HILBERT S 16TH PROBLEM AND BIFURCATIONS OF PLANAR POLYNOMIAL VECTOR FIELDS

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1 International Journal of Bifurcation and Chaos, Vol. 3, No. (003) c World Scientific Publishing Compan HILBERT S 6TH PROBLEM AND BIFURCATIONS OF PLANAR POLYNOMIAL VECTOR FIELDS JIBIN LI Center for Nonlinear Science Studies, School of Science, Kunming Universit of Science and Technolog, and Institute of Applied Mathematics of Yunnan Province, Kunming, Yunnan , P. R. China Received Februar, 00; Revised April 5, 00 The original Hilbert s 6th problem can be split into four parts consisting of Problems A D. In this paper, the progress of stud on Hilbert s 6th problem is presented, and the relationship between Hilbert s 6th problem and bifurcations of planar vector fields is discussed. The material is presented in eight sections. Section : Introduction: what is Hilbert s 6th problem? Section : The first part of Hilbert s 6th problem. Section 3: The second part of Hilbert s 6th problem: introduction. Section 4: Focal values, saddle values and finite cclicit in a fine focus, closed orbit and homoclinic loop. Section 5: Finiteness problem. Section 6: The weakened Hilbert s 6th problem. Section 7: Global and local bifurcations of Z q equivariant vector fields. Section 8: The rate of growth of Hilbert number H(n) with n. Kewords: Real algebraic curve; scheme of ovals; limit ccle; bifurcation of vector fields; configuration of limit ccles; Hilbert number H(n).. Introduction: What is Hilbert s 6th Problem? The following is the 6th of 3 problems posed b D. Hilbert at the Second International Congress of Mathematicians, Paris, in 900, and is still unsolved [Hilbert, 900]. 6. Problem of the topolog of algebraic curves and surfaces. The maimum number of closed and separate branches which a plane algebraic curve of the nth order can have has been determined b Harnack. There arises the further question as to the relative position of the branches in the plane. As to curves of the sith order, I have satisfied mself b a complicated process, it is true that of the eleven branches which the can have according to Harhack, b no means all can lie eternal to one another, but that one branch must eist in whose interior one branch and in whose eterior nine branches lie, or inversel. A thorough investigation of the relative position of the separate branches when their numberisthemaimumseemstometobeofver great interest, and not less so the corresponding Research is supported in part b the Natural Science Foundation of China and Yunnan Province. 47

2 48 J. Li investigation as to the number, form, and position of the sheets of an algebraic surface in space. Till now, indeed, it is not even known what is the maimum number of sheets which a surface of the fourth order in three-dimensional space can reall have. In connection with the purel algebraic problem, I wish to bring forward a question which, it seems to me, ma be attached b the same method of continuous variation of coefficients, and whose answer is of corresponding value for the topolog of families of curves defined b differential equations. This is the question as to the maimum number and position of Poincaré s boundar ccles (ccles limits) for a differential equation of the first order and degree of the form d d = Y X, where X and Y are rational integral functions of the nth degree in and. Written homogeneousl, this is ( X dz dt z d ) ( + Y z d ) dt dt dz dt ( + Z d dt d ) =0, dt where X, Y and Z are rational integral homogeneous functions of the nth degree in,, z and the latter are to be determined as functions of the parameter t. Clearl, Hilbert formulated his 6th problem b dividing it into two parts. The first part, which studies the mutual disposition of the maimal number (in the sense of Harnack) of separate branches of an algebraic curve, and also the corresponding investigation for nonsingular real algebraic varieties; and the second part, which poses the question of the maimal number and relative position of the limit ccles of the polnomial sstem d dt = P n(, ), d d = Q n(, ), (E n ) where P n and Q n are polnomials of degree n. For this problem, Llod [988] stated that the striking aspect is that the hpothesis is algebraic, while the conclusion is topological. Traditionall, the first part of Hilbert s 6th problem is the subject of stud for specialists of the real algebraic geometr, while the second part is investigated b the mathematicians of ordinar differential equations and dnamical sstems. Hilbert also pointed out that there eist possible connections between these two parts. We shall eplain some connections in detail below. With regard to the second part of Hilbert s 6th problem, Arnold [977, 983] posed the weakened Hilbert s 6th problem which will be detailed in Sec. 6. In addition, Smale in his two lectures Dnamics Restrospective: Great Problem, Attempts That Failed [Smale, 99] and Mathematical Problems for the Net Centur [Smale, 998] again posed the Hilbert s 6th problem as follows. Let P (, ), Q(, ) be real polnomials in two variables and consider the differential equation (E n ) in R. Is there a bound K = H(n) on the number of limit ccles of the form K n q, where n is the maimum of the degree of P and Q, andq is a universal constant? Smale stated that This is a modern version of the second half of Hilbert s siteenth problem. Ecept for the Riemann hpothesis, it seems to be the most elusive of Hilbert s problems. Clearl, Smale s problem is onl concerned with the half of the second part of Hilber s 6th problem. Of course, this is the important half. We need to notice that Hilbert also asked us to focus on another half of the second part in his 6th problem: suppose that there eists K = H(n) foragivenn, what schemes (i.e. relative positions) of limit ccles can be realized for ever number K, K i, i =,,..., K, respectivel. Like the first part of Hilbert s 6th problem where the distribution of ovals is to be considered, the distribution problem of limit ccles can also be ver interesting. Coleman [983] in his surve Hilbert s 6th problem: How Man Ccles? stated that For n > the maimal number of ees is not known, nor is it known just which comple patterns of ees within ees, or ees enclosing more than a single critical point, can eist. Here so-called ee means the limit ccle. In recent ears, a lot of new progression in this direction has been observed. In this paper, we first briefl discuss the stud results around the first part of Hilbert s 6th problem of which there eist ver good surves and articles: [Gudkov, 974; Wilson, 978; Rokhlin, 978; Viro, 984, 986; Oleinik, 969; Arnold & Oleinik, 979; Shustin, 99, 99; Korchagin, 996; Polotovskii, 989; Orevkov, 999]. We do not

3 Hilbert s 6th Problem and Bifurcations of Planar Polnomial Vector Fields 49 intend to attempt to list all references in the area of real algebraic curves. Interested readers should see the above papers and the references cited therein. We will be mainl interested in the stud of the second part of Hilbert s 6th problem. Since there are wide intersections and interactions between the stud of the geometr of polnomial vector fields and the development of bifurcation methods for analtic vector fields (see [Rousseau, 993]), we shall focus our attention on the stud of bifurcations of limit ccles. Some valuable surves and books: [Coppel, 966; Qin, 98; Chicone & Tian, 98; Cai, 989; Ye, 996; etc.] for quadratic sstems; [Coleman, 983; Llod, 988; Ilashenko, 99; Schlomiuk, 993; Roussarie, 998; Rousseau, 993; Yang, 99; Ye, 995; etc.] for cubic sstems and more general sstems. To our knowledge, there has been no surve article et that contains the studies for two parts of Hilbert s 6th problem. In this tutorial, we shall tr to present recent progress in two directions. However, this paper is not planned as an encclopaedic surve of the subject. Our aim is to onl describe some important progress and some aspects of the theor which have attracted our interest, and have been the object of our stud along with colleagues, friends and students during the last 0 ears. The choice of materials is ver much a personal choice. We shall omit technical details, but we will tr to provide the original literatures although we emphasize that we cannot make a complete literature list, because over the last 30 ears just for the stud of quadratic sstems, more than 000 papers have been published (e.g. see [Ren, 994]) and new published papers continuousl appear. In this paper, we ma fail to include man valuable contributions (for eample, the studies of Liénard equations, integrabilit and invariant algebraic curve solutions, etc.), for this we kindl apologize in advance.. The First Part of Hilbert s 6th Problem.. Formulation of the problem Let R be the real number set. The set R R = R is called the affine real plane. Let H(, ) be a nonconstant polnomial of degree m in (, ) R, its zero set A m = {(, ) R H(, )=0} is call an affine plane curve of degree m. A m is called irreducible if it is not the union of affine plane curves that do not properl contain it. For p( 0,0 ) A m,if( H( 0,0 )/ ) +( H( 0,0 )/ ) 0, then p is called a simple point of A m. Otherwise, p is called a singular point. If ever p in A m is simple, A m is called a nonsingular real affine algebraic curve. Let RP be the real projective plane and F ( 0,, )= i+j+k=m A ij k 0 i j be a real homogeneous polnomial of degree m. The set RA = {( 0,, ) RP F ( 0,, )=0} is called a real projective algebraic curve. If F is an irreducible polnomial, RA is called irreducible. The set p = {( 0,, ) RP 0 =0} is called the line at infinit of RP. The correspondence ( 0,, ) (, ) with = / 0, = / 0 defines a bijective mapping of RP /p onto the affine plane R with Cartesian coordinates (, ). The inverse mapping is the correspondence (, ) (,,). When we identif R with RP /p b means of this mapping, we regard RP as the completion of R b the line p at infinit. As a model of RP, we take the sphere = with diametricall opposite points identified as the projective sphere and consider the upper half-sphere 0 0. We project the points in this upper half-sphere orthogonall onto the disk +, 0 = 0. The resulting disk D,with diametricall opposite circumference points identified, is called the projective disk. Clearl, unlike R, the projective disk (the model of RP ) is compact. The curve RA is nonsingular if its corresponding affine curve A m is nonsingular. We assume that the curve RA has no singularities. Thus, RA is a compact onedimensional manifold, which is homeomorphic to a disjoint union of circles. There are two essentiall different embedding of a circle in the projective disk (i.e. RP ): two-sided (like the standard embedding in R ) or one-sided (the circle contains two points in + =, 0 = 0). The two-sided components of RA are traditionall called ovals. Obviousl, if the degree (m =k) is even, then the components are all positioned in RP double-sided. If the degree (m =k + ) is odd, then there is eactl one one-sided component.

4 50 J. Li We need to use the concept of isotop. Intuitivel speaking, we call two embedding curves RA and RB isotopic if one can be deformed (b a mapping f) to the other through embeddings; such a deformation f is called an isotop. If f is a diffeomorphism, then f is called an ambient isotop. B virtue of the general theorems of two-dimensional topolog, two curves of degree m have the same real scheme of mutual positions of their components if and onl if the sets of their real points are isotopic in RP. Such an isotop, connecting curves of degree m, is called real isotop. We notice that R and RP are the geometric spaces where A m and RA lie inside. Actuall, for a fied order m, the set of all real projective algebraic curves C m is the real projective space RP N depending on its N different coefficient parameters A ij, where N = m(m +3)/. We call RP N the space of curves C m of the mth order. Theorem. [Harnack, 876]. Let RA be a nonsingular algebraic curve of degree m in the real projective plane RP and n be the number of connected components of RA. Then, n (m )(m ) +. Thus, let l be the number of ovals of RA. Then l M (m 3m +3+( ) m ). () If the number of components of a curve RA of degree m is precisel M, it is called M-curves; if it is M i, it is called an (M i)-curve. The first part of Hilbert s 6th problem was stated in connection with the Harnack s Theorem.. We can now formulate as follows. Problem A. Describe which real schemes of ovals can be realized b a real algebraic curve of degree m. In other words, how can the complete isotop classification be realized for nonsingular real algebraic curves of degree m? Of course, broadl interpreted, Hilbert requests the stud of the topolog of real algebraic varieties. Since Problem A is alread ver difficult, presentl, this problem is onl completel solved for curves of degree 7 (see [Vio, 986; Orevkov, 999]). There are few chances of obtaining a complete answer for all m to this question in the near future. So, we do not touch on the subject of real algebraic varieties in this paper. However, Problem A has been attacked b some groups of mathematicians (in particular, Russian mathematicians). A lot of ver profound discoveries in this direction have been obtained. To stud Problem A, all stud activit around it can be roughl divided into two classes. (i) Prohibitions. We must prove theorems that restrict the possible relative positions of the ovals, i.e. prove that some isotop tpes of the ovals do not eist. (ii) Constructions. We must find methods of constructing curves of a given degree with a prescribed scheme, i.e. realize all isotop tpes of the curves which eist. Because M-curves usuall satisf the most interesting restriction conditions and if a M-curve is constructed, then corresponding curves with fewer components can be easil constructed, hence, the stud of M-curves occupies a center position... Definitions and notations. Nest of depth l. A set of l ovals totall ordered b inclusion is called a nest of depth l.. Even oval and odd oval. An oval is called even (or odd), if it lies inside an even (odd) number of other ovals. The number of even (odd) ovals is denoted b P (N). For eample, in the curve of Fig., we have P =5,N = 4. On the other hand, the curve F = 0 divides the plane into two regions defined b B + = {( i ) RP F ( i ) 0}, B = {( i ) RP F ( i ) 0}. We make the convention that F 0 in the region outside all the ovals. Then an oval is even (odd) if it is the eterior boundar of a component of B + (B ). Note that the Euler characteristic of B + is given b χ(b + )=P N. Fig.. A tpical plane curve F = 0. The regions F > 0, F<0 are indicated b +,.

5 Hilbert s 6th Problem and Bifurcations of Planar Polnomial Vector Fields 5 3. Elliptic (empt), parabolic and hperbolic oval. Even ovals and odd ovals can be divided into three classes as follows. Each oval bounds from one of the outside components of the complement of the curve; if the Euler characteristic of this component is positive (so that it is a topological disc), then the oval is called elliptic (or empt); if the Euler characteristic is zero (so that it is a topological annulus), then the oval is called parabolic; and if the Euler characteristic is negative (so that it is a topological disc with more than one hole), then the oval is called hperbolic. We denote b P +, P 0, P (respectivel) the number of even ovals in the three classes, so that P = P + + P 0 + P. We define N +, N 0, N similarl. For eample, in the curve of Fig., we have P + =3,P 0 =,P =;N + =3,N 0 =0,N =. 4. Real schemes of ovals. We use the notation given b [Polotovskii, 976; Viro, 980]. The smbol denotes the scheme consisting of one oval and the empt set b the smbol 0. Ifthesmbol s encodes some fragment of ovals, then A is the scheme consisting of all ovals of this fragment and one more oval, which encloses the fragment A. The smbol A B denotes the real scheme consisting of all ovals constituting the scheme A and the scheme B, where none of the ovals of one scheme lies inside an oval of the other. The smbol s denotes s empt ovals ling outside one another. The scheme consisting of s fragments A ling outside one another will be denoted for short b s A. The smbol A A A, where A is repeated n times, is abbreviated b the notation n A. Foreample, the scheme of the curve in Fig. is denoted b..3. Some restriction theorems on schemes of curves Nowadas, there are man old and new restriction theorems on schemes of curves [Wilson, 977; Viro, 986; Orevkov, 999; Paris, 999; etc.]. Here we onl introduce some important results, which do not concern with the comple topological characteristics of algebraic curves. We first notice that if there is a nest of depth s, then a straight line passing through the center of the nest intersects it at least s points. B using the fact that a straight line cuts a curve of degree m at most m points, the net theorem follows. Theorem. [Hilbert]. For a curve of degree m = k, the number of ovals in a nest (i.e. the depth), or in two disjoint nests, is at most k. Theorems. and. are enough to dispose of ovals for m =, 3, 4, 5. We will discuss these cases in the net subsections. Petrovskii [938] made the following remarkable contribution, which is the first theorem that imposes nontrivial restrictions on the mutual disposition of the ovals of plane curves. For an curve of de- Theorem.3 [Petrovskii]. gree k, we have 3 k(k ) P N 3 k(k ) +. () B using the inequalities (), for an M-curve of degree m = 3=6wehavek =3,P + N =, P N 0. It gives P 0.5. Thus, P, which easil leads to give the proof of Hilbert s assertion: a setic M-curve cannot have the scheme of ( empt ovals). Hilbert said that he arrived at this assertion b a complicated process. Theorem.4 (i) For an M-curve of degree m =k, we have P N k mod 8. (3) (ii) For an (M )-curve of degree m =k, we have P N k ± mod8. (4) Theorem.4 was proved b Rokhlin, Kharlamov, Gudkov and Krakhnov (see [Wilson, 977]). To show its application in our problem, we consider the setic M-curves. B Theorems. and.3, there eist M =ovalsandp. Thus, for the following logicall possible schemes of setic M-curves 0, 9, 8, 7 3, 6 4, 5 5, 4 6, 3 7, 8, 9, b using Theorem.4 (i), from P + N =, P N 9 mod 8, it follows that N =, 5, 9. In other words, the setic M-curves admit onl three real isotopic tpes: 9, 5 5, 9.

6 5 J. Li For the following logicall possible schemes of setic (M )-curves 9, 8, 7, 6 3, 5 4, 4 5, 3 6, 7, 8, 0, b using Theorem.4 (ii), from P + N = 0, P N 5 (or 0) mod 8, it follows that N =,4, 5, 8, 9. In other words, the setic (M )-curves do not admit four real isotopic tpes: 7, 6 3, In 97, Arnold discovered deep connections between the topolog of real plane algebraic curves and the topolog of four-dimensional manifolds and gave some new restriction theorems. Theorem.5 [Arnold, 97; Viro, 986]. curves of degree m =k, we have For an () P + P 0 (k 3k +3+( ) k )/; () N + N 0 (k 3k +)/; (3) If k is even and P +P 0 =(k 3k +4)/, then P = P + =0; (4) If k is odd and N + N 0 =(k 3k +)/, then N = N + =0and there is onl one eterior oval. B adding the inequalities () and () in Theorem.5, we get the following result. Corollar.6. For an M-curve of degree k, the number of empt ovals is at least k (in fact at least k + for odd k>). Notice that a curve of degree m need not be determined b the number m and the set of its real points. Generall, we need to see real algebraic curves as a comple object. Let C m be an algebraic curve of degree m, F a polnomial representing it, and RA the set of its real points. We denote b CA the set of comple points of C m, which is defined b CA = {( 0,, ) CP F ( 0,, )=0}, where CP is the comple projective plane. It is ver interesting to note the connection between the real scheme of a curve C m and the embedding of RA in the set CA of comple points of C m. In 970s, Rokhlin observed that a plane real algebraic curve is acquired under a natural assumption of a pair of opposite orientations from its compleification. Since then, these so-called comple orientations have plaed an important role in finding new restriction theorems on the real schemes of real algebraic curves. To understand these new developments of topological methods initiated b Arnold and Rokhlin, we need to use the knowledge of algebraic topolog. So, in the following we do not touch these themes..4. Some construction methods on schemes of curves and known results To answer the question what schemes can be realized b M-curves of a given degree m, firstl, we need to use the all known restriction theorems to remove impossible schemes in order to obtain all the logicall possible M-curves listed b some tables. Then, we must construct eamples of these curves, hoping eventuall to get all possibilities allowed b the tables. Thus, it is ver important to find some construction methods such that ever possible scheme of M-curves can be realized. Gudkov in his surve [Gudkov, 974] described the classical methods of construction of M-curves: the methods of Harnack, Hilbert, Brusotti and Wiman, which were called marking method or small parameter method. In [Polotovskii, 975] and [Korchagin, 978] there are tables of tpes of M-curves from degrees 6, constructed b these methods. The constructions were carried out in the following manner. Firstl, a pair of nonsingular curves (having lower orders k and m k, respectivel,) transversal to each other was constructed, and then the union of the curves was slightl perturbed, to remove the singularities such that a new M-curve of degree m was created. As an eample, we consider the Hilbert s method for construction of M-curves for m =6below. We take two ellipses, sa E and C,thatintersect at four real points z, z, z 3 and z 4, arranged on E and C in identical order [see Fig. (a)]. We form θ 4 = L L L 3 L 4 = 0, where L i is a line intersecting the arc z 3 z 4 of E at two points A i, A i (i =,, 3, 4), and denotes the product of two linear equations of L i and L j.then C 4 = E C + tθ 4 = 0, for the appropriate sign of t and sufficientl small t, consisting of four ovals, among which α intersects E at eight points A j arranged on α and E in identical order. Thus, b using a small variation method, C 4 can be constructed [see the broken curves in Fig. (a)].

7 Hilbert s 6th Problem and Bifurcations of Planar Polnomial Vector Fields 53 (a) (b) (c) Fig.. Hilbert s method for the construction of M-curves. (a) Small variation of E C ;(b)e and 4 ovals of C 4;(c)E and 4 ovals of C 4. (a) (b) Fig. 3. Hilbert s method for the construction of M-curves of C 6. (a) The tpe 9 of setic curves. (b) The tpe 9 of setic curves. The disposition of C 4 relative to E is indicated in Fig. (b). B a simplification of nodes from E C 4, we can obtain a curve C 6 with the tpe 9 [see Fig. 3(a)]. If at the relevant stage of the construction we choose A j on the arc z z 4 of E, we obtain a curve C 4 disposed relative to E as indicated in Fig. (c). Then a curve of C 6 with the tpe 9 [see Fig. 3(b)] can be obtained from E C 4. For m 5, using the classical small parameter method, all M-curves can easil be constructed. For m = 6, in 876, Harnack constructed a seticcurveoftpe 9. In 89, Hilbert constructed a setic curve of tpe 9 and made the conjecture that all M-curves can onl be of the above two tpes. In 900, he stated his conviction that to solve his 6th problem one must use a continuous change in the coefficients of the curve. However, Gudkov [974] pointed out that for the setic curves in RP, the M-curves of tpe 5 5 cannot be constructed b the classical method from curves of lower orders. He had to abandon the classical framework and perturb not a reducible curve but the image of a nonsingular curve under a quadratic transformation. However, as before, all the curves perturbed had onl nondegenerate double singularities (see [Gudkov, 974, p. 43]). Gudkov [969] solved the problem of the topological classification of the disposition of the ovals of a nonsingular setic curve in RP, i.e. he proved the following theorem. Theorem.7. Of 68 logicall possible tpes of nonsingular setic curves, 56 schemes can be realized. There eist three tpes of M-curves: 9, 9 and 5 5. In 980, Viro [980, 986] proposed a new construction method for perturbing a curve with a semiquashomogenuous singularit. It substitutes a curve fragment prepared beforehand for a small neighborhood of singularit. For some singularities, in particular for points of quadratic contact of three nonsingular branches, he obtained a complete

8 54 J. Li topological classification of their smoothing (i.e. of the curve fragments that appear in place of singularit after the perturbation). For nondegenerate fivefold points (singularit N 8 b Arnold s notation) and points of quadratic contact of four nonsingular branches, an ample suppl of smoothings was made. Shustin [983] completed the topological classification of smoothings of singularities N 8 and obtained new results on the smoothings of points of quadratic contact of four nonsingular branches. New development of the constructions of oval schemes was advanced essentiall b such smoothings. Thus, using the above new method of perturbation of curves with complicated singularities, Viro [980] solved the problem of the isotop classification of nonsingular curves of degree 7. Viro [984a] and Shustin [985] constructed some M-curves of degree 8. Theorem.8 [Viro, 980]. There eist nonsingular curves of degree 7 with the following real schemes: (i) α β with α + β 4, 0 α 3, β 3; (ii) α with 0 α 5; (iii). An nonsingular curve of degree 7 has one of these real schemes. For m = 8, Korchagin [989] stated that the 96 schemes of M-curves of degree 8 satisf all known restriction theorems, 78 schemes of ovals have been realized. For m = 9, according to Korchagin s stud results (see also [Korchagin, 989]), 7 schemes of M-curves of degree 9 satisf known restrictions, 400 schemes of ovals have been realized. For m = 0, Chislenko [984] continued Viro s work and has constructed man smoothings for points of quadratic contact of five nonsingular branches. As corollar, he obtained the known restrictions for M-curves of degree 0 that make possible about real schemes, more than 500 schemes have been realized b him. The greatest difficulties arise in the construction of all essential M-curves of a given degree m as m rises, since the number of isotop tpes grows rapidl. An available evidence suggests that the number of different M-curves approaches infinit as m. But, the number we are able to construct remains essentiall constant..5. A famil of real algebraic curves of degree m defined b planar polnomial Hamiltonian sstems of degree m Let H(, ) =H (, )+H (, )+ + H m (, ) = h 0 (5) be a real affine algebraic curve of degree m, where H i (, ) is a homogeneous polnomial of degree i, i =,,..., m and h 0 is a real constant. It is eas to see that the level curve H(, ) =h 0 is one of the famil H(, ) =h of integral curves defined b the planar Hamiltonian sstem of degree m d dt = H, d dt = H. (6) This gives the connection between the planar algebraic curves and orbits of the planar polnomial Hamiltonian vector fields. Obviousl, an oval of (5) corresponds to a periodic orbit of (6). Thus, for given m, if we know the topological classification of the phase portraits of (6), then in a particular case, we ma obtain all schemes of ovals of (5). Of course, this is also a ver difficult task. As simple eamples, we consider the cases m 5. Rokhlin [978] listed all real schemes of ovals with m 5. Leaving aside the simple case m =,, 3, we know that for m = 4, there are si real schemes: a nest of depth and five unnested schemes with l, l =0,,, 3, 4; for m = 5, there are eight real schemes: a scheme with a nest of depth, and seven unnested schemes with l, l =0,,, 3, 4, 5, 6. Theorem.9. For m = 4 and 5, ecept for si empt ovals, all real schemes of projective algebraic curves in RP can be realized b the orbits of Z q - equivariant Hamiltonian vector fields (q =, 3, 4, 5, see Sec. 7). In fact, for m = 4 we consider the following Z -equivariant cubic Hamiltonian vector fields: d dt = ( d a ), dt = ( c ), (7) with the Hamiltonian level curves H (, ) = 4 (c4 + a 4 )+ ( + )=h,

9 Hilbert s 6th Problem and Bifurcations of Planar Polnomial Vector Fields (a) (b) (c) (d) (e) (f) Fig. 4. Quartic algebraic curves defined b a Z -equivariant cubic sstem. (a) < 4h <0; (b) 0 < 4h </a; (c)4h =/a; (d) /a < 4h </c; (e)4h =/c; (f)/c < 4h </a +/c. or in the polar coordinates, dr dt = 4 r3 ((c a)+(c + a)cos θ)sin θ, dθ dt = + 8 r (3(a + c) +4(c a)cos θ +(c + a)cos 4θ), where ac >, a>c>0; and the Z 3 -equivariant cubic Hamiltonian vector fields: d dt = (a + b( + ) ), d dt = (a + b( + )+), or in the polar coordinates, dr dt = r3 sin 3θ, (8) dθ dt = a + br + r cos 3θ, (9) with the Hamiltonian level curves H 3 (r, θ) = 4 br4 + 3 r3 cos 3θ + ar = h, where b<0 <a. With h varing, the quartic algebraic curves defined b H (, ) =h have schemes shown in Fig. 4 (see [Li & Chen, 987a; Li & Huang, 987b]). For the sstem (8), we write = 4ab, α =( )/ b, α =( +)/ b, h = H 3 (α,π/3), h = H 3 (α, 0). Then, with h varing, the quartic algebraic curves defined b H 3 (, ) =h have schemes shown in Fig. 5 (see [Li & Wan, 99]). We see from Figs. 4 and 5 that the schemes of ovals,, 3, 4 and have been realized. For m = 5, we consider the following Z 5 - equivariant quartic Hamiltonian vector fields: dr dt = βr4 sin 5θ, (0)

10 56 J. Li 0 (a) (b) 0 (c) Fig. 5. Quartic algebraic curves defined b a Z 3-equivariant cubic sstem. (a) <h<0; (b) 0 <h<h ;(c)h = h ; (d) h <h<h. (d) (a) (b) Fig. 6. Fifth algebraic curves defined b a Z 5-equivariant quartic sstem. (a) The phase portrait of (0). (b) Five empt ovals when h (h,h ).

11 Hilbert s 6th Problem and Bifurcations of Planar Polnomial Vector Fields 57 dθ dt = δr + βr 3 cos 5θ, with the Hamiltonian level curves i.e. H 5 (r, θ) = r + δr4 5 βr5 cos 5θ = h, H 5 (, ) = ( + )+ δ( + ) 5 β( ( + ) +5( + ) )=h, where β, δ > 0. Taking δ = (β +), 0<β<, h = H 5 (, 0), h = H 5 ( β ( + 5), 0), then we have the phase portrait of (0) shown in Fig. 6(a). When h (h,h ), the fifth curve defined b H 5 (r, θ) =h realized the scheme of ovals 5 [see Fig. 6(b)]. 3. The Second Part of Hilbert s 6th Problem: Introduction The second part of Hilbert s 6th problem deals with the maimum number H(n) and relative positions of limit ccles of a polnomial sstem d dt = P d n(, ), dt = Q n(, ), (E n ) of degree n, i.e. ma(deg P, deg Q) = n. Hilbert conjectured that the number of limit ccles of (E n ) is bounded b a number depending onl on the degree n of the vector fields. Let χ N be the space of planar vector fields X =(P n = n i+j=0 a ij i j, Q n = n i+j=0 b ij i j ) with the coefficients (a ij,b ij ) B R N, for 0 i + j n, N =(n +)(n + ). The standard procedure in the stud of polnomial vector fields is to consider their behavior at infinit b etension to the Poincaré sphere. Consider the unit sphere in R 3 : S = { =(,, 3 ) =} and the tangent space Π = {(,, 3 ) R 3 : 3 =} at the north pole. Let L be a straight line connecting the origin (0, 0, 0) and a point p Π. L intersects S at two points p + and p, where p ± H ±, H ± = {(,, 3 ) S ± 3 > 0}. It gives via central projection the following two diffeomorphisms f ± :Π H ±, where f ± = ± (,, ) (), () =( + +) /. Suppose that X is a vector field in Π = R. Then we have X() =Df ± ()X(), = f ± () H ±, which is an induced vector field of X in H + H. For X χ N, X can be etended to the equator S = { S 3 =0} to become an analtic vector field ρ X in S and the equator is its invariant set, where ρ : S R given b ρ(,, 3 )=3 n. ρ X is called a compactification field of X. Most authors use the closed disc D as a model for the compactification field of X. The line at infinit is the boundar of the disc with diametricall opposite points identified. This is the so-called Poincaré compactification. Since the Poincaré sphere and disc are compact, all orbits of X are complete (the maimal time interval of ever orbit of X is equal to R). It is eas to see that if the number of limit ccles of an analtic vector field on S with isolated singular points is finite, then for an polnomial vector field on R, so is the number of limit ccles (see [Chicone & Sotomaor, 986; Li, 000]). B using the above discussion, we can see (E n ) as an analtic N-parameter famil of differential equations on S with the compact base B. Then, the second part of Hilbert s 6th problem ma be split into three parts: Problem B. Prove the finiteness of the number of limit ccles for an concrete sstem X χ N (given a particular choice for coefficients of (E n )) i.e {L.C. of (E n )} <. Problem C. Prove for ever n the eistence of an uniforml bounded upper bound for the number of limit ccles on the set B as the function of the parameters, i.e. n, (a ij,b ij ) B, H(n) such that {L.C. of (E n )} H(n), and find an upper estimate for H(n). Problem D. For ever n and known K = H(n), find all possible configurations (or schemes) of limit

12 58 J. Li ccles for ever number K, K i, i =,,..., K respectivel. Hence, the complete Hilbert s 6th problem consists of Problems A D. Problem B for polnomial and analtic differential equations have been alread solved b Ecalle [99] and Ilashenko [99] independentl. Of course, as Smale stated These two papers have et to be thoroughl digested b mathematical communit. We will state their main idea in Sec. 5. Up to now, there is no approach to the solution of Problem C, even for n =, which seem to be ver complicated. But there eists a similar problem, which seems to be a little bit easier. It is the weakened Hilbert s 6th problem proposed b Arnold [977]: Let H be a real polnomial of degree n and let P be a real polnomial of degree m in the variables (, ). How man real zeroes can the function I(h) = Pdd () H h have? The question is wh zeroes of the Abelian integrals I(h) is concerned with the second part of Hilbert s 6th problem? We will discuss it in Sec. 6. For a clear understanding of our topics, we now introduce some basic concepts in the qualitative theor of ordinar differential equations. An isolated periodic orbit Γ of a vector field in R is called a limit ccle which is the forward (ω-) or backward (α-) limit set of some orbits of (E n ) disjoint from Γ. To stud the number of bifurcations of limit ccles and their stabilit, the most basic tool is the Poincaré map or first return map. Let Γ be a periodic orbit of two-dimensional flow φ t ( 0, 0 ) in R of the vector field defined b (E n )andσ be a local cross-section line perpendicular to Γ at ( 0, 0 ). Then for an point (, ) Σ sufficientl near ( 0, 0 ), the solution of (E n ) through (, ) at t = 0, φ t (, ), will cross Σ again at P (, ) near ( 0, 0 ) (see Fig. 7). The map (, ) P (, ) is called the Poincaré map. Obviousl, the map P is analtic. A fied point of the Poincaré map, i.e. point (, ) Σ satisfing P (, ) = (, ), corresponds to a periodic orbit φ t (, ) of (E n ). A polnomial sstem (E n ) depends on its N parameters (a ij,b ij ). As these parameters are Fig. 7. The Poincaré map for a planar sstem, (, ) = (r 0 cos θ 0,r 0 sin θ 0). varied, changes of phase portraits ma occur for certain parameter values. These changes are called bifurcations and the parameter values are called bifurcation parameter values corresponding to bifurcation points (vector fields) which are structurall unstable sstems. B the theor of planar dnamical sstems [Andronov et al., 973; and so on], limit ccles of (E n ) can be created b (i) multiple Hopf bifurcation from a center or fine focus; (ii) homoclinic or heteroclinic bifurcations from some separatri loops; (iii) Poincaré Pontrjagin Andronov global center bifurcation from some period annulus; (iv) limit ccle bifurcations from some multiple limit ccles. This means that the development of analtic or smooth bifurcation theor for vector fields can give a deeper understanding of the eistence of limit ccles of polnomial vector field. To simplif the stud of bifurcations of vector fields, we shall tr to find additional coordinate transformations which simplif the analtic epression of the vector field on R, such that the essential features of the flow and the Poincaré mapsof the original vector field near a critical point (or a homoclinic orbit, a closed orbit, respectivel) become more evident. The resulting simplified vector fields are called normal forms. (see [Bruno, 989; Chow et al., 994; Wang, 990; Li, 000; etc.]) In the net section, we will use normal forms to discuss some bifurcation methods in order to approach Hilbert s 6 problem.

13 Hilbert s 6th Problem and Bifurcations of Planar Polnomial Vector Fields Focal Values, Saddle Values and Finite Cclicit in a Fine Focus, Closed Orbit and Homoclinic Loop In the last two decades, much progress on finite cclicit in a fine focus and homoclinic loop has been achieved. These are a series of concrete contributions on Hilbert s 6th problem. Roughl speaking, the so-called finite cclicit means that it can give rise to at most a finite number of limit ccles in some neighborhood of focus or homoclinic loop for small perturbations in the parameter space. 4.. Liapunov constants and small-amplitude limit ccles We consider the following polnomial sstem of degree n in which the origin is a nondegenerate center of its linearized sstem: where d dt = β + f(, ), d dt = β + g(, ), f(, )=f (, )+ + f n (, ), g(, )=g (, )+ + g n (, ), () f i, g i are homogeneous polnomials of order i. The problem of determining whether the origin of () is a center or a fine (or weak) focus is known as the criterion problem of center-focus. There are several was to solve this problem. Here, we are interested in the normal form method. Let =(/)( + ), =(/i)( ). Then, () becomes the comple form d dt = iβ + Y (, ), d dt = iβ + Y (, ), (3) where Y and Y are high order terms and Y = Y. There eists a formal normal transformation = u + h (u,u ), = u + h (u,u ), h = h (4) such that (3) reduces to the standard normal form du dt = iβu ( + P (u u )), du dt = iβu ( + P (u u )), (5) where P = P is a comple power series with respect to u u.write P (u u )=R(u u )+ig(u u ), P (u u )=R(u u ) ig(u u ), where G and R are real power series with respect to u u and R(0) = G(0) = 0. Let u = v + iv, u = v iv. From (5), we have the following real formal sstem which is called a real standard normal form of (). dv dt = β(v + v R(r )+v G(r )), dv dt = β(v + v R(r ) v G(r )), (6) where r = v + v.letg(r )= m= g m r m. First, suppose that there eist N satisfing g i =0,fori<N, but g N 0. Letv = r cos θ, v = r sin θ. Then, in polar coordinates, (6) has the form dr dt = g N rn+ + o(r N+ ), dθ dt = β( + o(r)). (7) Clearl, for small r, dr/dt has the same sign as g N. Hence the origin of (7) is a stable (unstable) focus if g N < 0(> 0). Second, suppose that g m = 0, for all m. Then, (7) becomes dr dt =0, Thus, the origin of (8) is a center. dθ dt = β( + R(r )). (8) Definition 4.. The constant g i, i is called the ith focal value of the origin of (). Corresponding to the first nonzero focal value g N,thenumberN is called the order of a fine focus. Notice that Andronov et al. [97] defined the notion of the i-focal value at the origin of () as the ith derivative v i = d (i) (0)/i! of the succession function d(r 0 )=P (r 0 ) r 0 where P is the Poincaré return map in a neighborhood of the origin. The first nonzero focal value v k+ of Andronov corresponds to an odd number i =k +.

14 60 J. Li On the other hand, we can construct a power series F (, )= + + F 3 (, ) + + F k (, )+ such that along orbits of () df dt = V ( + ) + V ( + ) 3 () + + V k ( + )k+ +. The constant V i is called Liapunov constant. The first nonzero V N gives the stabilit of the origin of (). It has been proved that the first nonzero Liapunov constant V N differs onl b a positive constant factor from the nonzero focal values g N and v N+ (0). (For eample, see [Gobber & Willamowskii, 979]). We see from (7) that under small perturbations of the coefficients of (), a N-order fine focus can create at most N limit ccles. Usuall, the appearance of these limit ccles is called multiple Hopf bifurcations. Obviousl, the origin of () is a center if and onl if V k = 0 for all k =,,... Since V, V k, are polnomials with rational coefficients in the coefficients of f and g, b well known Hilbert s basis theorem, the ideal generated b these polnomials has a finite basis B,B,..., B M. Hence we have a finite set of necessar and sufficient conditions for a center, i.e. B i = 0 for i =,,..., M. Of course, the calculation of this basis is ver difficult (see [Blows & Llod, 984; Llod & Pearson, 990, 996; etc.]). Theorem 4.. For an given number n, there eists a number C(n) such that () has no fine focus whose order k>c(n) and there eists a fied sstem () which has a C(n)-order fine focus and for this sstem, b small perturbations of the coefficients of (), at most C(n) small amplitude limit ccles can be created. This theorem tells us that the cclicit of fine focus is alwas finite. But, for a given n, howcan we determine C(n)? Thereareafewresults. The pioneering work given b [Bautin, 95] proved that C() = 3, namel, Bautin theorem. A fine focus or center of a quadraticsstemhascclicitatmost3. Up to now, we do not know what C(3) is. The sstems with homogeneous nonlinearities of degrees 3 5 have been investigated b [Sibirskii, 965; Chavarriga & Giné, 996, 997], respectivel. For some sstems of special form, for eample, Kukles sstem and quadratic-like cubic sstems, etc., the problem of cclicit in Hopf bifurcations was studied in [Christopher, 994; Chavarriga & Giné, 995; Blows & Llod, 984; Llod & Lnch, 988; Cima et al., 995; Lins et al., 977; etc.]. For the stud of calculations and properties of Liapunov constants, several have been published. See [Alwash, 998; Alwash & Llod, 987; Blows & Llod, 984; Llod & Pearson, 990; Gasull & Prohens, 997; Romanovskii, 99, 99; Wang & Mao, 994, 996; Jin & Wang, 990; Cima et al., 998; Hassard & Wan, 978; etc.]. 4.. Saddle values, separatri values and bifurcation of limit ccles from a homoclinic loop We net consider the sstem having a hperbolic saddle point d dt = + f(, ), d dt = + g(, ), (9) where f(, ), g(, ) are analtic functions with respect to, and f, g = O( + ). B using the transformation = + h (, ), = + h (, ), (9) can be of the Poincaré normal form (see [Arnold, 98; Amelikin et al., 98; Li, 000; etc.]) ( d dt = + d dt = ( + m= m= a m ( ) m ), b m ( ) m ). (0) We see from (0) that the following conclusion holds. Theorem 4.3. The sstem (9) has a nontrivial first integral if and onl if a m = b m for all m. Definition 4.4. Let α m = a m b m for m. The number α m is called m-order saddle value. Corresponding to the first nonzero saddle value α k,the saddle point is called k-order fine saddle point.

15 Hilbert s 6th Problem and Bifurcations of Planar Polnomial Vector Fields 6 Similar to Theorem 4., we have Theorem 4.5. For an given number n, there eists a number S(n) such that a polnomial sstem of degree n in R has a S(n)-order saddle point. In addition, for an polnomial sstem of degree n, if α = α = = α S(n) =0, then for all m S(n), α m =0, i.e. this sstem has an analtic nontrivial first integral in the neighborhood of saddle point. It is the same as C(n). How can we determine S(n)? This is a ver difficult problem. Up to now, we onl know that S() = 3. To discuss the bifurcations of limit ccles created from homoclinic loop, we net consider a famil of smooth vector fields in R : X ε : d dt = V (, ε), R,ε R l. () Suppose that for ε =0,X 0 has a hperbolic saddle point s 0, and that the stable and unstable manifolds of s 0 coincide to form a separatri loop (i.e. homoclinic orbit) Γ. Usuall, for small ε 0, Γ will be split and near Γ ma appear limit ccles of (). How man ccles can be created? This problem has been studied b [Andronov et al., 973; Leontovich, 95; Cherkas, 98; Feng & Qian, 985; Roussarie, 986; Han et al., 99; etc.]. It is well known that (i) if the rough saddle value α 0 of s 0 (the sum λ +λ of two characteristic roots of the linearilized sstem of () at s 0 )isnot equal to zero, then there eists at most one limit ccle of () near Γ for small ε 0 [Andronov et al., 973]. (ii) if α 0 = 0, but β = Γ div(x ε)dt 0, then there eist at most two limit ccles of () near Γ for small ε 0 [Cherkas, 973]. We consider the general case. Roussarie [986] used Dulac normal form [Dulac, 93] ( ) N ẋ =, ẏ = + α i () i i= to stud the Poincaré map (it is a composite map P = f consisting of a diffeomorphism f in the singular part of Γ and a transition map in the regular part of Γ (see Fig. 8). The following result was obtained. Proposition 4.6. Let P () be the Poincaré map in a neighborhood of Γ. Then,P() has the following four asmptotical epansions: (i) P () = c λ ( + o()), c > 0, where λ = (λ /λ ) 0,λ < 0 <λ ; Fig. 8. loop. The Poincaré map in a neighborhood of homoclinic (ii) there eists a natural number k such that P () = + β k k + o( k ), β k 0, () if k =,β > ; (iii) there eists a natural number k such that P () = + α k k ln + o( k ln ), α k 0; (3) (iv) for an natural number k, P () = + o( k ). In (3), the nonzero α k corresponds to (k )- order saddle value. In (), the nonzero β k is called k-order separatri value b Leontovich [95]. B using Proposition 4.6, we have (see [Ilashenko & Yakovenko, 99; Roussarie, 986; Li, 000]). Theorem 4.7. For the above C smooth famil X ε of vector fields, there eist a neighborhood U of the homoclinic orbit Γ and a neighborhood V of ε =0 such that (i) if P () has the form (), then for ε V, X ε has at most k limit ccles in U; (ii) if P () has the form (3), then for ε V, X ε has at most k limit ccles in U The singular point values and its constructive theorem We see from Secs. 4. and 4. that both focal and saddle values pla an important role in multiple Hopf bifurcations and homoclinic bifurcations. What is an universal law of these values? Liu and Li [990] answered this problem. The considered the following comple sstem dz dt = z + a αβ z α w β = Z(z, w), dw dt = w α+β= α+β= b αβ w α z β = W (z, w), (4)

16 6 J. Li where the functions Z(z, w) andw (z, w) areanaltic at a neighborhood of the origin of (4) and Z(0, 0) = W (0, 0) = 0. Variables z, w and T are comple. a αβ, b αβ are comple parameters. Let be the comple parameter space consisting of all a αβ and b αβ, where α and β are natural numbers with α + β. We perform transformations z = + i, w = i, T = it. It brings the sstem (4) to the form d = + h.o.t. = X(, ), dt (5) d = + h.o.t. = Y (, ), dt where i = and h.o.t. denotes all high order terms. We sa that the sstem (5) is the associated sstem of (4) and vice versa. Similar to the proof in [Amelikin et al., 98], we have Proposition 4.8. The sstem (4) can be reduced to the normal form dφ dt = φ p k (φψ) k, k=0 (6) dψ dt = ψ q k (φψ) k, k=0 b using a formal change of variables φ(z, w) =z + A αβ z α w β, ψ(z, w) =w + α+β= α+β= B αβ w α z β, where A k+,k = B k+,k = 0, p 0 = q 0 =, p k and q k are polnomials defined on with rational coefficients, and A αβ,b αβ can be determined uniquel, with k =,,... Definition 4.9. (i) The number µ k = p k q k (k =,,...), is called the k-order singular point value at the origin of (4). (ii) If µ 0 = µ = = µ k = 0, but µ k 0, then the origin of (4) is called a k-order fine critical singular point. (iii) If µ k =0forallk =,,..., then the origin of (4) is called an etended center. (In this case, sstem (4) is integrable). Remark 4.0. If sstem (4) is a real autonomous differential sstem, then µ k is the k-order saddle value of the origin as Definition 4.4. If sstem (5) is a real autonomous differential sstem with the associated sstem (4), then the kth focal value v k+ of the origin of (5) is equal to iµ k, i.e. v k+ = iµ k, for all k =,,...Thus, we can realize the unitar stud for both focal and saddle values b considering (4). We introduce a two-parameter transformation z = ρe iθ z, w= ρe iθ w, (7) where z and w are new variables, ρ and θ are comple parameters, ρ 0. B using (7), (4) becomes where d z dt = z + d w dt = w α+β= α+β= ã αβ z α w β, bαβ w α z β, ã αβ = a αβ ρ α+β e i(α β )θ, (8) bαβ = b αβ ρ α+β e i(α β )θ. Let f(a αβ,b αβ ) be a function of man variables defined on. Denotethat f = f(ã αβ, b αβ ), f = f(a αβ,b αβ ), where a αβ = b αβ, b αβ = a αβ, α 0, β 0, α + β. Suppose that f = ρ Is e iirθ f. I s and I r are called similar eponent and rotation eponent, respectivel. Definition 4.. (i) A function f(a αβ,b αβ ) is called a k-order invariant under (7), if f = ρ k f, i.e. I s =k, I r = 0. (ii) An invariant f is called a monomial invariant under (7), if f is a monomial on. (iii) A monomial invariant f is called an elementar invariant under (7), if it cannot be epressed as a product of two monomial invariants. It is eas to see that a k-order invariant multiplied b the other l-order invariant gives a (k + l)- order invariant. Proposition 4.. A monomial on given b g = nj= a ms= αj β j b usv s is a k-order invariant under (7) if and onl if n m (α j + β j ) + (u s + v s ) = k, j= s=

17 Hilbert s 6th Problem and Bifurcations of Planar Polnomial Vector Fields 63 n m (α j β j ) (u s v s ) = 0. j= s= Proposition 4.3. If a monomial g = g(a αβ,b αβ ) is a k-order invariant (or elementar invariant), then so is g = g(b αβ,a αβ ). We see from the above results that the following two conclusions hold. Theorem 4.4. The k-order singular point value of (4) at the origin is a k-order invariant under (7), i.e. µ k = ρ k µ k,k=,,..., and it has the inverse smmetr relation µ k = µ k, k =,,... Theorem 4.5 (The Constructive Theorem of Singular Point Values). A k-order singular point value of (4) at the origin can be represented as a linear combination of k order monomial invariants under (7) and their inverse smmetr forms, i.e. s µ k = γ kj (g kj gkj), k =,,..., (9) j= where s = s(k) is a natural number and γ kj arational number, g kj and gkj are k-order monomial invariants under (7). Hence, to compute singular point values, we need to find and combine elementar invariants under (7). For eample, we consider cubic sstem dz dt = z + a 0z + a zw + a 0 w + a 30 z 3 + a z w + a zw + a 03 w 3, (E 3 ) dw dt = w b 0w b wz b 0 z b 30 w 3 b w z b wz b 03 z 3, According to the different rotation eponents I r under (7), the coefficients of (E 3 ) can be divided into nine classes as follows: R 0 = {a,b }, R = {a 0,b }, R = {a 30,b }, R 3 = {b 0 }, R 4 = {b 03 }, R = {b 0,a }, R = {b 30,a }, R 3 = {a 0 }, R 4 = {a 03 }, where I r (R ±j )=±j, j =,,3,4. Clearl, a and b are two elementar invariant in the set R 0. Hence other elementar invariant of (E 3 ) cannot contain them. Define the set G = 4 α= R kα α 4 α= R lα α, α =,, 3, 4, where R α R β means that we take their coefficients for the multiplications. Thus, we have I r (G) = 4 α= αk α 4 α= αl α. Proposition 4.6. An element of G is monomial invariant under (7) if and onl if the following linear equation of integers holds: k +k +3k 3 +4k 4 = l +l +3l 3 +4l 4. (30) Let ζ = (k,k,k 3,k 4,l,l,l 3,l 4 ) be a nonzero vector solution of (30) in which ever component is a non-negative integer. Such ζ is called a normal solution of (30). We sa ζ is an elementar normal solution if it cannot be epressed as a sum of two normal solutions. Proposition 4.7. An element of G is an elementar invariant of (E 3 ) under (7) if and onl if ζ is an elementar normal solution of (30). Similarl, we define 4 4 G = Rα lα R α kα, α =,, 3, 4, α= α= and ζ =(l,l,l 3,l 4,k,k,k 3,k 4 ). Then, if ever element of G is elementar invariant of (E 3 ), then so is ever element of G. If ζ is an elementar normal solution of (30), then so is ζ. Proposition 4.8. Equation (30) has eactl 34 elementar normal solutions. B using known elementar normal solutions, we are able to construct sets of G k (k = 34). Ever element of G k (k =0 34, G 0 = R 0 ) is an elementar invariant under (7). Thus, we have Theorem 4.9. Sstem (E 3 ) has eactl 0 elementar invariants under (7) which are listed in Table in [Liu & Li, 990]. As special cases, we have the following two corollaries of the Theorem 4.9.