Integrals and phase portraits of planar quadratic differential systems with invariant lines of at least five total multiplicity

Transcription

1 Integrals and phase portraits of planar quadratic differential systems with invariant lines of at least five total multiplicity Dana SCHLOMIUK Nicolae VULPE CRM-3181 March 2005 Département de mathématiques et de statistique, Université de Montréal Work supported by NSERC and by the Quebec Education Ministry Institute of Mathematics and Computer Science, Academy of Science of Moldova Partially supported by NSERC

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3 Abstract In this article we prove that all real quadratic differential systems dx dy = p(x, y), = q(x, y) dt dt with gcd(p, q) = 1, having invariant lines of total multiplicity at least five and a finite set of singularities at infinity, are Darboux integrable having integrating factors whose inverses are polynomials over R. We also classify these systems under two equivalence relations: 1) topological equivalence and 2) equivalence of their associated cubic projective differential equations when cubic projective differential equations are acted upon by the group P GL(2, R). For each one of the 28 topological classes obtained we give necessary and sufficient conditions for such a system with invariant lines to belong to this class, in terms of its coefficients in R 12. Résumé Dans cet article nous prouvons que tous les systèmes différentiels réels dx dt = p(x, y), dy dt = q(x, y), gcd(p, q) = 1, ayant des droites invariantes de multiplicité totale au moins cinq, sont Darboux intégrable possédant des facteurs intégrant dont les inverses sont des polynomes sur R. Nous classifions ces systèmes par rapport à deux relations d équivalence : 1) l équivalence topologique et 2) l équivalence de leurs équations différentielles cubiques projectives sous l action du groupe P GL(2, R). Pour chaqune des 28 classes topologiques obtenues nous donnons des conditions nécessaires et suffisantes pour qu un tel système ayant des droites invariantes appartienne à cette classe, en terms de ses coefficients dans R 12.

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5 1 Introduction We consider here real planar differential systems of the form dx dt = p(x, y), dy dt = q(x, y), (1.1) where p, q R[x, y], i.e. p, q are polynomials in x, y over R, their associated vector fields D = p(x, y) x + q(x, y) y (1.2) and differential equations q(x, y)dx p(x, y)dy = 0. (1.3) We call degree of a system (1.1) (or of a vector field (1.2) or of a differential equation (1.3)) the integer m = max(deg p, deg q). In particular we call quadratic a differential system (1.1) with m = 2. Each such differential system generates a complex differential system when the variables range over C. In (cf. [7]) Darboux gave a method of integration of planar complex differential equations (1.3) using algebraic curves which are invariant for the equations (see Definition 1.5). Poincaré was enthusiastic about the work of Darboux [7], which he called admirable in [13]. This method of integration was applied to prove that several families of systems (1.1) are integrable. For example in [16] it was applied to show in a unified way (unlike previous proofs which used ad hoc methods) the integrability of planar quadratic systems possessing a center. A brief and easily accessible exposition of the method of Darboux can be found in the survey article [15]. In this article we consider another class of quadratic differential systems (1.1), namely the class of all quadratic systems which possess invariant lines of total multiplicity at least five. We work with the notion of multiplicity of an invariant line introduced by us in. The goal of this article is to present a full study of this class by: proving that all systems in this class are integrable via the method of Darboux yielding integrating factors whose inverses are polynomials over R and (elementary) Darboux first integrals (see Definition 1.8 below); constructing all the topologically distinct phase portraits of the systems in this class (we have 28 such phase portraits); giving invariant necessary and sufficient conditions, under the action of the affine group and time rescaling, in terms of the twelve coefficients of the systems, for which a specific phase portrait is realized; determining the representatives of the orbits of their associated projective differential equations under the action of the real projective group P GL(2, R). To do this we first need to recall some basic notions. In what follows we shall denote by K the field of real or complex numbers depending on the situation. Whenever a definition below is given for a system (1.1) or equivalently for a vector field (1.2), this definition could also be given for an equation (1.3) and viceversa. For brevity we sometimes state only one of the possibilities. Definition 1.1. A first integral on an open set U R 2 (respectively U C 2 ) of a real (respectively complex) system (1.1) is a C 1 function F : U K where K = R (respectively K = C) such that for all solutions (x(t), y(t)) of (1.1) with t in an open disk D 0 in K, such that (x(t), y(t)) U for all t D 0, we have F (x(t), y(t)) = constant. 1

6 Remark 1.1. We note that such a C 1 function F : U K is a first integral on U of (1.1) if and only if for all solutions (x(t), y(t)) with values in U of (1.1) defined when t is in an open disk D 0 in K, we have df (x(t), y(t)) = 0 for all t D 0, or equivalently dt DF p(x, y) F x + q(x, y) F y = 0. (1.4) Definition 1.2. An integrating factor of an equation (1.3) on an open subset U of R 2 is a C 1 function R(x, y) 0 such that the 1-form ω = Rg(x, y)dx Rp(x, y)dy is exact, i.e. there exist a C 1 function F : U K on U such that ω = df. (1.5) Remark 1.2. We observe that if R is an integrating factor on U of (1.3) then the function F such that ω = Rqdx Rpdy = df is a first integral of the equation w = 0 (or a system (1.1)). In this case we necessarily have on U: (Rq) = (Rp) (1.6) y x and developing the above equality we obtain R x p + R ( p y q = R x + q ) y or equivalently DR = R div D. (1.7) Hence R(x, y) is an integrating factor on U of (1.3) if and only if (1.6) (respectively (1.7)) holds. The method of Darboux yields for complex differential equations (1.3), first integrals of the form: F = e G(x,y) f 1 (x, y) λ1 f s (x, y) λs, G C(x, y), f i C[x, y], λ i C, (1.8) G = G 1 /G G i C[x, y], f i irreducible over C. It is clear that in general an expression (1.8) makes sense only for G 2 0 and for points (x, y) C 2 \ (G 2 (x, y) = 0} f 1 (x, y) = 0} f s (x, y) = 0}). The above expression (1.8) yields a multiple-valued function in U = C 2 \ (G 2 (x, y) = 0} f 1 (x, y) = 0} f s (x, y) = 0}). For every point (x 0, y 0 ) U, there is a neighborhood of (x 0, y 0 ) such that on this neighborhood the expression (1.8) yields an analytic function when arbitrarily choosing one of the many values of the logarithms of f i (x, y), i = 1,..., s in a small neighborhood of f i (x 0, y 0 ). So for specific open subsets U C 2 we have that the expression (1.8) is actually a (single-valued) analytic function. We shall call (1.8) a first integral of (1.1) on U in case DF = 0. The proper way of continuing this discussion is by introducing a bit of differential algebra. Indeed we ( are concerned here with differential field extensions of the differential field C(x, y), x, ). For example y f(x, y) 1/2 is an expression of the form (1.8), when f C[x, y]\0}. This function belongs to the differential ( field extension of C(x, y), x, ) obtained by adjoining to C(x, y) a root of the equation y u 2 f(x, y) = 0. This is an algebraic differential field extension. The expression (1.8) belongs in general to differential field extensions which are non-algebraic. 2

7 The long term goal of this work is to provide us with specific data to be used along with similar material for higher degree curves, for the purpose of solving some classical, very old problems among them the problem of Poincaré, stated by Poincaré in 1891 [13, 14]. This problem asks us to determine necessary and sufficient conditions for a planar polynomial differential system to have a rational first integral. Although much work has been done to solve it, this problem is still open. Partial results on this problem appeared in [10], [3] and [4]. Definition 1.3. A system (1.1) or a vector field (1.2) or a differential equation (1.3) has an inverse ( integrating factor V 0 in a differential field extension K of C(x, y), x, ) if V 1 is an integrating y factor of system (1.1) (or (1.2) or (1.3)). Definition 1.4. A first integral of (1.1) (or (1.2) or (1.3)) is a function F in a differential field extension ( K of C(x, y), x, ) such that y DF = 0. In 1878 Darboux introduced the notion of invariant algebraic curve. Definition 1.5 (Darboux [7]). An affine algebraic curve f(x, y) = 0, f C[x, y], deg f 1 is invariant for an equation (1.3) or for a system (1.1) if and only if f Df in C[x, y], i.e. k = Df C[x, y]. In this f case k is called the cofactor of f. Definition 1.6 (Darboux [7]). An algebraic solution of an equation (1.3) (respectively (1.1), (1.2)) is an algebraic curve f(x, y) = 0, f C[x, y] (deg f 1) with f an irreducible polynomial over C. Darboux showed that if an equation (1.3) or (1.1) or (1.2) possesses a sufficient number of such invariant algebraic solutions f i (x, y) = 0, f i C[x, y], i = 1, 2,..., s then the equation has a first integral of the form (1.8). Definition 1.7. An expression of the form F = e G(x,y), G(x, y) C(x, y), i.e. G is rational over C, is an exponential factor 5 for a system (1.1) or an equation (1.3) if and only if k = DF C[x, y]. In this case k F is called the cofactor of the exponential factor F. Proposition 1.1 (Christopher [6]). If an equation (1.3) admits an exponential factor e G(x,y) where G(x, y) = G 1(x, y) G 2 (x, y), G 1, G 2 C[x, y] then G 2 (x, y) = 0 is an invariant algebraic curve of (1.3). Definition 1.8. We say that a system (1.1) or an equation (1.3) has a Darboux first integral (respectively Darboux integrating factor) if it admits a first integral (respectively integrating factor) of the form e G(x,y) s f i (x, y) λ i, i=1 where G(x, y) C(x, y) and f i C[x, y], deg f i 1, i = 1, 2,..., s, f i irreducible over C and λ i C. Proposition 1.2 (Darboux [7]). If an equation (1.3) (or (1.1), or (1.2)) has an integrating factor (or first integral) of the form F = s i=1 f λ i i then i 1,..., s}, f i = 0 is an algebraic invariant curve of (1.3) ( (1.1), (1.2)). In [7] Darboux proved the following remarkable theorem of integrability using invariant algebraic solutions of differential equation (1.3): 5 Under the name degenerate invariant algebraic curve this notion was introduced by Christopher in [6]. 3

8 Theorem 1.1 (Darboux [7]). Consider a differential equation (1.3) with p, q C[x, y]. Let us assume that m = max(deg p, deg q) and that the equation admits s algebraic solutions f i (x, y) = 0, i = 1, 2,..., s (deg f i 1). Then we have: I. If s = m(m + 1)/2 then there exists (λ 1,..., λ s ) C s such that λ i 0 for i = 1,..., s and R = s f i (x, y) λ i i=1 is an integrating factor of (1.3). II. If s m(m + 1)/2 + 1 then there exists (λ 1,..., λ s ) C s such that λ i 0 for i = 1,..., s and is a first integral of (1.3). F = s f i (x, y) λ i i=1 Remark 1.3. We stated the theorem for the equation (1.3) but clearly we could have stated it for the vector field D (1.2) or for the polynomial differential system (1.1). We point out that Darboux s work was done for differential equations in the complex projective space. The above formulation is an adaptation of his theorem for the complex affine space. In [10] Jouanolou proved the following theorem which improves part II of Darboux s Theorem Theorem 1.2 (Jouanolou [10]). Consider a polynomial differential equation (1.3) and assume that it has s algebraic solutions f i (x, y) = 0, i = 1, 2,..., s (deg f i 1). Suppose that s m(m + 1)/ Then there exists (n 1,..., n s ) Z s \ 0} such that F = s i=1 f i(x, y) n i is a first integral of (1.3). In this case F is rational, i.e. F C(x, y). The above mentioned theorem of Darboux gives us sufficient conditions for integrability via the method of Darboux using algebraic solutions of systems (1.1). However these conditions are not necessary as can be seen from the following example. The system dx/dt = y x 2 + 2y dy/dt = x + 2xy has two invariant algebraic curves: the invariant line 1 + 2y = 0 and a conic invariant curve f = 12x 2 8y y 5 = 0. This system is integrable having as first integral F = (1 + 2y)f but here s = 2 < 3 = m(m + 1)/2. Sufficient conditions for Darboux integrability were obtained by Christopher and Kooij in [11] and Zoladek in [21]. Their theorems say that if a system has s invariant algebraic curves in generic position such that s i=1 deg f i = m+1 then the system has a Darboux integrating factor. Other sufficient conditions for integrability covering the example above were given in [5]. However to this day we do not have necessary and sufficient conditions for Darboux integrability and the search is on for finding such conditions. We can extend the problem of Poincaré as it was done in [16] where Schlomiuk called the resulting problem: The extended Poincaré problem: Give necessary and sufficient conditions for a polynomial system (1.1) to have: (i) a polynomial inverse integrating factor; (ii) an integrating factor of the form s i=1 f i(x, y) λ i ; (iii) a Darboux integrating factor (or a Darboux first integral); (iv) a rational first integral. In recent years there has been much activity in this area of research and there are several partial results which were obtained on the extended problem of Polincaré. A more systematic attempt to solve this problem is however needed. We need to collect in a systematic way information on this problem starting with the simplest kind of invariant algebraic curves of quadratic equations (1.3). It is thus natural to start by studying quadratic 4

9 systems having invariant lines of the maximum possible total multiplicity which for quadratic systems is six [1]. Among the things we prove in this article is the fact that all quadratic differential systems which have invariant lines of at least five total multiplicity are integrable. All those with invariant lines of total multiplicity 6 have polynomial inverse integrating factors and rational first integrals and are thus useful examples for studying Poincaré s problem. On the other hand all those with total multiplicity five (cf. Section 4) have a polynomial inverse integrating factor and are integrable via the method of Darboux with a first integral not necessarily rational, forming a very useful material for studying the problems arising from the work of Darboux, in particular the extended Poincaré problem (i), (ii) and (iii). Definition 1.9. We call configuration of invariant lines of a system (1.1) the set of all its invariant lines (real or/and complex), each endowed with its own multiplicity and together with all the real singular points of this system located on these lines, each one endowed with its own multiplicity. Theorem 3.1 Consider a quadratic system (1.1) possessing invariant lines whose total multiplicity is six, including the line at infinity and including multiplicities of the lines. Then this system has a polynomial inverse integrating factor which splits into linear factors over C and it has a rational first integral foliating the plane into conic curves. Furthermore under the action of the affine group and time rescaling a quadratic system (1.1) with invariant lines of total multiplicity six is equivalent to one of the eleven systems indicated in Table 1 which form a system of representatives of the orbits. This table sums up all possibilities we have for the configurations of invariant lines with their multiplicities, of systems in this family and it also lists the corresponding cofactors of the lines as well as the inverse integrating factors and first integrals of the systems. Remark 1.4. We note that nine of these cases are integrable via the integrability theorem of Christopher, Kooij and Zoladek on systems with invariant algebraic curves in general positions (see [11], [21]). Indeed, for the first case listed in Table 1, we only need to use the lines x y, x + 1, y 1 which are in generic position, yielding a Darboux first integral. In an analogous way we can find three lines in general position for all the remaining cases up to and including the ninth case. For the integrability problem only the cases 10 and 11 are not covered by their theorem. Theorem 3.2 Consider the action of the group of real affine transformations an time rescaling on the class of quadratic systems (1.1) possessing invariant lines whose total multiplicity is six. The orbits of the ( quotient of this class of systems under this action are classified by the multi integer-valued invariant NC, m G, N R, n, M(l R ), mg) R. The full classification of these orbits is given in Diagram 1 according to the possible values of this invariant. Diagram 1 also contains all the types of the configurations of the lines and all the corresponding phase portraits of such systems. Observation 1.5. In we have defined a number of integer-valued invariants attached to the systems which express geometric properties of the systems formulated in terms of the invariant lines with their respective multiplicities and their singularities (with their associated multiplicities). Are these invariants forced to satisfy certain restrictions when taken together? Evidently the restriction of the degree of the systems to be two forces other restrictions. For example the total multiplicity of the lines in a configuration cannot be greater than 6. We look for other relations of this kind. Notation 1.6. We denote by N C the number of all distinct invariant lines l 1,...,l NC of a system (2.1). We denote by M i the multiplicity of the line l i and by n Mi the number of the lines l j, j 1,..., N C } with multiplicity M i. We denote M = maxm i i = 1,..., N C }. Then we have the following formula: N C M i n Mi = 6. i=1 Notation 1.7. Let us denote by m S the maximum of the multiplicities of singularities of the system which are located on anyone of the lines and by n ms the number of the singularities of the system with multiplicity m S. 5

10 Theorem 3.3 Consider quadratic systems of the form (1.1) possessing invariant lines whose total multiplicity is six. Then M 3, m S 1, 3, 6}, all singularities are located on the invariant line Configurations and we have the following three possibilities: (i) M = 1 and in this case N C = 6 = n M. In this case m S = 1, we have 7 simple singular points in P 2 (C); each line contains exactly 3 singular points and through four of singularities there pass three invariant lines. (ii) M = 2 and in this case N C = 4, n M = 2, the remaining two lines are simple and m S = 4, n ms = 1. The singular point of multiplicity 4 is situated at the intersection of the two double lines. The remaining three singularities are simple. (iii) M = 3 and in this case N C = 2 = n M, m S = 6, n ms = 1 and the point of multiplicity 6 is at the intersection of the two triple lines. The only other singular point is simple, located on one of the two simple lines. Corollary 3.3 a) The only partitions of the number 6 which correspond to configurations of distinct invariant straight line of quadratic systems (1.1) possessing invariant lines whose total multiplicity is six are: (1, 1, 1, 1, 1, 1), (2, 2, 1, 1), (3, 3). b) The remaining partitions of the number 6 (2, 1, 1, 1, 1), (3, 1, 1, 1), (2, 2, 2), (3, 2, 1), (4, 1, 1), (4, 2), (5, 1), (6) cannot be realized in configurations of distinct invariant straight line of quadratic systems possessing invariant lines whose total multiplicity is six. Comment: It would be nice to have a direct geometrical reason explaining the above Corollary. 2 Divisors associated to invariant lines configurations Consider real differential systems of the form: dx = p 0 + p 1 (x, y) + p 2 (x, y) p(x, y), (S) dt dx = q 0 + q 1 (x, y) + q 2 (x, y) q(x, y) dt with p 0 = a 00, p 1 (x, y) = a 10 x + a 01 y, p 2 (x, y) = a 20 x 2 + 2a 11 xy + a 02 y q 0 = b 00, q 1 (x, y) = b 10 x + b 01 y, q 2 (x, y) = b 20 x 2 + 2b 11 xy + b 02 y 2. Let a = (a 00, a 10, a 01, a 20, a 11, a 0 b 00, b 10, b 01, b 20, b 11, b 02 ) be the 12-tuple of the coefficients of system (2.1) and denote R[a, x, y] = R[a 00, a 10, a 01, a 20, a 11, a 0 b 00, b 10, b 01, b 20, b 11, b 0 x, y]. Notation 2.1. Let us denote by a = (a 00, a 10..., b 02 ) a point in R 12. Each particular system (2.1) yields an ordered 12-tuple a of its coefficients. Notation 2.2. Let P (X, Y, Z) =p 0 (a)z 2 + p 1 (a, X, Y )Z + p 2 (a, X, Y ) = 0, Q(X, Y, Z) =q 0 (a)z 2 + q 1 (a, X, Y )Z + q 2 (a, X, Y ) = 0. We denote σ(p, Q) = w P 2 (C) P (w) = Q(w) = 0}. Definition 2.1. We consider formal expressions D = n(w)w where n(w) is an integer and only a finite number of n(w) are nonzero. Such an expression is called: i) a zero-cycle of P 2 (C) if all w appearing in D are points of P 2 (C); ii) a divisor of P 2 (C) if all w appearing in D are irreducible algebraic curves of P 2 (C); iii) a divisor of an irreducible algebraic curve C in P 2 (C) if all w in D belong to the curve C. We call degree of the expression D the integer deg(d) = n(w). We call support of D the set Supp (D) of w appearing in D such that n(w) 0. (2.1) 6

11 Notation 2.3. Let us denote by QS = (S) (S) is a system (2.1) such that gcd(p(x, y), q(x, y)) = 1 and max ( deg(p(x, y)), deg(q(x, y)) ) = 2 QSL = (S) QS (S) possesses at least one invariant affine line or the line at infinity with multiplicity at least two For the multiplicity of the line at infinity the reader is refereed. In this section we shall assume that systems (2.1) belong to QS. Definition 2.2. Let C(X, Y, Z) = Y P (X, Y, Z) XQ(X, Y, Z). D S (P, Q) = D S (C, Z) = D S (P, Q; Z) = D S (P, Q, Z) = w σ(p,q) w Z=0} w Z=0} w Z=0} I w (P, Q)w; I w (C, Z)w if Z C(X, Y, Z); I w (P, Q)w; ( ) I w (C, Z), I w (P, Q) w, } ; }. where I w (F, G) is the intersection number (see, [8]) of the curves defined by homogeneous polynomials F, G C[X, Y, Z] (deg(f ), deg(g) 1). Notation 2.4. n R =#w Supp D S (C, Z) w P 2 (R)}. (2.2) A complex projective line ux + vy + wz = 0 is invariant for the system (S) if either it coincides with Z = 0 or it is the projective completion of an invariant affine line ux + vy + w = 0. Notation 2.5. Let S QSL. Let us denote IL(S) = l l is a line in P 2 (C) such that l is invariant for (S) } ; M(l) = the multiplicity of the invariant line l of (S). Remark 2.6. We note that the line l : Z = 0 is included in IL(S) for any S QSL. Let l i : f i (x, y) = 0, i = 1,..., k, be all the distinct invariant affine lines (real or complex) of a system S QSL. Let l i : F i(x, Y, Z) = 0 be the complex projective completion of l i. Notation 2.7. We denote G : F i (X, Y, Z) Z = 0; Sing G = w G w is a singular point of G} ; i ν(w) = the multiplicity of the point w, as a point of G. Definition 2.3. D IL (S) = M(l)l, (S) QSL; l IL(S) Supp D IL (S) = l l IL(S)}. 7

12 Notation 2.8. M IL = deg D IL (S); N C =#Supp D IL ; N R =#l Supp D IL l P 2 (R)}; n R =#ω Supp D (P, Q) ω G G, σ S }; R 2 d R = I G, σ ω (P, Q); ω G R 2 m G = maxν(ω) ω Sing G C 2}; m R = maxν(ω) ω Sing G G R 2}. (2.3) 3 The class of quadratic systems with M IL = 6 Definition 3.1 (Poincaré [14]). Let F = F 1, where F 1, F 2 C[x, y], be a fist integral of a system (1.1). F 2 We call remarkable values in C for F, constants K C for which F 1 KF 2 are reducible. If K is such a remarkable value for F and F 1 KF 2 = u α 1 1 uα 2 2 uα k k where u i C[x, y] and not all α i s are 1, then such a K is called a critical value for F. Notation 3.1. We denote by QSL 6 the class of all quadratic differential systems (2.1) with p, q relatively prime (i.e. gcd(p, q) = 1), Z C, and possessing a configuration of invariant straight lines of total multiplicity M IL = 6 including the line at infinity and including possible multiplicities of the lines. 3.1 Darboux integrating factors and first integrals Theorem 3.1. Consider a quadratic system (2.1) in QSL 6. Then this system has a polynomial inverse integrating factor which splits into linear factors over C and it has a rational first integral foliating the plane into conic curves. Furthermore under the action of the affine group and time rescaling a system (2.1) in QSL 6 is equivalent to one of the eleven systems indicated in Table 1 which form a system of representatives of the orbits. This table sums up all possibilities we have for the configurations of invariant lines (including their multiplicities) of systems in this family and it also lists the corresponding cofactors of the lines as well as the inverse integrating factors and first integrals of the systems. Proof: Orbit representatives and invariant affine lines with their multiplicities which are listed here in the second columns of Table 1 were determined in. The corresponding co-factors, Darboux s integrating factors and first integrals from Table 1 are obtained via straightforward computations. We observe that the first integrals computed via the method of Darboux and which are listed in the last column of Table 1 yield foliations by conics of the plane in all cases. We note that in the cases 2) and 3) we first obtain first integrals with higher degree polynomials. To obtain the second first integral listed for the cases 2) and 3) in the last column of Table 1 we observe that K = 1 is a critical value for the first integral F = (x2 + 1)(y 2 + 1) (x y) 2. Indeed, we have (x 2 + 1)(y 2 + 1) (x y) 2 = (xy + 1) 2 = 0. Hence xy + 1 = 0 is an invariant conic of the system and by Darboux theory we obtain the first integral G = xy + 1 x y. In case 3) we obtain that K = 4 is a critical value for the first integral F = x4 + 2x 2 (y 2 + 1) + (y 2 1) 2 x 2. 8

13 Orbit Invariant lines and their representative multiplicities ẋ = 1 + x 1) x ± 1 (1), ẏ = 1 + y 2 y ± 1 (1), x y (1) ẋ = 1 + x 2) 2 x ± i (1),, ẏ = 1 + y 2 y ± i (1), x y (1) ẋ = 2xy, 3) ẏ = y 2 x 2 1 x ± i(y 1) (1), x ± i(y + 1) (1), x (1) ẋ = 2xy, x 1 ± iy (1), 4) ẏ = 1 x 2 + y 2 x + 1 ± iy (1), x (1) 5) ẋ = x ẏ = y 2 x (2), y (2), x y (1) ẋ = 2xy, 6) x ± iy (2), ẏ = x 2 + y 2 x (1) Respective cofactors x 1, y 1, x + y x i, y i, x + y y + 1 ix y 1 ix, y y i(x + 1), y i(x 1), y Inverse integrating factor (x 1)(x y) (y + 1) (x 2 + 1)(y 2 + 1) Table 1 (M IL = 6) First integral (x + 1)(y 1) (x 1)(y + 1) (x 2 +1)(y 2 +1) (x y) 2 or (xy+1)/(x y) [ x 2 + (y 1) 2] x 2 [ 2x 2 (y 2 +1)+ [ x 2 + (y + 1) 2] x 4 + (y 2 1) 2] or (x 2 +y 2 1)/x [ [ x 2 + (y 1) 2] x 2 + (y + 1) 2] (x 1) 2 + y 2 x x, y, x + y x 2 y 2 or xy(x y) (x y)x 1 y 1 (x y ix, x y 2 + y 2 ) 2 or x(x 2 + y 2 x ) 1 (x 2 + y 2 ) ẋ = 1 + x 7) x + 1 (1), y (1) x 1, 2 ẏ = 2y x 1 (2) x + 1 (x 2 (x + 1)y 1)y ẋ x 1 = 1 x 8) (x x ± 1 (2), y (1) ±x 1, 2x 2 1)y or x 2 1 ẏ = 2xy ẋ (x 2 1) 2 y = 1 + x 9) (x x ± i (2), y (1) x ± i, 2x 2 + 1)y or x ẏ = 2xy (x 2 + 1) 2 y 10) ẋ = x ẏ = 1 x (3) x x 2 (xy + 1)/x ẋ = x, 11) ẏ = y x 2 x (3) 1 x 2 (x 2 + y)/x Indeed, we have x 4 + 2x 2 (y 2 + 1) + (y 2 1) 2 4x 2 = (x 2 + y 2 1) 2 = 0. Then by theory of Darboux we obtain the first integral G = x2 + y 2 1 and hence in the case 3) too we have a foliation of the plane by x conics. 3.2 Phase portraits In order to construct the phase portraits corresponding to quadratic systems given by Tables 1 and 2 we shall use the CT -comitants constructed in [19] (see also ) as follows (for detailed definitions of the 9

14 notions involved see [19] and [17]). C i (a, x, y) = yp i (x, y) xq i (x, y), i = 0, 1, 2; D i (a, x, y) = x p i(a, x, y) + y q i(a, x, y), i = 1, 2; M(a, x, y) = 2 Hess ( C 2 (a, x, y) ) ; η(a) = Discriminant ( C 2 (a, x, y) ) ; K(a, x, y) = Jacob ( p 2 (x, y), q 2 (x, y) ) ; µ 0 (a) = Res x (p q 2 )/y 4 = Discriminant ( K(a, x, y) ) /16; H(a, x, y) = Discriminant (αp 2 (x, y) + βq 2 (x, y)) α=y, β= x} ; L(a, x, y) = 2K 4H M; K 1 (a, x, y) = p 1 (x, y)q 2 (x, y) p 2 (x, y)q 1 (x, y). (3.1) Remark 3.2. We note that by Discriminant (C 2 ) of the cubic form C 2 (a, x, y) we mean the expression given in Maple via the function discrim(c x)/y 6. In order to construct other necessary invariant polynomials let us consider the differential operator L = x L 2 y L 1 acting on R[a, x, y] constructed in [2], where L 1 = 2a 00 + a a 10 a 20 2 a b 00 + b a 11 b 10 b 20 2 b 01, b 11 L 2 = 2a 00 + a a 01 a 02 2 a b 00 + b a 11 b 01 b 02 2 b 10 as well as the classical differential operator (f, ϕ) (k) acting on R[x, y] 2 which is called transvectant of index k (see, for example, [9, 12]): b 11 (f, g) (k) = k ( ) k ( 1) h k f h x k h y h h=0 k g x h. (3.2) yk h So, by using these operators and the GL-comitants µ 0 (a), M(a, x, y), K(a, x, y), D i (a, x, y) and C i (a, x, y) we construct the following polynomials: µ i (a, x, y) = 1 i! L(i) (µ 0 ), i = 1,.., 4, κ(a) = (M, K) (2), κ 1 (a) = (M, C 1 ) (2), K 2 = 4 Jacob(J ξ) + 3 Jacob(C 1, ξ)d 1 ξ(16j 1 + 3J 3 + 3D 2 1), K 3 = 2C 2 2(2J 1 3J 3 ) + C 2 (3C 0 K 2C 1 J 4 ) + 2K 1 (C 1 D 2 + 3K 1 ), (3.3) where L (i) (µ 0 ) = L(L (i 1) (µ 0 )) and J 1 = Jacob(C 0, D 2 ), J 2 = Jacob(C 0, C 2 ), J 3 = Discrim(C 1 ), J 4 = Jacob(C 1, D 2 ), ξ = M 2K. Theorem 3.2. Consider the action of the group of real affine transformations and time rescaling on the class QSL 6. The orbits of the quotient of QSL 6 under this action are classified by the multi integervalued invariant ( N C, m G, N R, n, M(l R ), mg) R. The full classification of these orbits is given in Diagram 1 according to the possible values of this invariant. Diagram 1 also contains all the types of the configurations of the lines and all the corresponding phase portraits of such systems. Proof: Using the results of and we have obtained Table 1 where in column 2 we have the invariant lines of the systems. We shall examine step by step each orbit representative given by Table 1. 10

15 11 Diagram 1 (M IL = 6)

16 1) ẋ = 1 + x ẏ = 1 + y Finite singular points: M 1 ( 1, 1), M 2 (1, 1) are saddles; M 3 ( 1, 1), M 4 (1, 1) are nodes; 1.2. Infinite singular points: η = 1 > 0, µ 0 = 1 > 0 2) ẋ = 1 + x ẏ = 1 + y Finite singular points: There are no real singular points Infinite singular points: η = 1 > 0, µ 0 = 1 > 0 3) ẋ = 2xy, ẏ = 1 x 2 + y Finite singular points: M 1,2 (0, ±1) are nodes Infinite singular points: η = 4 < 0, µ 0 = 4 < 0 4) ẋ = 2xy, ẏ = 1 x 2 + y Finite singular points: M 1,2 (±1, 0) are centers Infinite singular points: η = 4 < 0, µ 0 = 4 < 0 5) ẋ = x ẏ = y Finite singular points: M 0 (0, 0) of multiplicity four. 12

17 5.2. Infinite singular points: η = 1 > 0, µ 0 = 1 > 0 6) ẋ = 2xy, ẏ = x 2 + y Finite singular points: M 0 (0, 0) of multiplicity four Infinite singular points: η = 4 < 0, µ 0 = 4 < 0 7) ẋ = 1 + x ẏ = 2y 7.1. Finite singular points: M 1 ( 1, 0) is a saddle and M 1 (1, 0) is a node Infinite singular points: η = 0, M = 8x 2 0, µ 0 = µ 1 = κ = κ 1 = 0, L = 8x 2 > 0, µ 2 = 4x 2 > 0, K 2 = 384x 2 > 0 8) ẋ = 1 x ẏ = 2xy 8.1. Finite singular points: M 1,2 (±1, 0) are nodes Infinite singular points: η = 0, M = 8x 2 0, µ 0 = µ 1 = κ = κ 1 = 0, L = 8x 2 < 0, µ 2 = 4x 2 < 0 9) ẋ = 1 x ẏ = 2xy 9.1. Finite singular points: There are no real singular points Infinite singular points: η = 0, M = 8x 2 0, µ 0 = µ 1 = κ = κ 1 = 0, L = 8x 2 < 0, µ 2 = 4x 2 > 0 10) ẋ = x ẏ = Finite singular points: There are no real singular points. 13

18 10.2. Infinite singular points: η = 0, M = 8x 2 0, µ 0 = µ 1 = µ 2 = µ 3 = κ = κ 1 = 0, µ 4 = x 4 > 0, K = 0, L = 8x 2 > 0, K 2 = 0 11) ẋ = x, ẏ = y x Finite singular points: M 1 (0, 0) is a node Infinite singular points: η = M = 0, C 2 = x 3 0, µ 0 = µ 1 = µ 2 = K = 0, µ 3 = x 3 < 0, K 1 = x 3 < 0, K 3 = 6x 6 > 0 As all the cases from Table 1 have been examined, the Theorem 3.2 is proved. As a byproduct of Diagram 1 we also obtain the following Theorem 3.3. Consider quadratic systems of the class QSL 6. Then M 3, m S 1, 3, 6}, all singularities are located on the invariant line Configurations and we have the following three possibilities: (i) M = 1 and in this case N C = 6 = n M. In this case m S = 1, we have 7 simple singular points in P 2 (C); each line contains exactly 3 singular points and through four of singularities there pass three invariant lines. (ii) M = 2 and in this case N C = 4, n M = 2, the remaining two lines are simple and m S = 4, n ms = 1. The singular point of multiplicity 4 is situated at the intersection of the two double lines. The remaining three singularities are simple. (iii) M = 3 and in this case N C = 2 = n M, m S = 6, n ms = 1 and the point of multiplicity 6 is at the intersection of the two triple lines. The only other singular point is simple, located on one of the two simple lines. Corrolary 3.3. a) The only partitions of the number 6 which correspond to configurations of distinct invariant straight line of quadratic systems (1.1) possessing invariant lines whose total multiplicity is six are: (1, 1, 1, 1, 1, 1), (2, 2, 1, 1), (3, 3). b) The remaining partitions of the number 6 (2, 1, 1, 1, 1), (3, 1, 1, 1), (2, 2, 2), (3, 2, 1), (4, 1, 1), (4, 2), (5, 1), (6) cannot be realized in configurations of distinct invariant straight line of quadratic systems possessing invariant lines whose total multiplicity is six. Comment: It would be nice to have a direct geometrical reason explaining the above Corollary. 3.3 The projective classification in P 2 (C) of the invariant lines configurations of systems in QSL 6 To a system (1.1) we can associate the equation (1.3) defined by the 1-form ω 1 = q(x, y)dx p(x, y)dy. We consider the map j : C 3 \ Z = 0} C given by j(x, Y, Z) = (X/Z, Y/Z) = (x, y) and suppose that max ( deg(p), deg(q) ) = m > 0. Since x = X/Z and y = Y/Z we have: dx = (ZdX XdZ)/Z dy = (ZdY Y dz)/z 14

19 the pull back form j (ω 1 ) has poles at Z = 0 and its associated equation j (ω 1 ) = 0 can be written as j (ω 1 ) = q(x/z, Y/Z)(ZdX XdZ)/Z 2 p(x/z, Y/Z)(ZdY Y dz)/z 2 = 0. Then the 1 form ω = Z m+2 j (ω 1 ) in C 3 \Z = 0} has homogeneous polynomial coefficients of degree m+1, and for Z 0 the equations ω = 0 and j (ω 1 ) = 0 have the same solutions. Therefore the differential equation ω = 0 can be written as AdX + BdY + CdZ = 0 (3.4) where A(X, Y, Z) = ZQ(X, Y, Z) = Z m+1 q(x/z, Y/Z), B(X, Y, Z) = ZP (X, Y, Z) = Z m+1 p(x/z, Y/Z), C(X, Y, Z) = Y P (X, Y, Z) XQ(X, Y, Z) and P (X, Y, Z) = Z m p(x/z, Y/Z), Q(X, Y, Z) = Z m q(x/z, Y/Z). Clearly A, B and C are homogeneous polynomials of degree m + 1 satisfying AX + BY + CZ = 0. The equation (3.4) becomes in this case P (Y dz ZdY ) + Q(ZdX XdZ) = 0 or equivalently (E 0 ) QZdX P ZdY + (Y P XQ)dZ = 0. (3.5) Notation 3.4. We consider the set EQ of all real differential equations (E) AdX + BdY + CdZ = 0 where A, B, C are cubic homogeneous polynomials in X, Y, Z over R subject to the identity AX + BY + CZ = 0. On EQ acts the group P GL(2, R) of projective transformations of P 2 (R). We consider the set Eq of all equations (E) in EQ of the form (E 0 ) obtained from equations (1.3). Notation 3.5. We denote by Eql 6 the class of all equations in Eq of the form (E 0 ) obtained from equations (1.3) possessing invariant lines of total multiplicity six. On Eq (and Eql 6 acts Aff(2, R). As Aff(2, R) is a subgroup of P GL(2, R), we may have two distinct orbits under the action of Aff(2, R) of systems in Eq contained in a single orbit of EQ under the action of the bigger group P GL(2, R). As we show in Theorem 3.4 below this indeed occurs. Definition 3.2. We define a projective invariant for equation in Eql 6 as follows: ( m R Sing = max # w l i w Sing G P2 (R) }). i 1,...,N R } Notation 3.6. Let us denote N m R Sing = the number of the real invariant lines on which lie exactly m R Sing real singularities of G. Theorem 3.4. We consider the systems in QSL 6 and their associated real equations in Eql 6. We consider the action of the group P GL(2, R) of real projective transformations of the plane on the class EQ. i) Two systems (S 1 ) and (S 2 ) in QSL 6 located on distinct orbits under the action of the real affine group and time rescaling could yield equations (E 1 ) and (E 2 ) located on the same orbit under the action of P GL(2, R) on EQ. ii) The classification of the orbits of equations (E 0 ) in EQ associated to systems (S) in QSL 6, under the action of P GL(2, R) on EQ is given in Diagram 2. 15

20 Diagram 2 Proof: i) Consider the systems (S 1 ) ẋ = 1 + x ẏ = 1 + y 2 and (S 2 ) ẋ = 2xy, ẏ = y 2 x 2 1 Their corresponding invariant line Configurations are: Config. 6.2 and Config. 6.3 which are not equivalent under the group of real affine affine transformation and time rescaling. Consider now their associated differential equations (3.5) in P 2 (R): (E 1 ) [ Z(Y 2 + Z 2 )dx Z(X 2 + Z 2 )dy + [ Y (X 2 + Z 2 ) X(Y 2 + Z 2 ) ] dz = 0 ] and (E 2 ) It can easily be checked that the real projective transformation of P 2 (R) X X 1 Y = Y 1 Z Z 1 transform (E 1 ) into (E 2 ). [ Z(Y 2 X 2 Z 2 )dx 2XY ZdY + X(X 2 + Y 2 + Z 2 )dz = 0. ii) We consider the eleven representatives of the orbits of QSL 6 under the action of the affine group and time rescaling and their respective configurations in Diagram 2. In i) we showed that the equations associated to systems with the distinct configurations Config. 6.2 and Config. 6.3 in Diagram 2, lie on the same orbit of the action of P GL(2, R) on EQ. We also show below that the equations associated to systems with Config. 6.i, i = 5, 7, 8 (respectively Config. 6.j, j = 6, 9 or Config. 6.j, j = 10, 11) are also located on 16 ]

21 the same orbit. The proofs are obtained in analogous way to the cases in i) and we only list below the corresponding transformations (the respective equations can be easily computed directly having canonical systems and the equation (3.5). 1) Config. 6.5 : 2) Config. 6.5 : 3) Config. 6.6 : 4) Config : ẋ = x ẏ = y 2 : ẋ = x ẏ = y 2 : ẋ = 2xy, ẏ = x 2 + y 2 : ẋ = x, ẏ = 1 : ẋ = 1 + x Config. 6.7 : ẏ = 2y ẋ = 1 x Config. 6.8 : ẏ = 2xy ẋ = 1 + x Config. 6.9 : ẏ = 2xy ẋ = 1 + x Config : ẏ = y x The class of quadratic systems with M IL = 5 Notation 4.1. We denote by QSL 5 the class of all quadratic differential systems (2.1) with p, q relatively prime ((p, q) = 1), Z C, and possessing a configuration of invariant straight lines of total multiplicity M IL = 5 including the line at infinity and including possible multiplicities of the lines. 4.1 Darboux integrating factors and first integrals Theorem 4.1. Consider a quadratic system (2.1) in QSL 5. Then this system has a polynomial inverse integrating factor which splits into linear factors over C and it has a Darboux first integral. Furthermore the quotient set by the action of the affine group and time rescaling of QSL 5 is formed by: (i) a discrete set of 19 orbits; (ii) a set of 11 one-parameter families of orbits. A system of representatives of the quotient is given in Table 2. This table sums up all possibilities we have for the configurations of invariant lines with their multiplicities, of systems in QSL 5 and it also lists the corresponding cofactors of the lines as well as the integrating factors and first integrals of the systems. Corrolary 4.2. All the systems in QSL 5 have elementary real first integrals. We only list below in Table 3 all real first integrals which correspond to those in the last column of Table 2 which are given there in complex form. 4.2 Phase portraits Notation 4.3. J f (S) = w σ(p,q) i w where i w is the Poincaré index of w. Theorem 4.2. Consider the systems of the form (2.1) in QSL 5. Then the action of the affine group and time rescaling yields a classification which corresponds to the distinct values of the invariant ( N C, m G, N R, n R, M(l ), n R G, σ, dr G, σ, J f (S)). This classification appears in Diagram 3 where all the types of the configurations of the lines and all the corresponding phase portraits of such systems are listed. Proof: Using the results of and we shall examine step by step each orbit representative given by Table 2. 17

22 Orbit Invariant lines and their representative multiplicities ẋ = (x+1)(gx+1), x + 1 (1), 1) ẏ = (g 1)xy + y gx + 1 (1), g(g 2 x y + 1 (1), 1) 0 y (1) ẋ=g(x 2 4), g 0 ±i(x 2)+ 2) ẏ =(g 2 4)+(g 2 +4)x y + g (1), x 2 + gxy y 2 x ± 2 (1) ẋ = 1+x 3) g = 0, 1 x ± 1 (1), ẏ = g(y 2 1), y ± 1 (1) ẋ = 1 + x 4) x ± 1 (1), ẏ = g(y 2 + 1), g 0 y ± i (1) ẋ = 1 + x 5) g = 0, 1 x ± i (1), ẏ = g(y 2 + 1), y ± i (1) I ẋ = 1 + 2xy, ± = x + c ± 6) ẏ = g x 2 + y 2 i(y, 1 2c ) (1), g = c 2 1/(4c 2 I ± = x c ± ) i(y + 1 2c ẋ ) (1) = 1 + x, x + 1 (1), y (1) 7) ẏ = xy + y 2 x y + 1 (1) ẋ = gx 8) g 0, 1 x (2), x y (1) ẏ = (g 1)xy + y 2 y (1) ẋ = 2x, 1 + y ± ix (1), 9) ẏ = 1 x 2 y 2 x (1) ẋ = gx 10) g 0 x ± iy (1) ẏ = x 2 + gxy y 2 x (2) ẋ = x 11) 2 + xy, x (2), y + 1 (1), ẏ = y + y 2 y (1) 14) ẏ = (g 1)xy, g(g 2 1) 0 ẋ = g(x 15) 2 + 1), ẏ = 2y, g 0 Respective cofactors gx + 1, g(x + 1), gx + y 1, (g 1)x+y (g ± i)x y + g 2i, g(x 2) x 1, g(y 1) x 1, g(y i) x i, g(y i) i(x c) +(y + 1 2c ), i(x + c) +(y 1 2c ) 1, y x, y + 1 gx, gx+y, (g 1)x+y 1 y ± ix, 2 (g i)x y, gx x + y, y, y + 1 Inverse integrating factor y(gx + 1) (x y + 1) (x + 2) [ (x 2) 2 + (y + g) 2 ) ] (x 2 1) (y 2 1) (x 2 1) (y 2 + 1) (x 2 + 1) (y 2 + 1) I + I I + I y(x y+1) xy(x y) x 2 +(y+1) 2 x(x 2 +y 2 ) x 2 y Table 2 (M IL = 5) First integral (g x + 1)y g (x y + 1) g (x + 2) 2 [i(x 2) + y + g] ig [i(2 x) + y + g] ig (x+1) g (y 1) (x 1) g (y+1) (x+1) g (y+i) i (x 1) g (y i) i (x+i) g (y i) (x i) g (y+i) (I +) 2c2 +i (I ) 2c2 i (I + )2c2 +i (I )2c2 i e x (x y + 1)/y xy g (x y) g e ix (1 + y ix) 1 + y + ix x 2 (ix + y) ig (ix y) ig y e (y+1)/x 12) ẋ = 1 + x ẏ = y 2 x ± 1 (1), y (2) x 1, y y 2 (x 2 1) e 2/y (x 1)(x+1) 1 ẋ = g(x 13) 2 1), x ± 1 (1), g(x 1), ẏ = 2y, g(g 2 y(x 1) 0 y (1) ) y g (x+1)(x 1) 1 ẋ = (x + 1)(gx + 1), x+1 (2), gx + 1, y(x + 1) (gx + 1)y g gx + 1 (1), g(x + 1), (gx + 1) (x + 1) y (1) (g 1)x, g x ± i (1) y (1) g(x i), 2 y(x 2 1) y g (x i) i (x+i) i 16) ẋ = 1 + x ẏ = y 2 x ± i (1), y (2) x i, y y 2 (x 2 + 1) e 2/y (x+i) i (x i) i 17) ẋ = x ẏ = 2y x (2), y (1) x, 2 x 2 y y e 2/x 1) ẋ = (x + 1)(gx + 1), ẏ = (g 1)xy + y g(g 2 1) Finite singular points: M 1 ( 1, 0) is a node, M 2 (g 1, 1) is a saddle; for g > 0 M 3 ( 1/g, 0) is a saddle, M 4 ((g 1)/g, 1/g) is a node; for g < 0 M 3 is a node, M 4 is a saddle;

23 Orbit representative Invariant lines and their multiplicities Respective cofactors Table 2 (M IL = 5)(continued) Inverse integrating factor First integral 18) ẋ = 1 + x, ẏ = xy x + 1 (2), y (1) 1, x (x + 1)y y e x (x + 1) 1 19) ẋ = x 2 + xy, ẏ = y 2 x (2), y (2) x + y, y x 2 y y e y/x 20) ẋ = 1 + x ẏ = 1 x ± 1 (1) x 1 x 2 1 e 2y (x+1)(x 1) 1 ẋ = 1 + x 21) x 1 (2), x + 1, (x 1) 1 (x 1) ẏ = x + 2y x + 1 (1) x 1 2 [ ] exp xy+y+1 x 1 ẋ = 1 x 22) x ± 1 (2) x ± 1 (x ẏ = 1 2xy 2 1) 2 exp [ 2x 4y ] x 2 1 (x 1)(x+1) 1 ẋ = 1 + x 23) x 1 (3), x + 1, (x + 1) (x + 1) ẏ = 3 + y x 2 + xy x + 1 (1) x 1 (x 1) 2 exp [ y 2] x+1 ) i ( i x 24) ẋ = 1 + x ẏ = 1 x ± i (1) x i x e 2y i + x ẋ = 1 x 25) x ± i (2) x ± i (x ẏ = 1 2xy 2 + 1) 2 exp [ 2x+4y ] x 2 +1 (i+x) i (i x) i ẋ = x, 26) ẏ = y x 2 x (1) 1 1 x(x 2 3y) ẋ = 1 + x, 27) ẏ = y x 2 x + 1 (2) x 1 (x + 1) 2 (x + 1) 2 exp [ x + y 1 ] x+1 ẋ = x 28) x (3) x x ẏ = 1 + x 2 x 1 exp [ y + 1 ] x ẋ = x 29) x (4) x x ẏ = 1 2xy 4 3xy 1 x ẋ 3 = 1, 30) ẏ = x 2 1 x 3 + 3y System 2) 4) 5) 6) 9) First integral (x + 2) 2 exp [ 2g arctg 2 x ] ln x+1 x 1 g + g+y 2 arctg (y) arctg (y) g arctg (x) F 2c2 1 F 2c2 2 exp2 arctg [ (4cx+8c 3 y) 4c 2 (x 2 + y 2 ) 4c 4 1)]} Table 3 xũ (1+y)ṽ xṽ + (1+y)ũ, ũ=cos x ṽ =sin x 2 System 10) 15) 16) 24) 25) First integral ln x +g arctg y x g ln y 2 arctg (x) F 1,2 = 4c 2 (x 2 + y 2 ) ± 8c 3 x 4cy + 4c /y arctg (x) y arctg (x) (x + 2y)/(x 2 +1)+ arctg (x) 1.2. Infinite singular points: η = 1 > 0, µ 0 = g 2 > 0 19

24 Diagram 3 (M IL = 5) 2) ẋ = g(x 2 4), g 0, ẏ = (g 2 4) + (g 2 + 4)x x 2 + gxy y Finite singular points: M 1 (2, g) is a node, M 2 (2, 3g) is a saddle.

25 Diagram 3 (M IL = 5) (continued) 2.2. Infinite singular points: η = 4 < 0, µ 0 = g 2 > 0 21

26 Diagram 3 (M IL = 5) (continued) 22

27 3) ẋ = 1 + x ẏ = g(y 2 1), g Finite singular points: M 1,2 (±1, ±1) are saddles for g < 0 and nodes for g > 0; M 3,4 (±1, 1) are nodes for g < 0 and saddles for g > 0; 3.2. Infinite singular points: η = g 2 > 0, µ 0 = g 2 > 0 4) ẋ = 1 + x ẏ = g(y 2 + 1), g Finite singular points: There are not real singular points Infinite singular points: η = g 2 > 0, µ 0 = g 2 > 0 5) ẋ = 1 + x ẏ = g(y 2 + 1), g = 0, Finite singular points: There are no real singular points Infinite singular points: η = g 2 > 0, µ 0 = g 2 > 0 6) ẋ = 1 + 2xy, ẏ = g x 2 + y g R 6.1. Finite singular points: ( M 1,2 (± 2g + 2 1/2, g 2 + 1) ( g/2 + ) ) 1/2 g 2 + 1/2 are foci Infinite singular points: η = 4 < 0, µ 0 = 4 < 0 7) ẋ = 1 + x, ẏ = xy + y Finite singular points: M 1 ( 1, 1) is a saddle and M 2 ( 1, 0) is a node. 23

28 7.2. Infinite singular points: η = 1 > 0, µ 0 = µ 1 = κ = 0, µ 2 = y(y x) = L/8, µ 2 L > 0 8) ẋ = gx g(g 2 1) 0, ẏ = (g 1)xy + y Finite singular points: the systems are homogeneous and the singular point M 1 (0, 0) of multiplicity four has 2 parabolic and 2 hyperbolic sectors Infinite singular points: η = 1 > 0, µ 0 = g 2 > 0 9) ẋ = 2x, ẏ = 1 x 2 y Finite singular points: M 1 (0, 1) is a saddle and M 2 (0, 1) is a (dicritical) node Infinite singular points: η = 4 < 0, µ 0 = µ 1 = 0, κ = 0, µ 2 = 4(x 2 + y 2 ) 0 10) ẋ = gx g 0, ẏ = x 2 + gxy y Finite singular points: the systems are homogeneous and the singular point M 1 (0, 0) of multiplicity four has 2 hyperbolic sectors Infinite singular points: η = 4 < 0, µ 0 = g 2 > 0 11) ẋ = x 2 + xy, ẏ = y + y Finite singular points: M 1 (0, 1) is a node, M 2 (1, 1) is a saddle, M 3 (0, 0) is a saddle-node Infinite singular points: η = 0, M = 8x 2 0, µ 0 = 1 > 0 12) ẋ = 1 + x ẏ = y Finite singular points: M 1 ( 1, 0) and M 2 (1, 0) are both saddlenodes 24

29 12.2. Infinite singular points: η = 1 > 0, µ 0 = 1 > 0 13) ẋ = g(x 2 1), ẏ = 2y, g(g 2 1) Finite singular points: for g > 0 M 1 ( 1, 0) is a saddle and M 2 (1, 0) is a node; for g < 0 M 1 ( 1, 0) is a node and M 2 (1, 0) is a saddle Infinite singular points: η = 0, M = 8g 2 x 2 0, µ 0 =µ 1 =κ=κ 1 =0, L=8g 2 x 2 >0, µ 2 = 4g 2 x 2 > 0, K 2 = 384g 4 x 2 > 0 14) ẋ = (x + 1)(gx + 1), ẏ = (g 1)xy, g(g 2 1) Finite singular points: M 1 ( 1, 0) is a node; M 2 ( 1/g, 0) is a saddle for g > 0 and it is a node for g < Infinite singular points: η = 0, M = 8x µ 0 = µ 1 = κ = κ 1 = 0, µ 2 = g(g 1) 2 x L = 8gx K 2 = 48(g 1) 2 (g 2 g + 2)x 2. We shall consider two subcases: g > 0 and g < 0. a) g >0: η = µ0 =µ 1 =κ=κ 1 =0, M 0, µ 2 >0, L>0, K 2 >0 b) g <0: η = µ0 =µ 1 =κ=κ 1 =0, M 0, µ 2 <0, L<0 15) ẋ = g(x 2 + 1), ẏ = 2y, g Finite singular points: There are no real singular points Infinite singular points: η = 0, M = 8g 2 x 2 0, µ 0 =µ 1 =κ=κ 1 =0, L=8g 2 x 2 >0, µ 2 =4g 2 x 2 >0, K 2 = 384g 4 x 2 <0 16) ẋ = x 2 + 1, ẏ = y Finite singular points: There are no real singular points 25

30 16.2. Infinite singular points: η = 1 > 0, µ 0 = 1 > 0 17) ẋ = x ẏ = 2y Finite singular points: M 1 (0, 0) is a saddle-node Infinite singular points: η = 0, M = 8x 2 0, µ 0 =µ 1 =κ=κ 1 =0, L=8x 2 >0, µ 2 =4x 2 >0, K 2 =0 18) ẋ = 1 + x, ẏ = xy Finite singular points: M 1 ( 1, 0) is a node (dicritical) Infinite singular points: η = 0, M = 8x 2 0, µ 0 =µ 1 =µ 2 =κ=κ 1 =0, L=0, µ 3 = x 2 y, K 1 = x 2 y, µ 3 K 1 > 0 19) ẋ = x 2 + xy, ẏ = y Finite singular points: the systems are homogeneous and the singular point M 1 (0, 0) of multiplicity four has 2 parabolic and 2 hyperbolic sectors Infinite singular points: η = 0, M = 8x 2 0, µ 0 = 1 > 0 20) ẋ = 1 + x ẏ = Finite singular points: there are no singular points Infinite singular points: η = 0, M = 8x 2 0, L=8x 2 >0, µ 0 =µ 1 =µ 2 =µ 3 =κ=κ 1 =0, µ 4 =x 4 >0, K =0, K 2 =384x 2 > 0 21) ẋ = 1 + x ẏ = x + 2y Finite singular points: M 1 ( 1, 1/2) is a saddle and M 2 (1, 1/2) is a node. 26

31 21.2. Infinite singular points: η = 0, M = 8x 2 0, µ 0 =µ 1 =κ=κ 1 =0, µ 2 =4x 2 >0, L=8x 2 >0, K 2 =384x 2 > 0 22) ẋ = 1 x ẏ = 1 2xy Finite singular points: M 1 (1, 1/2) and M 2 ( 1, 1/2) are nodes Infinite singular points: η = 0, M = 8x 2 0, µ 0 =µ 1 =κ=κ 1 =0, µ 2 = 4x 2 <0, L= 8x 2 <0 23) ẋ = 1 + x ẏ = 3 + y x 2 + xy Finite singular points: M 1 (1, 2) is a (dicritical) node Infinite singular points: η = 0, M = 0, µ 0 =µ 1 =µ 2 =0, µ 3 = 8x 3, K =2x K 3 = 24x 6 > 0 24) ẋ = 1 + x ẏ = Finite singular points: There are no singular points Infinite singular points: η =0, M = 8x 2 0, µ 4 =x 4 >0, µ 0 =µ 1 =µ 2 =µ 3 =κ=κ 1 =0, K =0, L=8x 2 >0, K 2 = 384x 2 <0 25) ẋ = 1 x ẏ = 1 2xy Finite singular points: There are no real singular points Infinite singular points: η = 0, M = 8x 2 0, µ 0 = µ 1 = κ = κ 1 = 0, µ 2 = 4x 2 > 0, L = 8x 2 < 0 26) ẋ = x, ẏ = y x Finite singular points: M 1 (0, 0) is a saddle. 27

32 26.2. Infinite singular points: η = 0, M = 0, C 2 = x 3 0, µ 0 = µ 1 = µ 2 = 0, µ 3 = x 3 0, K = 0, K 1 = x 3, µ 3 K 1 < 0 27) ẋ = 1 + x, ẏ = y x Finite singular points: M 1 ( 1, 1) is a node Infinite singular points: η = 0, M = 0, C 2 = x 3 0, µ 0 = µ 1 = µ 2 = 0, K = 0, µ 3 K 1 = x 6 > 0, K 3 = 6x 6 > 0 28) ẋ = x ẏ = 1 + x Finite singular points: There are no singular points Infinite singular points: η = 0, M = 8x 2 0, L=8x 2 >0, µ 0 =µ 1 =µ 2 =µ 3 =κ=κ 1 =0, µ 4 =x 4 >0, K =0 = K 2 29) ẋ = x ẏ = 1 2xy Finite singular points: There are no singular points Infinite singular points: η = 0, M = 8x 2 0, µ 0 =µ 1 =µ 2 =µ 3 =κ=κ 1 =0, µ 4 =x 4 >0, L= 8x 2 <0, 30) ẋ = 1, ẏ = x Finite singular points: There are no singular points Infinite singular points: η = 0, M = 0, C 2 = x 3 0, µ 0 = µ 1 = µ 2 = µ 3 = 0, µ 4 = x 4 > 0, K 3 = 0 As all the classes from Table 2 are examined, Theorem 4.2 is proved. 28