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On the global analysis of the planar quadratic vector fields
Pergamon
Nonlinear Analysis, Theory, Methods & Applications, Vol. 30, No. 3, pp. 142%1437,1997 Proc. 2rid World Congress of Nonlinear Analysts
© 1997ElsevierScienceLtd
PII: S0362-546X(97)00196-X
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ON THE GLOBAL ANALYSIS OF THE PLANAR QUADRATIC VECTOR HELDS. Dana Schlomiuk 1 D6partementde Math6matiqueset de Statistique,Universit6de Montr6al,Montr6al,(Qu6bec), Canada, H3C 3J7 Key words and phrases: Vector field, bifurcation,quadratic system, projective curve, intersectionmultiplicity,
center, Hamiltoniansystem, real projective space, foliationwith singularities. 1. INTRODUCTION The study of planar polynomial vector fields involves a diversity of methods (analytic, topological, algebraic, numerical calculation, etc). Ideas and tools from algebraic geometry appear also to be necessary. In [1], [2], [3] we considered some problems on polynomial vector fields which are to a certain extent, of an algebraic nature: the problem of the existence of algebraic invariant algebraic curves or existence of a rational first integral of such systems, etc. Naturally, algebro-geometric concepts fit very well these problems. But an algebro-geometric frame is also useful for other problems and especially so for those which have a global content, for example the problem of classifying low degree systems. The literature abounds in works on special classes of low degree polynomial systems but, at least in some of these works, a global picture does not emerge. Some bifurcation diagrams are like a labyrinth and organizing principles to lead us to more understanding are often lacking. There is a need for more global thinking which goes beyond the listing of specific phase portraits and types of bifurcations involved. Algebraic geometry could be of help, at least in some of the stages of the classification of low degree systems. This article shows how by using the concept of intersection multiplicity of projective curves we can shed light and construct organizing tools for the understanding of bifurcation diagrams of families of planar quadratic vector fields. The ideas also apply to higher degrees systems. The work is organized as follows: In §2 we motivate our discussion by giving some examples of problems for which an algebro-geometric frame is necessary. In §3 we briefly describe the concept of intersection multiplicity of planar projective curves. In §4 we give some results concerning classification problems for some families of quadratic systems and show how the notion of intersection multiplicity helps us to obtain a clear global picture. Ideas in this section apply to other classes of polynomial systems. 2. PROBLEMSON PLANARPOLYNOMIALVECTORFIELDSWHERE ALGEBRO-GEOMETRICCONCEPTSARE HELPFUL. The study of the behaviour of the integral curves of a system
1 This work was partially supported by the NSERC and by the Quebec Education Ministry. 1429
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(2.1)
with P, Q, polynomials with real coefficients around singular point A e R 2 of (2.1) (i.e. P(A) = 0 = Q(A)), in the situation where both eigenvalues are zero, involves the blow-up construction (cf. [4]). In this construction one replaces the point A of the affine plane with the projective line so as to obtain a new two-dimensional manifold. This manifold is constructed as an algebraic surface in the product space R2xRP(I). (Actually in many articles the blow-up is done by using polar coordinates. While this is efficient for constructing phase portraits, the changes are transcendental and the quadratic transformations used in the first construction are algebraic and simpler). One singularity may require several such consecutive blow-ups, landing us on a complicated surface. The building blocks of the blow-up construction are algebraic varieties connected by morphisms of such varieties. A shorter, simpler proof of the theorem of resolution of singularities of vector fields (cf.[5]), entirely based on the algebro-geometric concept of intersection multiplicity of projective curves was given by van den Essen in [6]. This is an instance where we see the usefulness of an algebro-geometric viewpoint. Other instances are the problem of algebraic integrability of systems (2.1) (cf.[7-8]) or of the existence of algebraic invariant curves of (2.1) (cf. [2]), etc. For these algebraic problems methods of algebraic geometry are clearly natural. But even when the problems are not algebraic, like the problem of classifying quadratic vector fields with a weak focus, in some stages of the classification an algebro-geometric viewpoint is helpful. There are numerous articles on planar quadratic vector fields. Some of these deal with perturbations of singular cycles of such systems. In [9], [10] the authors show that "the existential part" of Hilbert's 16th problem for quadratic systems (i.e.: prove that H(n) is finite where H(n) is the lowest cardinal number which bounds the number of limit cycles for any polynomial vector field of degree n) reduces to showing the finite cyclicity of 121 cases of singular cycles (so far only some 55 of them have been shown to have finite cyclicity). Assuming all 121 cases are shown to have finite cyclicity, to find H(2) (i.e. to solve Hilbert's 16th problem) we need to use different kind of methods which would enable us to calculate effectively this number. A detailed and systematic work on the global geometric behaviour of these systems needs to be done. Since the bifurcation diagram of this class must have a very large number of distinct phase portraits, conceptual unifying themes or guiding principles through this labyrinth are essential. Algebro-geometric concepts, in particular the intersection multiplicity of algebraic curves, prove to be helpful at least in the initial stages of this task of classsifying low degrees polynomial systems. 3. INTERSECTIONMULTIPLICITYOF PROJECTIVECURVES The singularities of (2.1) with P,Q polynomials in x,y with real coefficients are points of intersection of the two algebraic curves P(x,y) = 0 and Q(x,y) = 0. (3.1) in the affine plane. The number N of distinct such points depends on how these curves intersect, whether they are tangent or not and if they are how many tangents at a point do they have in common and with what multiplicity. The concept which captures how the curves intersect at a point is the intersection multiplicity of the two curves at that point. Roughly speaking it indicates how many points the curves have in common at that point. For example the intersection multiplicity of y = 0 with y - x 2 = 0 at (0,0)
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is two, since the line is tangent to the parabola at (0,0). For a quick understanding of the concept of intersection multiplicity via resultants see for example [11]. The definition which extends easily to intersection multiplicity at a point p of a number m of algebraic curves with m > 2 is via the dimension of a quotient ring of the local ring at p (cf.[12]). DEFINITION 3.1. Let K be an algebraically closed field and let K[x,y] be the ring of polynomials in x,y with coefficients in K. Let F,G ~ K[x,y] such that F,G have no nonconstant factors in common. The intersection multiplicity of the affine algebraic curves F = 0 and G = 0 at a point p of the affine plane A 2 = K 2 is a nonnegative integer number Ip(P,Q) with Ip(P,Q) = dimK(Op(A2)/(F,G))
(3.2)
where Op(A 2) is the local ring of A 2 at p and (F,G) is the ideal generated by F and G in K[x,y]. We recall that the local ring Op(A 2) at p is the ring of all rational functions f(x,y) = p(x,y)/q(x,y) on A 2 which are defined at p i.e. q(p) ;e 0. It is convenient to consider the projective completions of the curves (3.1) in the complex projective space CP(2): P*(x,y,z) = 0, Q*(x,y,z) = 0,
(3.3)
where P*(x,y,z) = znp(x/z,y/z), Q*(x,y,z) = zmQ(x/z,y/z) assuming that n = deg(P), m = deg(Q). In studying real systems (2.1) we simultaneously use two distinct compactifications of the real affine plane and we also use the complex projective plane. Indeed, for real polynomial systems (2.1) one considers the PoincarE's compactification of such systems on the sphere (cf.[13]) which is obtained by central projection on the sphere of the plane (x,y) identified with the plane z = 1 in (x,y,z) space. For the resulting system one also has singular points "at infinity" located on the equator of the sphere which are usually studied separately. Another approach is to associate to (2.1) the differential equation P(x,y)dy - Q(x,y)dx = 0 (3.4) and consider its induced foliation with singularities on the real (resp. complex) projective plane RP(2) (resp. CP(2)). By the singularities of (2.1) or (3.4) it is meant the solutions of the equations (3.1). The infinite singularities correspond to points on the equator of the Poincar6 sphere and they are usually studied separately. It turns out that they are points located at the intersection of the line Z = 0 with the curve YP* - XQ* = 0 in the real projective plane. 4. APPLICATIONSOF THE CONCEPTOF INTERSECTIONMULTIPLICITYOF PROJECTIVECURVES TO THE GLOBALANALYSISOF FAMILIESOF PLANARQUADRATICVECTORFIELDS The class of quadratic Hamiltonian systems with a center is perhaps the simplest class of nonlinear vector fields which are algebraically integrable: they have a first integral which is a cubic polynomial. These systems were discussed in a number of works in the literature, for example in [14] and [15] we see phase portraits of these systems but these are not assembled in a bifurcation diagram so as to be able to follow easily the changes in the systems as parameters vary. In [16], a Ph.D. thesis which appeared in russian and was not published, the bifurcation diagram of all quadratic systems with a center was given, hence in particular for the Hamiltonian case, but these diagrams were not realized in the more convenient parameter space RP(4) i.e. the four-dimensional real projective space, and for the Hamiltonian case, the
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real projective plane RP(2). Also, a summing up of results in a clear statement in geometric terms of the global phenomena involved is absent in [16]. In [17] such a summing up is given in terms of the geometry of projective curves and the bifurcation diagram is done on a disk representing the projective plane when opposite points on the circumference are identified. Here we want to go further than [17] and give a more unified picture of the global dynamics of this class of systems by making use of intersection multiplicity of three projective curves. To do this we consider the canonical form of the Hamiltonian systems with a center placed at the origin which is: d x / d t = - y + k x 2 + n y 2, d y / d t = x + a x 2 - 2 k x y (QHC) This system depends on the parameter ~, = (a,k,n) ~ R 3 and has the following Hamihonian Hk(x,y) = - (ax 3 - ny3)/3 + kx2y - (x 2 + y2)/2
(4.1)
For a nonlinear system we must have (a,k,n) 4:0 and hence the systems can be rescaled using homotheties of the axes. Thus, instead of considering the parameter space as being R 3, we may consider it as the real projective plane RP(2). This condenses the bifurcation diagram on a disk with the identification of opposite points on the circumference. We may place n = 0 as the line at infinity of R 2, i.e. the circumference of the disk. We observe that the following identity holds for the systems (QHC): H(_a,k,n)(x,y) = H(a,k,n)(-x,y). (4.2) Thus discussing the systems (QHC) in a semidisk corresponding to a < 0 is sufficient. For the systems (QHC) we have a cubic Hamiltonian, hence there are no limit cycles. Their study reduces to the study of their singularities and their saddle to saddle connections. As it is customary, the Poincar6 compactification is used (cf. [13]) and the phase portraits of the systems (QHC) are pictured on disks. The finite singular points of (QHC) are the intersection points of the conic curves P(x,y)= -y+kx 2+ny 2=0, Q(x,y)=x+ax 2-2kxy=0. (4.3) Let N(~,) be the number of distinct singular points in R 2 of the system (QHC) for the parameter ~ = (a,k,n) e R 3. N also yields a function on RP(2) whose value at a point [a:k:n] of RP(2) of homogeneous coordinates (a,k,n) we denote by N[a,k,n]. The function N which depends on the relative position of the curves (4.3) captures part of the dynamics of the systems. N(~.) is in fact the number of points of intersection of the projective completions of the curves (4.3) which are not located at infinity for (4.3) i.e. which lie on both curves: P*(x,y,z) = - yz + kx 2 + ny 2 = 0, Q*(x,y,z) = xz + ax 2 - 2kxy = 0 (4.4) but not onthe line z = 0. B6zout's theorem (cf. [11] or [12]) says that the number of points of intersection in CP(2), counted with multiplicities, of two projective curves f(x,y,z) = 0, g(x,y,z) = 0, where f, g are relatively prime homogeneous polynomials in x,y,z with complex coefficients, is n.m, where n = deg(f), m = deg(g). Therefore the number of intersection points of (4.4) in CP(2)), counted with multiplicities, is four. Hence 1 < N(a,k,n) <_ 4. N(a,k,n) = 4 if and only if all singular points are real, the curves (4.3) have no point of intersection "at infinity" and there is no singular point p of (QHC) with Ip(P,Q) > 1. N(~,) <_ 3 if and only if one of the intersection points in (4.4) is at infinity or there is no common point at infinity of the curves (4.3) but there is at least one point in the finite plane with higher intersection multiplicity than one (i.e. where at least two singular points coalesce). Using these facts, calculations in [17] yielded the following two results:
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T H E O R E M 4.1. ([17[) Consider the family (QHC) for parameter values 2, = [a:k:n]. 2, is a bifurcation point for singularities for the family (QHC) if and only if it lies on the real projective curve C l : nC~5(n - 2k) = 0 (Here C = a2n + 4k 2 and ~5 = a 2 - 4kn + 8k2).
The points on nC = 0 are characterized as correponding
to systems for which there exists a singular point p at infinity such that Ip(P*,Q*) > 1 and the points on 8(n - 2k) = 0 are those for which the systems have a finite singular point with Ip(P*,Q*) > 2. T H E O R E M 4.2. ([17])
2, = [a:k:n] is a bifurcation point of saddle to saddle connections for the family
of real vector fields (QHC) if and only if [a:k:n] lies on the real projective curve C 2 : alan - (n + k)(n 2k)] = 0 and if 2, is on a = 0, then ~5 > 0. The analysis of the nature of singularities and of the saddle-to-saddle connections yields a bifurcation diagram which is given in Fig.l of [17]. C O R O L L A R Y 4.1. 2' = [a:k:n] is a bifurcation point of (QHC) if and only if 2, lies on the curve C: nC~5(n - 2k)a[an - (n + k)(n - 2k)] = 0 and if a = 0, then 8 > 0. R E M A R K 4.1.
All points of bifurcation of singularities of system (QHC) are characterized in terms of
intersection multiplicities at finite or infinite singular points, but the lower bounds on the values of Ip(P*,Q*) are different in the two cases. By introducing the foliation with singularities associated with the systems (QHC) (cf.[18]) and the concept of intersection multiplicity of m curves with m > 2 we first succeed in unifying further the results the results of Theorem 4.1 by placing finite and infinite singularities on equal footing.
A foliation with
singularities is associated to a polynomial vector field P(x,y)/)/~x + Q(x,y)/9/t3y on C 2 or the associated differential system (2.1) as follows: Let us suppose that P and Q have no nonconstant c o m m o n factor. We consider the associated differential 1-form to 1 = Q(x,y)dx - P(x,y)dy and the differential equation col = 0. The affine plane R 2 (respectively C 2) is compactified to the real (resp. complex) projective space RP(2) = (R 3 - {0})/-, (resp. CP(2) = (C 3 - {0})/-), where (x,y,z) ~ ( x ' , y ' , z ' ) if and only if ( x ' , y ' , z ' ) = u(x,y,z) for some u ¢: 0, u real (resp. complex). Let [X:Y:Z] be the equivalence class of (X,Y,Z). Clearly the restriction of the equation co 1 = 0 on nonsingular points of (2.1) defines a foliation on this submanifold of C 2. Thus a foliation with singularities on C 2 is obtained. This foliation can be extended to a singular foliation on CP(2) and the one-form col can be extended to a meromorphic one-form on C P ( 2 ) (cf.[18]). A n a l o g o u s l y to the way we can describe a plane p r o j e c t i v e curve by a single h o m o g e n e o u s equation in x,y,z, we can describe this meromorphic one-form by a single one-form A*(X,Y,Z) dX + B*(X,Y,Z)dY + C*(X,Y,Z)dZ with homogeneous polynomial coefficients. Indeed, let us suppose that P and Q have no c o m m o n nontrivial factors and max(deg(P),deg(Q)) = n > 0 and consider the application r: C 3 \ {(X,Y,Z) I z = 0} ---> C 2 given by r(X,Y,Z) = (X/Z,Y/Z). X/Z, y = Y/Z it follows that dx = (ZdX - XdZ)/Z 2, dy = (ZdY - YdZ)/Z 2.
From the relations x =
The differential form ~ =
r*(col) has poles at Z = 0 and the equation col = 0 can be written in coordinates X,Y,Z as = Q(X/Z,Y/Z)(ZdX - XdZ)/Z 2 - P(X/Z,Y/Z)(ZdY - YdZ)/Z 2 = 0.
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Then to = Z n+ 2 ~ has polynomial coefficients and for Z ¢ 0, the equations to = 0 and ~ = 0 have the same solutions. The differential equation to = 0 is: ZnQ(X/Z,Y/Z)(ZdX - XdZ) - z n p ( x / z , Y / Z ) ( Z d Y - YdZ) = 0. Hence, regrouping the terms in dX, dY, dZ respectively, we have: to = ZQ*(X,Y,Z)dX - ZP*(X,Y,Z)dY + (YP*(X,Y,Z) - XQ*(X,Y,Z))dZ where P*(X,Y,Z) = z n p ( x / z , Y / Z ) ,
Q*(X,Y,Z) = ZnQ(X/Z,Y/Z).
We have thus obtained a polynomial
one-form to = A*(X,Y,Z)dX + B*(X,Y,Z)dY + C*(X,Y,Z)dZ with A*(X,Y,Z) = ZQ*(X,Y,Z), B*(X,Y,Z) = - ZP*(X,Y,Z), C*(X,Y,Z) = YP*(X,Y,Z) - XQ*(X,Y,Z) where A*,B*,C* are homogeneous polynomials of degree n + 1 in the variables X,Y,Z.
The form co yields an equation to = 0 on C 3. The singularities of
the foliation induced by tol on CP(2) are the intersection points of the curves A* = 0, B* = 0, C* = 0. We consider the intersection multiplicity at p, Ip(A*, B*, C*) of the three curves A* = 0, B* = 0, C* = 0. D E F I N I T I O N 4.1.
The intersection multiplicity I p ( f l , f 2 ..... fn) at a point p of the algebraic curves
C I,C 2 ..... C n in C 2 where C i : fi(x,y) = 0 is defined as being zero if the curves do not intersect, infinity if the curves have a c o m m o n component and otherwise we use the definition Ip(fl,f2 ..... fn) = d i m c O p / ( f l , f 2 ..... fn)
(4.5)
where O p is the local ring of the affine complex plane A 2 ( C ) = C 2 at p. P R O P O S I T I O N 4.1
If p is a finite or infinite singular point of a polynomial system (2.1) and A*, B*, C* are as defined above, we have: Ip(A*,B*,C*) = Ip(Z,YP* - XQ*) + Ip(P*,Q*). Ip(P*,Q*) (= Ip(P,Q))
if p is finite
Ip(Z,YP* - XQ*) + Ip(P*,Q*)
if p is infinite.
(4.6)
Ip(A*,B*,C*) =
(4.7)
Proof: Ip(A*,B*,C*) = Ip(ZP*, - ZQ*, YP* - XQ*) -- Ip(ZP*,ZQ*,YP* - XQ*), = Ip(Z,YP* - XQ*) + Ip(P*,Q*,YP* - XQ*) = Ip(Z,YP* - XQ*) + Ip(P*,Q*). If p is finite, then Z ~: 0 and Ip(Z,YP* - XQ*) = 0 and Ip(A*,B*,C*) = Ip(P*,Q*) = Ip(P,Q). T H E O R E M 4.3.
~, is a bifurcation point of singularities of the family of nonlinear systems (QHC) <-->
there exists a point p ~ CP(2), p a singular point of the system, finite or infinite, such that Ip(A*, B*, C*) _> 2. i) ~, is a codimension one bifurcation point of singularities
<--> there exists a point p ~ CP(2), p
singular point of the system (finite or infinite), such that Ip(A*,B*, C*) = 2. ii) ~. is a codimension two bifurcation point of singularities <--> there exists a point p c CP(2), p a singular point of the system (finite or infinite), such that Ip(A*,B*, C*) > 3. Proof: we see from the formula (4.7) that if p is finite Ip(A*,B*,C*) = Ip(P*,Q*).
Using T h e o r e m 4.1
we obtain the result in this case. If p is infinite, from (4.7) we have Ip(A*,B*,C*) = Ip(Z,YP* - XQ*) + Ip(P*,Q*).
F r o m T h e o r e m 4.1 we have a point p with Ip(P*,Q*) >_ 1 and the result follows since
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Ip(Z,YP* - XQ*) > 1. The second part is obtained using the bifurcation diagram in Fig.1 in [17]. REMARK 4.2. Unlike the Theorem 4.1 which separates the discussion in two types of singular points, finite and infinite, the above theorem places on equal footing all singularities by using Ip(A*,B*,C*). DEFINITION 4.2. For a polynomial system (2.1), we call algebraic multiplicity of the point p, finite or infinite, the number Ip(A*,B*,C*) where A*, B*, C* are as defined before. NOTATION 4.1. Let NSC(X) be the number of singular invariant complex cubic curves of the family (QHC) at the parameter value X = [a:k:n]. PROPOSITION 4.4 For systems (QHC), and C: nCS(n - 2k)a[an - (n + k)(n - 2k)] = 0, we have: (i NSC(X) =
iff iff
~'~C X • C\Sing(c)
iff
X • Sing(C)
The proposition follows from the values of the function N given by the formula below, by considering the curves passing through the singular points and remarking that on the curve 0 2 we have singular cubics which are reducible. As shown in [17] the values of N are: i N(a,k,n) =
iff n C S ( n - 2 k ) ~ 0 a n d S > 0 . iff only one of the equations n = 0, 8 = 0, C = 0, n - 2k = 0 is satisfied. iff 8 < 0 or two distinct ones of the equations n = 0,8 = 0,C = 0, n - 2k = 0 are satisfied
If)~ ~ C, each of the four singular cubics has the following properties: it is irreducible, it has an ordinary double point which is also a singular point of the system, it is transversal to the line at infinity. For X • C a loss in the number of singular invariant cubics occurs and for at least one of the remaining ones a loss in the properties mentioned above for the generic curves occurs: one of the curves may become reducible with either two or three singularities or it remains irreducible but its singularity may become a cusp or the curve may become tangent to the line at infinity. DEFINITION 4.3. An invariant cubic curve of (QHC) is called a simple singular cubic curve for the family (QHC) if and only if it is a singular cubic curve, its singularities are all ordinary double points located in the finite plane and the curve is transversal to the line at infinity. The bifurcations are determined by the singular invariant cubic curves which are not simple. DEFINITION 4.4. A singular invariant cubic curve C O of (QHC) for the value X0 of the parameter is a
multiple curve of multiplicity m if and only if by perturbing the system in the family (QHC), there are exactly m simple singular cubic curves in (QHC) for all values X near X0 which all approach the curve C O when X tends to X0. Finally we announce another result which integrates bifurcations of singularities with bifurcations of
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THEOREM 4.4. ~, = [a:k:n] is a bifurcation point of the family of systems (QHC) if and only if there exists a multiple singular invariant cubic curve with multiplicity greater than one. In this case the curve is unique. ~, is a codimension one bifurcation point of the family (QHC) if and only if there exists a singular invariant cubic curve of multiplicity two. ~, is a codimension two bifurcation point of the family (QHC) if and only if there exists a singular invariant cubic curve which is of multiplicity 3. The proof follows from the theorems 4.1 and 4.2, the proposition 4.1 and the definitions 4.3 and 4.4. REMARK 4.3. The bifurcation locus is characterized above in terms of multiplicity of singular invariant cubics only. This is a unified way of placing on equal footing bifurcation points of singularities, be they finite or infinite and bifurcations of saddle-to saddle connection by using multiple invariant singular cubic curves for the family (QHC). This notion may be generalized in the natural way to the notion of multiple invariant curve for a certain property P with respect to a family of systems F. In the case above, P is the property of being a simple singular cubic curve, a multiple curve of multiplicity m producing by perturbation m curves with the property P. The discussion we made above applies also for other classes of quadratic systems. The numbers Ip(A*,B*,C*) plays an important role since jumps in these numbers correspond to bifurcation points. The study of quadratic Hamiltonian systems with a saddle point is done in an analogous manner by making use of these functions and results about this class along the lines given here were obtained in [19]. The concepts we used have a wider domain of application. Indeed, for general problems of classification, in the initial stages, the basic algebro-geometric concepts we used play an analogous role. To solve Hilbert's 16th problem for quadratic systems means to find the lowest cardinal number H(2) which bounds the number of limit cycles for planar quadratic vector fields of degree 2. A limit cycle of such a vector field necessarily surrounds a single singular point which is a focus (cf.[20]). For nonlinear, quadratic vector fields with a focus or a center placed at the origin, we may use the following canonical form: dx/dt = ~,x -y + kx 2 + mxy + ny 2,
dy/dt = x + ax 2 + bxy.
(QFC)
The ideas in this work can be applied to the global analysis of the family of quadratic systems with a weak focus (QWF) (i.e. we take ~, = 0 in (QFC)). This is the approach taken in the Ph.D thesis [21] which is in preparation. The functions N(~), I p ( A * ( ~ ) , B * ( ~ ) , C * ( ~ ) ) , for o~ = [a:b:k:m:n] varying in the parameter space
RP(4) since for a nonlinear system (QWF), (a,b,k,m,n) must be nonzero, so the systems
can be rescaled. REFERENCES 1. SCHLOMIUK, D. Elementaryfirst integrals and algebraic invariant curves of differential equations, Expo. Math. 11,433454 (1993). 2. SCHLOMIUK,D. Algebraic and Geometric Aspects of the Theory of Polynomial Vector Fields, in Bifurcations and Periodic Orbits of Vector Fields, (Edited by D. Schlomiuk), 429-467, Kluwer Academic Publishers, (1993).
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3. SCHLOMIUK, D. Algebraic particular integrals, integrability and the problem of the center, Transactions of the A.M.S., 338, Number 2, August, 799-841, (1993). 4. CAMACHO, C. Complex foliations arising from Polynomial Differential Equations, Notes by Maria lzabel Camacho, in Bifurcations and Periodic Orbits of Vector Fields, (Edited by D. Schlomiuk), 1-19, Kluwer Academic Publishers, (1993).
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VULPE, N.I. Affine-invariant conditions for the topological discrimination of quadratic systems with a center, Translated from Differentsial'nye Uravneniya, 19, No.3, 371-379, (1983).
15. ARTES, J.C. and LLIBRE J. Sistemes quadratics Hamiltonians, (preprint) 60 pages, (1992). 16. ANDRONOVA, E. A., Decomposition of the parameter space of a quadratic equation with a singular point of center type and topological structures with limit cycles. Ph.D. Thesis (in russian) Gorky, Russia, 114 pages, (1988). 17. PAL, J. and SCHLOMIUK, D. Summing up of the dynamics of Quadratic Hamiltonian Systems with a center, Canadian Journal of Mathematics, (to appear).
18. CAMACHO, C., LINS NETO, A, SAD, P., First integrals of Algebraic Differential Equations, Lecture Notes for participants to the School on Dynamical Systems, ICTP, Trieste, 35 pages, (1992).
19. DUPUIS, Y., SCHLOMIUK, D. Summing up the dynamics of quadratics systems with a trace zero integrable saddle, (article in preparation). 20. COPPEL, W. A., A survey of quadratic systems, Journal ofDiff Eq. 2, 293-304 (1966). 21. PAL, J. Quadratic systems with a weak focus. Ph.D Thesis, Universit6 de Montreal, (in preparation).
Nonlinear Analysis, Theory, Methods & Applications, Vol. 30, No. 3, pp. 142%1437,1997 Proc. 2rid World Congress of Nonlinear Analysts
© 1997ElsevierScienceLtd
PII: S0362-546X(97)00196-X
Printed in Great Britain. All rights reserved 0362-546X/97 $17.00 + 0.00
ON THE GLOBAL ANALYSIS OF THE PLANAR QUADRATIC VECTOR HELDS. Dana Schlomiuk 1 D6partementde Math6matiqueset de Statistique,Universit6de Montr6al,Montr6al,(Qu6bec), Canada, H3C 3J7 Key words and phrases: Vector field, bifurcation,quadratic system, projective curve, intersectionmultiplicity,
center, Hamiltoniansystem, real projective space, foliationwith singularities. 1. INTRODUCTION The study of planar polynomial vector fields involves a diversity of methods (analytic, topological, algebraic, numerical calculation, etc). Ideas and tools from algebraic geometry appear also to be necessary. In [1], [2], [3] we considered some problems on polynomial vector fields which are to a certain extent, of an algebraic nature: the problem of the existence of algebraic invariant algebraic curves or existence of a rational first integral of such systems, etc. Naturally, algebro-geometric concepts fit very well these problems. But an algebro-geometric frame is also useful for other problems and especially so for those which have a global content, for example the problem of classifying low degree systems. The literature abounds in works on special classes of low degree polynomial systems but, at least in some of these works, a global picture does not emerge. Some bifurcation diagrams are like a labyrinth and organizing principles to lead us to more understanding are often lacking. There is a need for more global thinking which goes beyond the listing of specific phase portraits and types of bifurcations involved. Algebraic geometry could be of help, at least in some of the stages of the classification of low degree systems. This article shows how by using the concept of intersection multiplicity of projective curves we can shed light and construct organizing tools for the understanding of bifurcation diagrams of families of planar quadratic vector fields. The ideas also apply to higher degrees systems. The work is organized as follows: In §2 we motivate our discussion by giving some examples of problems for which an algebro-geometric frame is necessary. In §3 we briefly describe the concept of intersection multiplicity of planar projective curves. In §4 we give some results concerning classification problems for some families of quadratic systems and show how the notion of intersection multiplicity helps us to obtain a clear global picture. Ideas in this section apply to other classes of polynomial systems. 2. PROBLEMSON PLANARPOLYNOMIALVECTORFIELDSWHERE ALGEBRO-GEOMETRICCONCEPTSARE HELPFUL. The study of the behaviour of the integral curves of a system
1 This work was partially supported by the NSERC and by the Quebec Education Ministry. 1429
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Second WorldCongress of NonlinearAnalysts dx/dt = P(x,y), dy/dt = Q(x,y)
(2.1)
with P, Q, polynomials with real coefficients around singular point A e R 2 of (2.1) (i.e. P(A) = 0 = Q(A)), in the situation where both eigenvalues are zero, involves the blow-up construction (cf. [4]). In this construction one replaces the point A of the affine plane with the projective line so as to obtain a new two-dimensional manifold. This manifold is constructed as an algebraic surface in the product space R2xRP(I). (Actually in many articles the blow-up is done by using polar coordinates. While this is efficient for constructing phase portraits, the changes are transcendental and the quadratic transformations used in the first construction are algebraic and simpler). One singularity may require several such consecutive blow-ups, landing us on a complicated surface. The building blocks of the blow-up construction are algebraic varieties connected by morphisms of such varieties. A shorter, simpler proof of the theorem of resolution of singularities of vector fields (cf.[5]), entirely based on the algebro-geometric concept of intersection multiplicity of projective curves was given by van den Essen in [6]. This is an instance where we see the usefulness of an algebro-geometric viewpoint. Other instances are the problem of algebraic integrability of systems (2.1) (cf.[7-8]) or of the existence of algebraic invariant curves of (2.1) (cf. [2]), etc. For these algebraic problems methods of algebraic geometry are clearly natural. But even when the problems are not algebraic, like the problem of classifying quadratic vector fields with a weak focus, in some stages of the classification an algebro-geometric viewpoint is helpful. There are numerous articles on planar quadratic vector fields. Some of these deal with perturbations of singular cycles of such systems. In [9], [10] the authors show that "the existential part" of Hilbert's 16th problem for quadratic systems (i.e.: prove that H(n) is finite where H(n) is the lowest cardinal number which bounds the number of limit cycles for any polynomial vector field of degree n) reduces to showing the finite cyclicity of 121 cases of singular cycles (so far only some 55 of them have been shown to have finite cyclicity). Assuming all 121 cases are shown to have finite cyclicity, to find H(2) (i.e. to solve Hilbert's 16th problem) we need to use different kind of methods which would enable us to calculate effectively this number. A detailed and systematic work on the global geometric behaviour of these systems needs to be done. Since the bifurcation diagram of this class must have a very large number of distinct phase portraits, conceptual unifying themes or guiding principles through this labyrinth are essential. Algebro-geometric concepts, in particular the intersection multiplicity of algebraic curves, prove to be helpful at least in the initial stages of this task of classsifying low degrees polynomial systems. 3. INTERSECTIONMULTIPLICITYOF PROJECTIVECURVES The singularities of (2.1) with P,Q polynomials in x,y with real coefficients are points of intersection of the two algebraic curves P(x,y) = 0 and Q(x,y) = 0. (3.1) in the affine plane. The number N of distinct such points depends on how these curves intersect, whether they are tangent or not and if they are how many tangents at a point do they have in common and with what multiplicity. The concept which captures how the curves intersect at a point is the intersection multiplicity of the two curves at that point. Roughly speaking it indicates how many points the curves have in common at that point. For example the intersection multiplicity of y = 0 with y - x 2 = 0 at (0,0)
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is two, since the line is tangent to the parabola at (0,0). For a quick understanding of the concept of intersection multiplicity via resultants see for example [11]. The definition which extends easily to intersection multiplicity at a point p of a number m of algebraic curves with m > 2 is via the dimension of a quotient ring of the local ring at p (cf.[12]). DEFINITION 3.1. Let K be an algebraically closed field and let K[x,y] be the ring of polynomials in x,y with coefficients in K. Let F,G ~ K[x,y] such that F,G have no nonconstant factors in common. The intersection multiplicity of the affine algebraic curves F = 0 and G = 0 at a point p of the affine plane A 2 = K 2 is a nonnegative integer number Ip(P,Q) with Ip(P,Q) = dimK(Op(A2)/(F,G))
(3.2)
where Op(A 2) is the local ring of A 2 at p and (F,G) is the ideal generated by F and G in K[x,y]. We recall that the local ring Op(A 2) at p is the ring of all rational functions f(x,y) = p(x,y)/q(x,y) on A 2 which are defined at p i.e. q(p) ;e 0. It is convenient to consider the projective completions of the curves (3.1) in the complex projective space CP(2): P*(x,y,z) = 0, Q*(x,y,z) = 0,
(3.3)
where P*(x,y,z) = znp(x/z,y/z), Q*(x,y,z) = zmQ(x/z,y/z) assuming that n = deg(P), m = deg(Q). In studying real systems (2.1) we simultaneously use two distinct compactifications of the real affine plane and we also use the complex projective plane. Indeed, for real polynomial systems (2.1) one considers the PoincarE's compactification of such systems on the sphere (cf.[13]) which is obtained by central projection on the sphere of the plane (x,y) identified with the plane z = 1 in (x,y,z) space. For the resulting system one also has singular points "at infinity" located on the equator of the sphere which are usually studied separately. Another approach is to associate to (2.1) the differential equation P(x,y)dy - Q(x,y)dx = 0 (3.4) and consider its induced foliation with singularities on the real (resp. complex) projective plane RP(2) (resp. CP(2)). By the singularities of (2.1) or (3.4) it is meant the solutions of the equations (3.1). The infinite singularities correspond to points on the equator of the Poincar6 sphere and they are usually studied separately. It turns out that they are points located at the intersection of the line Z = 0 with the curve YP* - XQ* = 0 in the real projective plane. 4. APPLICATIONSOF THE CONCEPTOF INTERSECTIONMULTIPLICITYOF PROJECTIVECURVES TO THE GLOBALANALYSISOF FAMILIESOF PLANARQUADRATICVECTORFIELDS The class of quadratic Hamiltonian systems with a center is perhaps the simplest class of nonlinear vector fields which are algebraically integrable: they have a first integral which is a cubic polynomial. These systems were discussed in a number of works in the literature, for example in [14] and [15] we see phase portraits of these systems but these are not assembled in a bifurcation diagram so as to be able to follow easily the changes in the systems as parameters vary. In [16], a Ph.D. thesis which appeared in russian and was not published, the bifurcation diagram of all quadratic systems with a center was given, hence in particular for the Hamiltonian case, but these diagrams were not realized in the more convenient parameter space RP(4) i.e. the four-dimensional real projective space, and for the Hamiltonian case, the
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real projective plane RP(2). Also, a summing up of results in a clear statement in geometric terms of the global phenomena involved is absent in [16]. In [17] such a summing up is given in terms of the geometry of projective curves and the bifurcation diagram is done on a disk representing the projective plane when opposite points on the circumference are identified. Here we want to go further than [17] and give a more unified picture of the global dynamics of this class of systems by making use of intersection multiplicity of three projective curves. To do this we consider the canonical form of the Hamiltonian systems with a center placed at the origin which is: d x / d t = - y + k x 2 + n y 2, d y / d t = x + a x 2 - 2 k x y (QHC) This system depends on the parameter ~, = (a,k,n) ~ R 3 and has the following Hamihonian Hk(x,y) = - (ax 3 - ny3)/3 + kx2y - (x 2 + y2)/2
(4.1)
For a nonlinear system we must have (a,k,n) 4:0 and hence the systems can be rescaled using homotheties of the axes. Thus, instead of considering the parameter space as being R 3, we may consider it as the real projective plane RP(2). This condenses the bifurcation diagram on a disk with the identification of opposite points on the circumference. We may place n = 0 as the line at infinity of R 2, i.e. the circumference of the disk. We observe that the following identity holds for the systems (QHC): H(_a,k,n)(x,y) = H(a,k,n)(-x,y). (4.2) Thus discussing the systems (QHC) in a semidisk corresponding to a < 0 is sufficient. For the systems (QHC) we have a cubic Hamiltonian, hence there are no limit cycles. Their study reduces to the study of their singularities and their saddle to saddle connections. As it is customary, the Poincar6 compactification is used (cf. [13]) and the phase portraits of the systems (QHC) are pictured on disks. The finite singular points of (QHC) are the intersection points of the conic curves P(x,y)= -y+kx 2+ny 2=0, Q(x,y)=x+ax 2-2kxy=0. (4.3) Let N(~,) be the number of distinct singular points in R 2 of the system (QHC) for the parameter ~ = (a,k,n) e R 3. N also yields a function on RP(2) whose value at a point [a:k:n] of RP(2) of homogeneous coordinates (a,k,n) we denote by N[a,k,n]. The function N which depends on the relative position of the curves (4.3) captures part of the dynamics of the systems. N(~.) is in fact the number of points of intersection of the projective completions of the curves (4.3) which are not located at infinity for (4.3) i.e. which lie on both curves: P*(x,y,z) = - yz + kx 2 + ny 2 = 0, Q*(x,y,z) = xz + ax 2 - 2kxy = 0 (4.4) but not onthe line z = 0. B6zout's theorem (cf. [11] or [12]) says that the number of points of intersection in CP(2), counted with multiplicities, of two projective curves f(x,y,z) = 0, g(x,y,z) = 0, where f, g are relatively prime homogeneous polynomials in x,y,z with complex coefficients, is n.m, where n = deg(f), m = deg(g). Therefore the number of intersection points of (4.4) in CP(2)), counted with multiplicities, is four. Hence 1 < N(a,k,n) <_ 4. N(a,k,n) = 4 if and only if all singular points are real, the curves (4.3) have no point of intersection "at infinity" and there is no singular point p of (QHC) with Ip(P,Q) > 1. N(~,) <_ 3 if and only if one of the intersection points in (4.4) is at infinity or there is no common point at infinity of the curves (4.3) but there is at least one point in the finite plane with higher intersection multiplicity than one (i.e. where at least two singular points coalesce). Using these facts, calculations in [17] yielded the following two results:
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T H E O R E M 4.1. ([17[) Consider the family (QHC) for parameter values 2, = [a:k:n]. 2, is a bifurcation point for singularities for the family (QHC) if and only if it lies on the real projective curve C l : nC~5(n - 2k) = 0 (Here C = a2n + 4k 2 and ~5 = a 2 - 4kn + 8k2).
The points on nC = 0 are characterized as correponding
to systems for which there exists a singular point p at infinity such that Ip(P*,Q*) > 1 and the points on 8(n - 2k) = 0 are those for which the systems have a finite singular point with Ip(P*,Q*) > 2. T H E O R E M 4.2. ([17])
2, = [a:k:n] is a bifurcation point of saddle to saddle connections for the family
of real vector fields (QHC) if and only if [a:k:n] lies on the real projective curve C 2 : alan - (n + k)(n 2k)] = 0 and if 2, is on a = 0, then ~5 > 0. The analysis of the nature of singularities and of the saddle-to-saddle connections yields a bifurcation diagram which is given in Fig.l of [17]. C O R O L L A R Y 4.1. 2' = [a:k:n] is a bifurcation point of (QHC) if and only if 2, lies on the curve C: nC~5(n - 2k)a[an - (n + k)(n - 2k)] = 0 and if a = 0, then 8 > 0. R E M A R K 4.1.
All points of bifurcation of singularities of system (QHC) are characterized in terms of
intersection multiplicities at finite or infinite singular points, but the lower bounds on the values of Ip(P*,Q*) are different in the two cases. By introducing the foliation with singularities associated with the systems (QHC) (cf.[18]) and the concept of intersection multiplicity of m curves with m > 2 we first succeed in unifying further the results the results of Theorem 4.1 by placing finite and infinite singularities on equal footing.
A foliation with
singularities is associated to a polynomial vector field P(x,y)/)/~x + Q(x,y)/9/t3y on C 2 or the associated differential system (2.1) as follows: Let us suppose that P and Q have no nonconstant c o m m o n factor. We consider the associated differential 1-form to 1 = Q(x,y)dx - P(x,y)dy and the differential equation col = 0. The affine plane R 2 (respectively C 2) is compactified to the real (resp. complex) projective space RP(2) = (R 3 - {0})/-, (resp. CP(2) = (C 3 - {0})/-), where (x,y,z) ~ ( x ' , y ' , z ' ) if and only if ( x ' , y ' , z ' ) = u(x,y,z) for some u ¢: 0, u real (resp. complex). Let [X:Y:Z] be the equivalence class of (X,Y,Z). Clearly the restriction of the equation co 1 = 0 on nonsingular points of (2.1) defines a foliation on this submanifold of C 2. Thus a foliation with singularities on C 2 is obtained. This foliation can be extended to a singular foliation on CP(2) and the one-form col can be extended to a meromorphic one-form on C P ( 2 ) (cf.[18]). A n a l o g o u s l y to the way we can describe a plane p r o j e c t i v e curve by a single h o m o g e n e o u s equation in x,y,z, we can describe this meromorphic one-form by a single one-form A*(X,Y,Z) dX + B*(X,Y,Z)dY + C*(X,Y,Z)dZ with homogeneous polynomial coefficients. Indeed, let us suppose that P and Q have no c o m m o n nontrivial factors and max(deg(P),deg(Q)) = n > 0 and consider the application r: C 3 \ {(X,Y,Z) I z = 0} ---> C 2 given by r(X,Y,Z) = (X/Z,Y/Z). X/Z, y = Y/Z it follows that dx = (ZdX - XdZ)/Z 2, dy = (ZdY - YdZ)/Z 2.
From the relations x =
The differential form ~ =
r*(col) has poles at Z = 0 and the equation col = 0 can be written in coordinates X,Y,Z as = Q(X/Z,Y/Z)(ZdX - XdZ)/Z 2 - P(X/Z,Y/Z)(ZdY - YdZ)/Z 2 = 0.
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Then to = Z n+ 2 ~ has polynomial coefficients and for Z ¢ 0, the equations to = 0 and ~ = 0 have the same solutions. The differential equation to = 0 is: ZnQ(X/Z,Y/Z)(ZdX - XdZ) - z n p ( x / z , Y / Z ) ( Z d Y - YdZ) = 0. Hence, regrouping the terms in dX, dY, dZ respectively, we have: to = ZQ*(X,Y,Z)dX - ZP*(X,Y,Z)dY + (YP*(X,Y,Z) - XQ*(X,Y,Z))dZ where P*(X,Y,Z) = z n p ( x / z , Y / Z ) ,
Q*(X,Y,Z) = ZnQ(X/Z,Y/Z).
We have thus obtained a polynomial
one-form to = A*(X,Y,Z)dX + B*(X,Y,Z)dY + C*(X,Y,Z)dZ with A*(X,Y,Z) = ZQ*(X,Y,Z), B*(X,Y,Z) = - ZP*(X,Y,Z), C*(X,Y,Z) = YP*(X,Y,Z) - XQ*(X,Y,Z) where A*,B*,C* are homogeneous polynomials of degree n + 1 in the variables X,Y,Z.
The form co yields an equation to = 0 on C 3. The singularities of
the foliation induced by tol on CP(2) are the intersection points of the curves A* = 0, B* = 0, C* = 0. We consider the intersection multiplicity at p, Ip(A*, B*, C*) of the three curves A* = 0, B* = 0, C* = 0. D E F I N I T I O N 4.1.
The intersection multiplicity I p ( f l , f 2 ..... fn) at a point p of the algebraic curves
C I,C 2 ..... C n in C 2 where C i : fi(x,y) = 0 is defined as being zero if the curves do not intersect, infinity if the curves have a c o m m o n component and otherwise we use the definition Ip(fl,f2 ..... fn) = d i m c O p / ( f l , f 2 ..... fn)
(4.5)
where O p is the local ring of the affine complex plane A 2 ( C ) = C 2 at p. P R O P O S I T I O N 4.1
If p is a finite or infinite singular point of a polynomial system (2.1) and A*, B*, C* are as defined above, we have: Ip(A*,B*,C*) = Ip(Z,YP* - XQ*) + Ip(P*,Q*). Ip(P*,Q*) (= Ip(P,Q))
if p is finite
Ip(Z,YP* - XQ*) + Ip(P*,Q*)
if p is infinite.
(4.6)
Ip(A*,B*,C*) =
(4.7)
Proof: Ip(A*,B*,C*) = Ip(ZP*, - ZQ*, YP* - XQ*) -- Ip(ZP*,ZQ*,YP* - XQ*), = Ip(Z,YP* - XQ*) + Ip(P*,Q*,YP* - XQ*) = Ip(Z,YP* - XQ*) + Ip(P*,Q*). If p is finite, then Z ~: 0 and Ip(Z,YP* - XQ*) = 0 and Ip(A*,B*,C*) = Ip(P*,Q*) = Ip(P,Q). T H E O R E M 4.3.
~, is a bifurcation point of singularities of the family of nonlinear systems (QHC) <-->
there exists a point p ~ CP(2), p a singular point of the system, finite or infinite, such that Ip(A*, B*, C*) _> 2. i) ~, is a codimension one bifurcation point of singularities
<--> there exists a point p ~ CP(2), p
singular point of the system (finite or infinite), such that Ip(A*,B*, C*) = 2. ii) ~. is a codimension two bifurcation point of singularities <--> there exists a point p c CP(2), p a singular point of the system (finite or infinite), such that Ip(A*,B*, C*) > 3. Proof: we see from the formula (4.7) that if p is finite Ip(A*,B*,C*) = Ip(P*,Q*).
Using T h e o r e m 4.1
we obtain the result in this case. If p is infinite, from (4.7) we have Ip(A*,B*,C*) = Ip(Z,YP* - XQ*) + Ip(P*,Q*).
F r o m T h e o r e m 4.1 we have a point p with Ip(P*,Q*) >_ 1 and the result follows since
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Ip(Z,YP* - XQ*) > 1. The second part is obtained using the bifurcation diagram in Fig.1 in [17]. REMARK 4.2. Unlike the Theorem 4.1 which separates the discussion in two types of singular points, finite and infinite, the above theorem places on equal footing all singularities by using Ip(A*,B*,C*). DEFINITION 4.2. For a polynomial system (2.1), we call algebraic multiplicity of the point p, finite or infinite, the number Ip(A*,B*,C*) where A*, B*, C* are as defined before. NOTATION 4.1. Let NSC(X) be the number of singular invariant complex cubic curves of the family (QHC) at the parameter value X = [a:k:n]. PROPOSITION 4.4 For systems (QHC), and C: nCS(n - 2k)a[an - (n + k)(n - 2k)] = 0, we have: (i NSC(X) =
iff iff
~'~C X • C\Sing(c)
iff
X • Sing(C)
The proposition follows from the values of the function N given by the formula below, by considering the curves passing through the singular points and remarking that on the curve 0 2 we have singular cubics which are reducible. As shown in [17] the values of N are: i N(a,k,n) =
iff n C S ( n - 2 k ) ~ 0 a n d S > 0 . iff only one of the equations n = 0, 8 = 0, C = 0, n - 2k = 0 is satisfied. iff 8 < 0 or two distinct ones of the equations n = 0,8 = 0,C = 0, n - 2k = 0 are satisfied
If)~ ~ C, each of the four singular cubics has the following properties: it is irreducible, it has an ordinary double point which is also a singular point of the system, it is transversal to the line at infinity. For X • C a loss in the number of singular invariant cubics occurs and for at least one of the remaining ones a loss in the properties mentioned above for the generic curves occurs: one of the curves may become reducible with either two or three singularities or it remains irreducible but its singularity may become a cusp or the curve may become tangent to the line at infinity. DEFINITION 4.3. An invariant cubic curve of (QHC) is called a simple singular cubic curve for the family (QHC) if and only if it is a singular cubic curve, its singularities are all ordinary double points located in the finite plane and the curve is transversal to the line at infinity. The bifurcations are determined by the singular invariant cubic curves which are not simple. DEFINITION 4.4. A singular invariant cubic curve C O of (QHC) for the value X0 of the parameter is a
multiple curve of multiplicity m if and only if by perturbing the system in the family (QHC), there are exactly m simple singular cubic curves in (QHC) for all values X near X0 which all approach the curve C O when X tends to X0. Finally we announce another result which integrates bifurcations of singularities with bifurcations of
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THEOREM 4.4. ~, = [a:k:n] is a bifurcation point of the family of systems (QHC) if and only if there exists a multiple singular invariant cubic curve with multiplicity greater than one. In this case the curve is unique. ~, is a codimension one bifurcation point of the family (QHC) if and only if there exists a singular invariant cubic curve of multiplicity two. ~, is a codimension two bifurcation point of the family (QHC) if and only if there exists a singular invariant cubic curve which is of multiplicity 3. The proof follows from the theorems 4.1 and 4.2, the proposition 4.1 and the definitions 4.3 and 4.4. REMARK 4.3. The bifurcation locus is characterized above in terms of multiplicity of singular invariant cubics only. This is a unified way of placing on equal footing bifurcation points of singularities, be they finite or infinite and bifurcations of saddle-to saddle connection by using multiple invariant singular cubic curves for the family (QHC). This notion may be generalized in the natural way to the notion of multiple invariant curve for a certain property P with respect to a family of systems F. In the case above, P is the property of being a simple singular cubic curve, a multiple curve of multiplicity m producing by perturbation m curves with the property P. The discussion we made above applies also for other classes of quadratic systems. The numbers Ip(A*,B*,C*) plays an important role since jumps in these numbers correspond to bifurcation points. The study of quadratic Hamiltonian systems with a saddle point is done in an analogous manner by making use of these functions and results about this class along the lines given here were obtained in [19]. The concepts we used have a wider domain of application. Indeed, for general problems of classification, in the initial stages, the basic algebro-geometric concepts we used play an analogous role. To solve Hilbert's 16th problem for quadratic systems means to find the lowest cardinal number H(2) which bounds the number of limit cycles for planar quadratic vector fields of degree 2. A limit cycle of such a vector field necessarily surrounds a single singular point which is a focus (cf.[20]). For nonlinear, quadratic vector fields with a focus or a center placed at the origin, we may use the following canonical form: dx/dt = ~,x -y + kx 2 + mxy + ny 2,
dy/dt = x + ax 2 + bxy.
(QFC)
The ideas in this work can be applied to the global analysis of the family of quadratic systems with a weak focus (QWF) (i.e. we take ~, = 0 in (QFC)). This is the approach taken in the Ph.D thesis [21] which is in preparation. The functions N(~), I p ( A * ( ~ ) , B * ( ~ ) , C * ( ~ ) ) , for o~ = [a:b:k:m:n] varying in the parameter space
RP(4) since for a nonlinear system (QWF), (a,b,k,m,n) must be nonzero, so the systems
can be rescaled. REFERENCES 1. SCHLOMIUK, D. Elementaryfirst integrals and algebraic invariant curves of differential equations, Expo. Math. 11,433454 (1993). 2. SCHLOMIUK,D. Algebraic and Geometric Aspects of the Theory of Polynomial Vector Fields, in Bifurcations and Periodic Orbits of Vector Fields, (Edited by D. Schlomiuk), 429-467, Kluwer Academic Publishers, (1993).
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3. SCHLOMIUK, D. Algebraic particular integrals, integrability and the problem of the center, Transactions of the A.M.S., 338, Number 2, August, 799-841, (1993). 4. CAMACHO, C. Complex foliations arising from Polynomial Differential Equations, Notes by Maria lzabel Camacho, in Bifurcations and Periodic Orbits of Vector Fields, (Edited by D. Schlomiuk), 1-19, Kluwer Academic Publishers, (1993).
5. SEIDENBERG, A, Reduction of singularities if the differential equation Ady = Bdx, Am. Journal of Math. 90, 248-269, (1968). 6. VAN DEN ESSEN, A. Reduction of singularities of the differential equation Ady = Bdx, Springer Lecture Notes in Math. 712, 44-59, (1975). 7. POINCARI~, H. Sur l'int6gration alg6brique des 6quations differentielles, C. R. Acad. Sci. Paris 112, 761-764, (1891). 8. POINCARI~, H. Sur l'int6gration alg6brique des 6quations differenetielles du premier ordre et du premier degr6, Rendiconti del Circolo Matematico di Palermo 5, 160-191, (1891).
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DUMORTIER, F., ROUSSARIE, R., and ROUSSEAU, C.
Hilbert's 16th problem for quadratic systems, Journal of
Differential Equations, 110, No 1, 86-133, (1994).
10. DUMORTIER, F., F., ROUSSARIE, R., and ROUSSEAU, C. Elementary graphics of cyclicity 1 and 2, Nonlinearity, 7, 1001-1003, (1994). 11. KIRWAN, F. Complex algebraic curves, London Mathematical Society Student Texts 23, Cambridge University Press, 264 pages, (1992). 12. FULTON, W. Algebraic curves, W. A. Benjamin, Inc., 226 pages, (1969). 13. GONZALES VELASCO, E. A.
Generic properties of polynomial vector fields at infinity.
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