
It has been known for some time that if P and Q are symmetric cubics (i.e. there are no quadratic terms) then (1.1) has at most five smallamplitude limit cycles [2]. Various classes of cubic systems with several limit cycles are described in [12]. In particular, an example is given in which six limit cycles bifurcate from the origin; Wang gives another such example in [18]. More recently, instances of cubic systems with seven smallamplitude limit cycles have been given [1,9]. In [5] we describe such an example in which there is only one quadratic term, so The main part of [5] is concerned with the description of a class of cubic
that the introduction of this one term increases the number of possible bifurcating limit cycles from five to seven. systems with eight smallamplitude limit cycles. Systems of the form
= ;Lx+y+p2(x,y)+xs(x,y), Y =x+~.y+q2(x,y)+ys(x,y ), where P2,q2 and s are homogeneous quadratic forms, have also been investigated recently. James and Yasmin [6] give necessary and sufficient conditions for a centre and have shown that the maximum number of bifurcating limit cycles is again five (note that there are nine parameters in the nonlinear part of the equations). papers [10,11] and the references contained therein. The structure of this paper is that in Section 2 we give a brief description of the technique which we use to investigate bifurcating limit cycles, while Section 3 consists of an account of our investigation of systems of the form (1.2) with a2 = O. For a description of other recent developments on the number of limit cycles of polynomial systems, we refer to the survey
2. Smallamplitude limit cycles
We consider polynomial systems in which the origin is a critical point of focus type. canonical coordinates such systems are of the form In
232
:x = Xx+y+p(x,y), and we write
y =x+Xy+q(x,y),
(2.1)
p(x,y) = P2(x,y)+P3(x,y)+...+Pn(X,y), q(x,y) = q2(x,y)+q3(x,y)+...+qn(x,y), where Pk and qk are homogeneous polynomials of degree k. Recall that the origin is a fine
focus if ;L=O. It is well known that there is a function V defined in a neighbourhood of the origin
such that V, its rate of change along orbits, is of the form
~/= "q2r2 + q2r4+ ....
where r2 = x2 + y2.
The coefficients T12k are polynomials in the coefficients arising in p and q,
and are the focal values. It is easily verified that q2 = X. The origin is a fine focus of orderkif
q2j = 0 for 1 <j < k but T12k+2 ~ 0. Clearly, the stability of the origin is determined by the first nonvanishing focal value. Consequently the significant quantities are the reduced focal values or Liapunov quantifies L(0),L(1) ..... These are the nontrivial expressions obtained by computing each rl2k subject to positive multiplicative numerical factors can be ignored. The
the conditions ~2j = 0 for j < k;
origin is a centre if all the focal values are zero;
moreover, for a given class C of systems,
there is K(C) such that the origin is a centre if Tl2 j = 0 for j _< ~:. Let K be the smallest such ~; then the origin is a fine focus of order at most K1 for systems in C. The idea is to start with a fine focus of as high order as possible, and then to introduce perturbations into p and q each of which reverses the stability of the origin and reduces its order as a fine focus by one. It is
easily seen that if the origin is a fine focus of order k, then at most k smallamplitude limit cycles can bifurcate from it [2]; however there is no guarantee that it is possible to bifurcate
this maximum n u m b e r  an example where it is not possible is given in [11] for instance. The details of this technique are described in several other papers; we refer the reader to [10,13,14], for instance. elements: As explained in these papers, the procedure consists of four
233
(1) The calculation of the focal values. (2) The reduction of the focal values. (3) Verification of the maximum possible order of the fine focus. (4) Introduction of appropriate perturbations.
In phase (1) it is necessary to use an appropriate Computer Algebra system. algorithm and the computational issues involved are described in [14]. write
V = ~ V, .xly ]
The
The basic idea is to
i+ j>_2
13
and use the equations (2.1) to obtain an expression for V; comparing coefficients, sets of
linear equations for the focal values and the Vij are obtained and these are solved symbolically The implementation of the algorithm, called F[NDETA, uses REDUCE, and the computations described in this paper have been performed on the Amdaht 5890 at the Manchester Computing Centre which we access via the JANET network. and has been designed to be very 'user friendly'. Phase (2) of the procedure also requires the use of REDUCE. Section 3, rational substitutions are taken from the relations tl 2 = q4 . . . . . As we shall see in T12k = 0 to 'reduce' A F[NDETA is described in detail in [14]
"r12k+2. This process is continued until it appears that all subsequent focal values are zero. proposed value of K is thus obtained.
In phase (3) it is confirmed that this proposed value of K is indeed the correct one. This is done by proving that the origin is a centre if t12k = 0 for k < 1 + K. Deriving conditions for a
critical point to be a centre is often a difficult problem, and necessary and sufficient conditions are known for only a few classes of systems: conditions for quadratic systems are known and for symmetric cubic systems [16], but in very few other instances. number of cases. The final phase of the procedure involves an appropriate selection of perturbations each of which reduces the order of the fine focus by one and reverses its stability. Thus, if we start with a fine focus of order k, the first step is to arrange for a perturbation such that 112. . . . = "r12k2 = 0 and "r12kT12k+2 < 0; we then continue with a sequence of perturbations until k limit cycles bifurcate. The approach using invariant algebraic curves has enabled us to derive necessary and sufficient conditions in a
234 The computations described in this paper were done using REDUCE 3.2. timings are given in [14]). We reverted
to version 3.2 after finding that version 3.3 was considerably slower for our purposes (some The version of F[NDETA used here utilises the REDUCE functions
COEFFN and SOLVE, and much use is made of the FACTORIZE facility in the second phase of the programme described above. Since completing the work described in this paper a new This
version of F[NDETA has been written in which the use of the function SOLVE is avoided.
and other improvements have made the procedure significantly more efficient, and have reduced the required cpu time by more than a half in cases such as those considered in this paper.
3.
The Kukles conditions We consider systems of the form
)~= ~.x+y, ~, =x+~,y+a 1x2+a2xy+a3y2+a4x3+a5x2y+a6xy2+a7Y 3.
(3.1)
The conditions proposed by Kukles, and given on page 124 of Nemytskii and Stepanov [15], are that the origin is a centre if X = 0 and one of the following four conditions are satisfied. (K1) M 1 = M2 = M 3 = M 4 = 0, where M 1 = a4a22 + a5~, M2 = (3a7~+l~2+a6a22)a5  3a71~2  a6a221~, M 3t~+ a l a 2 + a 5 , M4 = 9a6a22 + 2a24 + 9t~2 + 27a7~, and I~ = 3a 7 + a2a3; (K2) a 7 = M I = M 2 = M 3 = 0 ; (K3) a 7 = a 5 = a 2 = 0 ; (K4) a 7 = a 5 = a 3 = a 1 = 0 . In [7] Jin and Wang report their computation of focal values for certain systems of the form (3.1). Forthe system
= y,
~' =  x + a I x 2  2a 1y2 _ al x2y/3 _ 3a7x2y + a7y3
(3.2)
235
with 18a72 = a 14 and a 7 ~ 0, they found that T12k= 0 for k < 9, and rightly regarded this as evidence that the origin is a centre even though the system is not covered by any of the conditions (K1)  (K4). 18a72 = a 14. In [3] we proved that the origin is indeed a centre for (3.2) when
Doubt having been cast on the conditions given by Kukles, it was clearly Jin and Wang considered the subclass of
necessary to investigate the whole question afresh.
systems of the form (3.1) with a 7 = 0, and found that ~12k = 0 for k _<6 if and only if one of (K2), (K3) or (K4) holds. In [3] we also considered the case a 7 = 0 and derived the following result,
which implies that the Kukles conditions are in fact complete for this particular subclass. Theorem$.l [3] Let a 7 = 0 in system (3.1).
The origin is a centre if and only if X = O a n d
one of the following conditions holds.
(i) a 2 = a 5 = 0, (ii) a 1 = a 3 = a 5 = 0 ,
(iii) a 4 = a 5 = a 6 = 0 , a l + a 3 = 0 , (iv) a 4 = (a I +a3)a 3, a 5 =  ( a 1 +a3)a 2, a 6 =  ( a I +a3)a32(a1 +2a3) 1 .
Furthermore, at most five limit cycles can bifurcate from the origin and this maximum is attained.
We now consider another subclass of (3.1), namely that in which a2 = 0, noting that this contains the example given by Jin and Wang. We shall give necessary and sufficient The conditions for a centre and show that up to six limit cycles can bifurcate from the origin.
interesting feature of this result is that there are as many smallamplitude limit cycles as there are parameters in the system. In most cases which have been studied before there are many The possibility a 2 = 0 was excluded in the
fewer bifurcating limit cycles than parameters.
investigation of the full system (3.1) because the required substitutions for the reduction of the focal values are defined only for a 2 ~ 0. Significantly, if a2 = 0 then the system (3.1) without the cubic terms is symmetric about the xaxis, and so the origin is a centre. such conditions are given in [4] when a 2 ~ 0. We therefore consider systems of the form
° 3 2 2 ~=y, y = _ x + al x 2 +a3Y 2 +a4x +a5x Y+a6xY +a7Y 3
As noted in
Section 1, the centre conditions are not covered by (K1)  (K4), and we emphasise that other
(3.3)
236
and suppose throughout that a 7 ~ 0. w h e r e b l = a l + a 3 and b 2 = 5 a 1+3a 3,
It is convenient to replace a 1 and a 3 by b 1 and b 2,
We use F[NDETA to compute the focal values.
First we find that L(1) = a5+3a 7.
For a
fine focus of order greater than one we must have L(1) = 0; we take
a 5 =  3 a 7. Further computation then gives L(2) = a7(3a4+b 1 b2+a6). two we need L(2) = 0; since a 7 ~ 0, we take
(3.4) For a fine focus of order greater than
a 4 =  ( b 1 b2+a6)/3.
(3.5)
Continuing with FINDETA we find that L(3) = a7(36a72+f(a6,b1 ,b2) ), where
f(a6,b 1 ,b2) = _4a62+4a6b22+8a6b1 b2_ 104a6b12_2b23b1 +9b22b12+130b2b13_ 135bl 4. Since a 7 ~ 0, for a fine focus of order greater than three we take
a72 = f(a6,b 1 ,b2)/36, noting that a6,b 1 and b 2 must be such that f > 0;
(3.6) in particular, b 1 and b 2 cannot both be zero.
Further computation now gives L(4) = a7(Aa62+Ba6+C ), where
A =  8 0 ( b 2  4 b 1 )2, B = 2(b24b 1 )(6b23+58b22b 11102b2b 121935b 13), C =  b 2 ( b 2  b 1 )(6b2416b23b 11325b22b 12+6165b2b 13+26730bl 4). For a fine focus of order greater than four, we must, of course, have L(4) = 0. B24AC > 0, and suppose that We need
a 6 = (B+D)/2A,
(3.7)
237 where D 2 = B24AC, noting for future reference that D has two possible values. b3 = b24b 1. We compute that B24AC = b32~(b 1 ,b2), where Let
~(bl,b2)= 653625b16+291060b15b2+43416b14b225148b13b23 900b12b24+24blb25+4b26. We exclude for the present the possibility that b3 = 0. Continuing the computation of focal
1
values using FINDETA, we find that L(5) =a7(GD+H)b 3 1 where
G =(184b262892b25b131596b24b12+356142b23b13+1551708b22b144210434b2b15 25431345b16 ) and
H = 368b294680b28b1123600b27b12+1091796b26b13+14126832b25b14 59449766b24b15648352836b23b16355378374b22b17 +8039416860b2b18+20526992325b19.
For a fine focus of order greater than five we need L(5) = 0, and we therefore have D = H/G if G ~ 0, or G=H=0 otherwise. Using the RESULTANT function in REDUCE, the resultant of G
and H is nonzero unless b I = b2 = 0, a possibility which we have already excluded. It follows that G and H cannot be zero simultaneously, and so we suppose that G ~ 0. two expressions for D: for consistency we require that B2  4AC = (H/G)2; (3.8) There are now
moreover, D has the sign ofH/G. and only if
A straightforward calculation tells us that (3.8) is satisfied if
V(bl ,b2) = b 16(b2b 1 )F(b 1 ,b2) = 0,
238 where F(bl,b2) = 3186845749080b111

+ 1048502846790b110b2
+ 193976300259b19b22
54272788740b18b23 22059722583b17b24 + 7321569165b16b25 570046311b15b26  34456359b14b27 + 8127591b13b28  495966b12b29 + 12302bib210  92b211

Henceb 1 = 0
(andb 2 ~ 0 ) orb l = b 2 ~ 0 o r F ( b 1,b2)=0.
We suppose for the moment that
b 1 ~ 0 and b l  b 2 ~ 0. F[NDETA, and find that
With F(b 1,b2) = 0, we continue the calculation of focal values using
L(6) = aTb 16(b2b 1)x(b 1 ,b2)/b32G, where %(bl,b2) = 7124b211 951494b210b1 + 39115942b29b12  737410267b28b13 + 6661763643b27b14 18667396833b26b15 130220619285b25b16 + 1047738480831b24617 2171066759280b23b18 + 4793413413537b22b1936632809942470b2b110 + 80202151292760bl 11 We again use the RESULTANT function of REDUCE and find that the resultant of F and ~ is nonzero unlessb l = b 2 = 0 . Thusx~0whenF=0.
We summarise the discussion so far in the following lemma. Lemma 3.2 Suppose that the following conditions hold. (1) a7 ~ 0, b I ~ 0, b 1b 2 ~ 0, 4b 1b 2 ~ 0, f(a6,b I ,b2) > 0, ~(b 1 ,b2) > 0, (2) a 5 =  3 a 7, (3) a4 =  ( b 1b2+a6)/3, (4) a72 = f(as,b 1 ,b2)/36 , (5) as = (B+D)/2A, (6) F(b 1 ,b2) = 0.
Then the origin is a fine focus of order six. If any of the conditions (2)  (6) are violated, then
the order of the fine focus at the origin is less than six.
239
We shall see later that these conditions can be satisfied simultaneously. we consider the cases excluded in Lemma 3.2.
First, however, Referring to
Suppose that b 1 = 0, but b2 ~ 0.
the expressions given above for the Liapunov quantities and recalling that a 7 ;~ 0, for a fine focus of order at least four we have a 5 =  3 a 7, a 4 = a6/3, a72 = 4a6(b22a6)/36. L(4) = a 7 ( 1 2 b 2 2  8 0 a 6 ) b 2 2 a 6 . hypothesis; Then
Now a 6 = 0 implies that a 7 = 0, which is contrary to We compute L(5) = a7b28. Since both a 7 and b 2
hence we take a 6 = 3b22/20.
are nonzero by hypothesis, the origin is of order at most five.
In the second excluded case, b I = b2 ~ 0. a 1 =  a3/2 ? 0. For a fine focus of order at least four,
In terms of a 1 and a 3 this means that
a 5 =  3 a 7, a 4 =  ( 4 a 6 + a 3 2 ) / 1 2 ,
288a72 =(32a62+184a6a32a34).
Computing TI10, we have
L(4) = a32a6a7(160a6991 a52).
If a 3 = 0 then a 1 = 0, which is exclude?l. a 6 = 0; it follows that
If a6 = 991a32/160 we compute that L(5) = a38a 7,
and since neither a 3 nor a 7 is zero, fo" a fine focus of order greater than five we must have
a 1 =  a 3 / 2 , a 4 =a32/12, a 5 =  3 a 7 , a72 = a14/18. This is the situation covered by the example of Jin and Wang, which was proved in [3] to imply that the origin is a centre.
It remains to consider the case in which b3=0; this means that a l = a 3. If L(1) = L(2) = 0, we h a v e a 5 =  3 a 7 and a 4 =  ( ~ 6 + 1 6 a 3 2 ) / 3 . From L(3) = 0 we h a v e 9a72 = Now a3 = 0 implies that
(a62+8a6a3160a34), and further computation gives L(4) = a36a 7. a 1 = 0 and 9a72 =  a 6 2 , which is impossible. four. We therefore have the following result.
Thus the origin is a fine focus of order at most
240
Theorem 3.$
Suppose that a 7 ~ 0.
The ongin is a fine focus of order at most six.
It is of
order six if f(a6,b 1 ,b2) > 0, ?b(b1 ,b2) > 0 and conditions (2) to (6) of Lemma 3.2 are satisfied. The origin is a centre if and only ff
a 3 =  2 a 1, a 4 =  a 1 2 / 3 , a 5 =  3 a 7, a 6 = 0, a72 = a14/18. We proceed to show that there are indeed systems of the form (3.3) for which all the conditions of Lemma 3.2 are satisfied, and then show how six limit cycles can bifurcate. The first task is to locate the zeros of F(b 1 ,b2), which is a homogeneous polynomial of degree 11. Real zeros of polynomials in one variable can be located by means of the classical theorem of Sturm (see [17], page 220, for example). STURM, was written. The associated computations are impossible to complete by hand in the case of polynomials of high degree, and a REDUCE procedure, calted This takes as parameters a polynomial p, the independent variable and an interval I, and returns the number of zeros of p in I. Let fl(u) = F(1,u) and f2(v) = F(v,1). Using STURM, fl has seven roots in the interval (100,100), all of which are simple, and no roots in (1,1); furthermore, there are only seven zeros of f2 in (1,1). Hence F has exactly seven distinct real factors, all of multiplicity one.
More precisely, one root of fl is located in each of the following intervals: (10,9), (3,4), (4,5), (10,12), (12,15), (15,20) and (78,79). each of these intervals. Since D has the sign of G/H, we consider G and H in We select the interval (12,15), and after
Using STURM, it transpires that GH is of one sign in three of them:
GH>0 in (3.4), while GH<0 in (12,15) and (78,79). 13.9.
some experimentation using REDUCE find that the zero of fl in (12,15) lies between t3.7 and Since we simply seek one instance where the conditions for a free focus of order six are satisfied, we do not need to enter into more precise calculations in the other intervals.
We fix an arbitrary nonzero b 1 and let b2 = ub 1 ; we then regard functions of b 1 and b 2 as functions of u alone. [ = (13.7,13.9). For a fine focus of order six, u = u,, where u, is the unique zero of fl in In addition,
Since G and H are of opposite sign for u e I, we have that D > 0.
A < 0, so a 6 is given by the smaller root of Aw2+Bw+C.
Furthermore, again using STURM, we Next we consider
find that ~ is nonzero for u ~ [, and it is easily checked that ~ > 0.
f(a6,b 1 ,b2); with a 6 as given by (3.7) we compute that f = (c~D+~)/7, where 7 = 9600b32 and
=  6 0 9 5 b 1 3 + 258b12b2 + t 3 8 b l b 2 2  34b23 , = 4863675bl 6  955620b15b2 + 126552b14b22 + 25444b13b13  9300b12b24 72blb25+68b26.
241
We first confirm by means of STURM that neither o~ nor ~ is zero for u ? I, and we check that cc < 0 and 13 > 0 in this interval; therefore c~D  ~ < 0. consider o~2D2B 2 and find that this is negative for u e I. are given by (3.4), (3.5), (3.6) and (3.7), respectively. To determine the sign of ~D+I3, we Hence f(a6,b 1 ,b2) > 0. With b2 as
chosen above we therefore find that the origin is a fine focus of order six when a5,a4,a 7 and a 6
It is now possible to describe how six limit cycles can bifurcate out of the origin.
We
start, of course, with a fine focus of order six, and suppose that the coefficients a i are as chosen above. We have already seen that F and X do not have a common zero; however, at this
stage it is necessary to be more precise, for we need to know the sign of X when u = u . It turns out that 7. has a zero wIth u E I, and so we locate u, more precisely than hitherto. u, e [1 = (13.7, 13.81) and note that ~ < 0 in t 1. The first perturbation is of b 2. We find that
Since G > 0, initially L(6) has the sign of a 7. Now
We require L(5)L(6) < 0, and so (GD+H) > 0.
G2D2H 2 = b 16(b2b 1)F(b I ,b2), and it is readily verified that F changes sign from positive to negative as u increases through u,; hence G2D2H 2 is decreasing at u = u , . But, since G > 0 Consequently, GD+H To ensure
and H < 0 for u ? 11,we have D > 0 , and so GDH > 0 f o r u e [1"
decreases as u increases through u,, and a limit cycle bifurcates if b 2 is decreased.
that the origin is a fine focus of order five after perturbation, a5,a4,a7 and a 6 are adjusted in accordance with the relations (3.4)  (3.7). The second perturbation is of a6: we require L(4)L(5) < 0, that is, Aa62+Ba6+C > 0. Now A < 0 and initially a 6 is the smaller root of Aw2+Bw+C; therefore we increase a 6, and at the same time adjust a5,a 4 and a 7 so that (3.4)  (3.6) continue to hold. Thus the stability of A limit cycle
the origin is again reversed, and its order as a fine focus is reduced by one. The third limit cycle is generated by perturbing a7; L(3)L(4) has the sign of (Aa62+Ba6+C)(36a72+f(a6,b 1 ,b2)). We therefore increase a7 and adjust a 5 to maintain (3.4). appear by perturbing a4,a 5 and introducing a nonzero X.
bifurcates and provided that the perturbation of a 6 is small enough the first limit cycle persists.
The remaining three limit cycles It may be checked that a 4 is
increased, a 5 is increased or decreased according to whether a 7 < 0 or a 7 > 0, and X is chosen so that Xa 7 > 0. We arrive at the following result. Theorem 3.4
There are systems of the form (3.3) with six smallamplitude fimit cycles, and
this is the maximum possible number.
242 References 1. J M Abdutrahman, Bifurcation of limit cycles of some polynomial systems (PhD thesis, The University College of Wales, Aberystwyth, 1989). 2. T R Blows and N G Lloyd, 'The number of limit cycles of certain polynomial differential equations', Proc. Roy. Soc. Edinburgh Sect. A 98 (1984) 215239. 3. C J Christopher and N G Lloyd, 'On the paper of Jin and Wang concerning the conditions for a centre in certain cubic systems', Bull. London Math. Soc., 94 (1990) 512. 4. C J Christopher and N G Lloyd, 'Invariant algebraic curves and conditions for a centre', pre pr)nt, The University College of Wales, Aberystwyth, 1989. 5. E M James and N G Ltoyd, 'A cubic system with eight smallamplitude limit cycles', preprint The University College of Wales, Aberystwyth, 1990. 6. E M James and N Yasmin, 'Limit cycles of a cubic system', preprint, The University College of Wales, Aberystwyth, 1989. 7. Jin Xiaofan and Wang Dongmin, 'On Kukles' conditions for the existence of a centre', Bull London Math. Soc. , 94 (1990) 14. 8. I S Kukles, 'Sur quelques cas de distinction entre un foyer et un centre', Dokt. Akad. Nauk. SSSR 42 (1944) 208211. 9. Li Jibin and Bai Jinxin, 'The cyclicity of multiple Hopf bifurcation in planar cubic differential systems: M(3) _>7', preprint, Kunming Institute of Technology, 1989. 10. N G Lloyd, 'Limit cycles of polynomial systems', New directions in dynamical systems, London Math. Soc. Lecture Notes Series No.127 (ed. T Bedford and J Swift, Cambridge University Press, 1988), pp.t92234. 11. N G Lloyd, 'The number of limit cycles of polynomial systems in the plane', Bull. Inst. Math. AppL 24 (1988), 161165. 12. N G Lloyd, T R Blows and M C Kalenge, 'Some cubic systems with several limit cycles', Nonfinearity 1 (1988) 653669. 13. N G Lloyd and S Lynch, 'Small amplitude limit cycles of certain Lienard systems', Proc. Royal Soc. London Ser A 418 (1988) t99208. 14. N G Lloyd and J M Pearson, 'REDUCE and the bifurcation of limit cycles', J. Symbolic Comput., to appear. 15. V V Nemytskii and V V Stepanov, Qualitative theory of differential equations (Princeton University Press, 1960). 16. K S Sibirskii and V A Lunkevich, 'On the conditions for a centre', Differenciarnye Uravneniya 1 (1965) 5366. 17. B L van tier Waerden, Modern Algebra, Volume 1 (English translation, Frederick Ungar, New York, 1949). 18. Wang Dongmin, 'A class of cubic differential systems with 6tuple focus', Technical report 8847.0, RISC, University of Linz, 1988.