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Quadratic vector fields in the plane have a finite number of limit cycles

Quadratic vector fields in the plane have a finite number of limit cycles QUADRATIC VECTOR FIELDS IN THE PLANE HAVE A FINITE NUMBER OF LIMIT CYCLES by Ronmoo BAMON INTRODUCTION An isolated periodic orbit of a vector field in R z is called a limit cycle. Part of Hilbert's x6-th problem is to find an upper bound for the number of limit cycles of polynomial vector fields of a given degree. Still today, very little is known about these upper bounds. Moreover it is not known if an arbitrary polynomial vector field has a finite number of limit cycles. In x923, Dulac [D] claimed that all graphs (see definitions in Chapter I) of analytic vector fields in the plane arefinite (i.e. they are not accumulated by limit cycles). From this result follows the finiteness of limit cycles for polynomial vector fields. Recently, II'y~enko [I] gave a strikingly simple counterexample to one of Dulac's main assertions, and gave a correct proof for the fact that all hyperbolic graphs (see Chapter x) of analytic vector fields are finite. This represents a major step and is essential for the result in this paper. Around I956 Petrovskii and Landis [P-Lt, P-L2] claimed that quadratic vector fields in the plane have at most 3 limit cycles. In I959 they withdrew their proof [P-Ls]. Later in I979, the chinese mathematician Shi Song Ling [Shi, Sh~] produced examples of quadratic vector fields with 4 limit cycles, disproving the estimate of Petrovskii and Landis. For our work we start from the fact that a polynomial vector field with infinitely many limit cycles must contain a graph (bounded or unbounded, see Chapter i) which is accumulated by limit cycles. This follows from the Poincar&Bendixon Theorem. Taking into account very special qualitative properties of quadratic systems, all of which are recalled in Chapter I, we prove the following theorem The author acknowledges the very kind hospitality and a fine mathematical atmosphere provided by IMPA/CNPq during the preparation of this paper. This work was partially supported by CNPq (Brasil), CONICYT (Chile) and PNUD-UNESCO. 111 I12 RODRIGO BAM6N Theorem B. -- All graphs (bounded or unbounded) of quadratic vector fields in R ~ are finite. From the fact we mentioned above follows immediately Theorem A. -- Every quadratic vector field in R e has a finite number of limit cycles. In I983, Chicone and Shafer [Ch-S] proved the finiteness of bounded graphs. Here we give an alternative proof. In Chapter i we give definitions and general information. We also recall pro- perties of quadratic systems. In Chapter 2 we prove Theorem B. We will denote by Z e the space of quadratic vector fields endowed with the topology of the coefficients. Acknowledgements. -- I am indebted to a number of people for helping me in several ways to achieve the results in this paper, among them A. Lins Neto, R. Moussu, J. PalLs, J. Sotomayor. I also acknowledge N. Yus for his earlier stimulus in the area of Dyna- mical Systems. x. Pre *hmlnarles I.I. General definitions Let X be a differentiable vector field in R e. Definition I. -- An orbit q~(t) = (x(t),y(t)) of X is called a separatrix of X if its t0-1imit set (or its a-limit set) is a singular point p = (xo,yo) of X and ~im x(t)-x~ (or lim x(t)--~o ) x) y(t) --Yo t~'-~Y(t) ~ exists and belongs to R w {+ oo}; 2) there exists r >~ and T>o (T<o) with 19(t)--P[<e 1 for all t>t T (t~< T) such that for all e>o there exist an orbit d/(t) of X and T>T (T<T) such that ]~b(t) -- q~(t) l < ~ for all t e [o, T] (t ~ IT, o]) and ]+(~) -- p] > r Definition 2. -- A closed curve F in R e is called a graph of X if it can be parametrized by a: [o, i] ~R * of class C 1 with ct(o) = g(I) satisfying: x) if a'(t) = o then X(ct(t)) = o; 2) if ~'(t) ~: o then a(t) belongs to a separatrix of X and there exists ?~ > o such that We will say that a graph I" of X has a return map if for all cross sections Y~ of X intersecting F, there exists p e Z such that to(p) n Y. :~ q) or ~(p) n Y. ~: ~. I . 2. Poincart' s compactification Let S 2 ---- { (x,y, z) ~ R 3 I xe + ye + z e = I } and let H + and H- be the hemispheres { (x, y, z) e S" ] z > o] and { (x,y, z) e S e [ z < o }, respectively. 112 QUADRATIC VECTOR FIELDS xx 3 To a differentiable vector field X in R 2 we can associate vector fields X + and X- in H + and H- in the following way: first we define in {(x,y, z)[ z = i } the vector field X(x,y, z) ---- (X(x,y), o) and then we project it by central projection onto H + and H-. When X is a polynomial vector field of degree n, it happens that multiplying X + and X- by z"- a we obtain a vector field in H + u H- that can be continuously extended to S x={(x,y,z) ~S *[z=o} in such a way that the resulting vector field on S * is analytic. This is the so-called Poincarg compactification of X and is denoted by ~(X). This construction allows us to study the flow of a polynomial vector field far away from the origin. The equator S 1 of S ~ represents the " points at infinity " and it happens (by construction) that points at infinity remain at infinity under the flow of :~(X) (i.e. S x is invariant for t~(X)). A graph of X containing separatrices (and thus singularities) at infinity will be called an unbounded graph. To study a polynomial vector field at infinity we consider the following coordinates u 1 = y/x and v 1 = I/x v~ = Ily. Geometrically they can be represented as follows: ?)lt If X~(2 is given by {y=P(x,Y)=o~+mx+ny+ax2+bxy+cy ' (~) x : Q(x,y) = ~ + ~x + ~y + ~x, + ~xy + ~y, then in the coordinates ul, vt it is expressed in the form (2) X, : [ '~' = Pl(ul, vl) ~t = Q.,(ut, ~t) with PI(ua,vl) =d+ (b--a) u I +Nv I+ (~--b) u~+ (ff--m) u tv 1 Qx(ut, vx) = -- avt -- but v x - m~ - cu~ v, - nut ~ - e~ 15 it4 RODRIGO BAMON and in the coordinates u2, v s it is expressed in the form u2 = Ps(us, vs) (~) xs : is = Q2(us, v,) with Ps(u2, vz) =c+ (b--c) u s +nv 2 + (a--b) u~+ (m--g) u sv s Lemma x.x. ~ Every X ~ X z has one, two or three pairs of symmetric singularities at infinity or the whole infinity is filled up with singularities. Proof. -- If X e X s is given by (i), using the expressions of X 1 and X z we have that the singularities at infinity are obtained by solving Px(u,,o) =a + (-b--a) u x + (i--b) u]--cuat = o and Ps(us, o) = c + (b -- ~) u s + (a -- b) u~ -- ~u~ ----- o. From these expressions the lemma follows immediately. Note that u x 4: o and u s = u~-t represent the same point at infinity. [] Remark. ~ By a rotation of coordinates we can always carry one of the pairs of symmetric singularities at infinity to the pair of points p : (o, l,o) and -- p = (o, -- I, o). Hence we can always suppose that the origin in the (us, vs)-plane is a singularity of X s or, in other words, we can always suppose c --- o. The following corollary is now clear. Corollary x.2. -- Let X e x s be given by (x) with c = o. Then: I) X has one pair of symmetric singularities at infinity if and only if [(5 -- a) ~ -- 4a(c -- b) < o] or [b--a =~--b =o and ~4: o]; II) x has two pairs of symmetric singularities at infinity if and only if [(b-- a) s-4n(c- b) ---- o and c-- b4: o] or [g-- b =o and b -- a 4= o]; III) X has three pairs of symmetric singularities at infinity if and on(~ if (b -- a) ~ -- 4d(i -- b) > o and i -- b = o; IV) the whole infinity is filled with singularities of X if and only if ~ -- b = b -- a = ~ = o. The proof of Theorem B in the next chapter will consider separately each one of the cases arising from this corollary. I. 3. General properties of quadratic vector field in R s Among the properties of planar quadratic vector fields there is a simple but basic one that we will use throughout the paper. It will be refered to as the periodic orbit property and says the following: 114 QUADRATIC VECTOR FIELDS ss5 If a planar quadratic vector field has a periodic orbit Y, then in the compact region bounded by Y there is a unique singularity. Moreover, the linear part of the vector field at this singularity has conjugate complex eigenvalues. This property can be used to find simple expressions for the vector field. In fact, if a quadratic vector field X has a periodic orbit, it is clear that there exists an afine change of coordinates such that X: {y=mX--y+P2(x,y) = x _ my + Q2(x,y), where P~ and Q.2 are homogeneous polynomial vector fields of degree two. In some cases we will use the fact that this form of X is invariant under rotations. Let us recall another basic property of planar quadratic vector fields. Contact property. -- Let X s )~2 be a quadratic vector field in R ~. Then every straight line t in R ~ is either invariant or has at most two contacts with X (i.e. points where t is not transversal to X). This property enables us to find geometric properties for bounded or unbounded graphs of quadratic systems. For example we can prove: (i) all graphs with return map enclose a convex region; (ii) a graph with return map and with at least two singularities must contain the straight line segment joining two adjacent singularities. In this case the quadratic vector field has an invariant line. With these properties we obtain simple expressions for the vector field. In fact, if a quadratic vector field X has an invariant line, then there exists an afine change of coordinates such that {~x(m+ax+by) x: = Q(x,y) and if X has two transversal invariant lines then {y~X(m-k-ax+by) x : + + Whenever necessary we will use these forms of X. Let us recall two more properties of quadratic systems. Invariant line property. ~ If a quadratic system has an invariant line then it has at most one limit cycle. Two invariant lines property. ~ The quadratic system k ---- x(m -~- ax q- by) .} -----y(K -k bx "k- cy) has no limit cycle. 115 xl6 RODRIGO BAM6N Although the invariant line property may substantially simplify the proofs, we are not going to use it, because we do not know of a good reference for its proof. However, we shall make use of the two invariant lines property, first proved by Bautin (see [C]). I .4. Singularities and periodic orbits Let p be a singular point of a vector field X in R * (i.e. X(p) = o) and let )'1, )', be the eigenvalues of DX(p). We say that p is hyperbolic if Re ),i ue o, i = z, 2, that it is semi-hyperbolic if X1 X2 ---- o but )~1 + )'~ + o and that it is a degenerate singularity if k 1 ---- ),, = o. Moreover, we say thatp is a center type singularity ifX 1 and ),~ are complex conjugate with zero real part. For the moment we are interested in the description of the flow of X in a neigh~ borhood of p (the topological type of p). First we define some types of singularities by their geometrical features: \t/ Jr< %1{ repellor saddle-node saddle attractor If the orbits of an attractor (repellor) spiral around the singularity we speak of a focus; if not we speak of a node. Let us now relate hyperbolicity with the above topological types. Ifp is a hyperbolic singularity of X and X1, )', are the eigenvalues of DX(p), then p is a saddle if )~1 )'2 < o, an attractor if Re ~ < o, i = I, 2, and a repellor if Re )~ > o, i = I, 2. Moreover, p is a focus if and only if X1 and ),, are complex conjugate numbers. These are simple basic facts about dynamical systems and can be seen for example in [P-M]. Ifp is a semi-hyperbolic singularity of X there are two local invariant differentiable curves intersecting transversally at p, such that the behavior of X along these curves determines the topological type of p. The tangent lines to these curves at p are the lines generated by the eigenvectors of DX(p). The invariant curve whose tangent line at p is generated by the eigenvector associated with the vanishing eigenvalue is called the center manifold ofp. The flow of X on the center manifold ofp is given by the first nonzero derivative fCk/(o) of an associate one-dimensional differential equation ~? =f(x) for which f(o) =f'(o) = o. The flow on the other invariant curve is determined by the sign of ),, the nonzero eigenvalue. With this, the topological type ofp is a saddle- node if k is even, a saddle if k is odd and )~.f(k)(o) < o and a node if k is odd and k./(k)(o) > o (attractor if ~. < o and repellor if ), > o). The center manifold theory needed here can be found in [H-P-S] or [Ca]. 116 QUADRATIC VECTOR FIELDS II7 The topological type in the degenerate case can be obtained by means of the blowing-up method. This method consists in " opening " (blowing-up) the singularity into a circle using for example the map ~): R X S 1 -+R 2 (r, 0) ~ (r cos 0, r sin 0). (We suppose that the vector field X is defined in an open set ofR z and has the degenerate singularity p at the origin.) An important fact is that there exists a vector field X in R x S x (the blow-up of X) verifying Dq~cr, 0)(X(r , 0)) = X(q~(r, 0)) for (r, 0) ~ U and leaving { o } � S 1 invariant. The set U is open in R � S 1 and contains { o } � S t. If we know the flow of .X in a neighborhood of { o ) � S x (for example if all the singu- larities of X in {o} � S 1 are hyperbolic or semi-hyperbolic), then " blowing down " X we obtain the topological type ofp. IfX has degenerate singularities along {o} � S 1, we blow-up again each one of these singularities and observe if we can determine the corresponding flows. If not, we blow-up again and again. Fortunately this process ends; in fact (X being analytic) we know that after a finite number of blowings-up we only get hyperbolic and semi-hyperbolic singularities. This allows us to describe the topological type of p. In sections 2. i. i, 2.2 (1.2) and 2.2 (II.2), we will give the topological types of all degenerate singularities which will be needed. The blowing-up method can be seen in detail in [A], [Du] and [T]. Finally, we recall that the topological type of a center type singularity p of an analytic vector field is either a focus or a center (all orbits in a neighborhood of p are periodic). A periodic orbit y is called an attractor (repellor) if it is the o-limit set (,t-limit set) of all points in a neighborhood of y. Let X be a vector field in R ~ and let ? be a periodic orbit of X of period T. The number c = div X(y(t)) dt is called the characteristic exponent of y. It is a well known fact (see [A], [S]) that for c > o the orbit "~ is a repellor and for c< o it is an attractor. I. 5. ll'yMenko's Theorem ([I]) Definition. -- A graph I ~ of a vector field X in R ~ is called a hyperbolic graph of X if all singularities of X contained in 1 ~ are hyperbolic. Theorem (II'ya~enko). -- Every hyperbolic graph of an analytic vector field in R 2 is finite (i.e. not accumulated by limit cycles). 117 x~8 RODRIGO BAMON This theorem is crucial for our result because it allows us to consider only the non-hyperbolic graphs. i. 6. Dulac' s Proposition Definition. -- A semi-hyperbolic singularity p of a vector field X in R 2 is called contractive if div X(p)< o and expansive if div X(p)> o. Notice that in this case div X(p) is the nonzero eigenvalue of DX(p). Proposition (Dulac [D]). -- Every graph of an analytic vector field in R ~ which contains only hyperbolic or contractive (expansive) semi-hyperbolic singularities is finite. The proof of this fact is straightforward. 2. Proof of Theorem B 2.z. We will first prove that all bounded graphs of quadratic vector fields are finite. To do this we observe that a quadratic vector field has at most four singularities in the plane. Since we are interested in periodic orbits we may suppose that one of the singularities is a focus or a center (see I. 3 and 1.4). Hence bounded graphs contain one, two or three singularities (this is true for every quadratic vector field; see Berlinskii's Theorem in [C]). 2.x.x. Let us first consider graphs with one singularity. If the singularity is either hyperbolic or semi-hyperbolic the graph is finite. This follows from II'ya~enko's Theorem and Dulac's Proposition, respectively. Suppose now that the singularity is at the origin and that both eigenvalues are zero. If the linear part of the vector field at (o, o) is identically zero then the vector field is homogeneous and there is no limit cycle. We may then suppose that after a linear change of coordinates the vector field has the form i k =y + ax ~ + bxy + cy z Following the blowing-up method we observe that if ~ # o the topological type of the origin is ~>o /~<o Since the line y = o is transversal except in (o, o) this singularity does not belong to any graph. Thus, necessarily ff ---- o. 118 QUADRATIC VECTOR FIELDS If9 In this case (~ ---- o) the line y ---- o is invariant and the existence of a periodic orbit implies both b 4= o and the existence of another singularity which must be a focus or a center. Changing coordinates by (x,y) ~-~ ([~x + ~y, ~,y) with an appro- priate ~,, we obtain the following form for X: i fc=ny+ax 2 +bxy--ny 2 X: . y = xy with (o, o) and (o, x) as singularities. For (o, I) to be a focus or a center it is necessary that b ~<4 n. We also need a4= o. Suppose b = o. Then the vector field verifies A, X = -- X for A(x,y) = (--y, x). It follows that (o, i) is a center and that there is no limit cycle. Now denote by X b the vector field Xb:l; ~yg. Then, X b = X 0 +b (xy) and det (X0, Xb) = -- bx'yL It follows that the orbits of X b are topologically transverse to the ones of X 0 and since X 0 has a center and is symmetric with respect to the y-axis we see that Xb, b + o, does not have any periodic orbit. Moreover, calculating the topological type of (o, o) (by the blowing-up method) we conclude that X 0 has the following bounded graphs according to the values of the coefficient a. , @ a<o o<a<l/2 Also, the vector fields X 0 for a/> I/2 and X n for b 4= o do not have any bounded graph. This settles the case of bounded graphs with one singularity. 2.x.2. Let us now consider bounded graphs with two singularities. Since we are interested in graphs with return map we may suppose (by the contact property) that both singularities are points of an invariant line for the vector field. By a linear change of coordinates we can carry these points to (o, o) and (o, I). The vector field then takes the form i k~- x(m q- ax + by) X: !,~ ---- k-y(y -- i) + -mx + ax' + bxy Roe o. The eigenvalues of the linear part of X are {m, -- ~} at (o, o) and {m + b, ~} at (o, x). Since ~ 4 o each singularity is either hyperbolic or semi-hyperbolic. If =x.y.ny~axqt-[-bxy 120 RODRIGO BAMON at least one of them is hyperbolic then the graph is finite (by II'ya~enko's Theorem or by Dulac's Proposition). If both singularities are semi-hyperbolic then m = b = o. But in this case J = ax ~ and there is no bounded graph. 2. x.3. Finally, if there is a graph with three singularities and with return map then, by the contact property, there are three invariant lines and by the two invariant lines property we know that there is no limit cycle. This conclude the proof for bounded graphs. 2.2. Now we prove that all unbounded graphs are finite. We proceed by consi- dering separately each one of the relations in Corollary 1.2. 2.2(I.I) (b--a) 2-4~(~-b)< 0 Let X e Z ~ be given by (I) with c = o and verify the relation above. In this case p = (o, I, o) and -- p are the unique singularities at infinity. Since X is expressed in coordinates (u~, v~) (see x.2) as ~= (b--~) us + nv~ + ... X2 b2 = -- ~v~ + ... the point p is hyperbolic for X restricted to infinity (u,-axis) and it is hyperbolic if and only if ~4: o. Lemma 2. I. -- If X has an unbounded graph I', then: (i) p and -- p are saddles (and so b * o) and they belong to r. An arc at infinity joining p and --p must be contained in F. (ii) The line t : x = -- n/b is invariant and contained in I'. (iii) There exist coordinates in which we can write i ic = xy with b 2 -- 4a(C- I) < O. X: t.Y = Q.(x,y) Proof. -- Part (i) is clear. To prove (ii) we notice that (X(-- n/b,y), (I, o)) = o~ -- mn/b q- an~/b ~ proving that ,e is invariant or transversal to the flow. If it is transversal, the separatrices of the saddlesp and -- p must be on different sides of/(see the figure below). Therefore, there is no unbounded graph 120 QUADRATIC VECTOR FIELDS We now prove (iii). By translating coordinates we carry t to the line x = o. The vector field is now given by k = x(m + ax + by) t-} = Q(x,y). Since b + o (p is a saddle), we obtain the desired form for X by changing coordinates: (x,y) ~ (x, m + ax -t- by) [] Let X e ff be given by k = xy b~ -- 4ti(E- x) < o. x: t) = Q(x,y) Since in this case ,~,=,,,[I -- i-- b,, -- ~, -- i 4-~u,v2-~41 X2 : ~,- = Qa(u,-, v~) = - iv, + ... we conclude that the v~-axis is invariant and that the flow along this line is given by = Q,-(o, v,) = - - - Thus, the origin (u2, v2) = (o, o) (and hence p) is a saddle if and only if (I -- i) E> o (hyperbolic case) or C---- ~i =- o and ~ > o (non-hyperbolic ease). For X e ff as above we will prove the following scheme: : K S- 4~> o : there is no unbounded graph i IK+ o : graph as in Fig. x ~(I --E)>o ti,---4~E=ol i ~ + o : graph as in Fig. 2 I n=~ tb=o : graph as in Fig. 3 Kz 4~ < o : graph as in Fig. 4 =/i=o and i>o! ~ + o : graph as in Fig. 5 = o : graph as in Fig. 6 -,.,, ~*"~ Fro. L -- The graph does not have a return map and so it is finite Fro. 2. -- Calculating the characteristic exponent it follows that three is no periodic orbit 16 RODRIGO BAMON Flo. 3. -- There is syrrunetry, therefore the graph is finite Flo. 4. -- By II'ya~enko's theorem, the graph is finite Fro. 5. -- Same as in Fig. 2 lhG. 6. -- Same as in Fig. 3 Let us consider the case ~(1 -- c) > 0. We first look at the singularities of X on the invariant line t : x = o. These singularities are given by the roots of Q(o,y) = ~ + n) + ~y~ = o. If K S -- 4fi~ > 0, there are two singularities on t which are hyperbolic fi~r X restricted to t. In this case there is no unbounded graph. For K S -- 4fii = 0, there is a unique singularity on t, namely P0 --- (o, -- n/~i), which has eigenvalues -- K]2c and o. Thus, for ff + o, P0 is a saddle-node and we obtain the graph of Figure ]. For K := o, we necessarily have ~ = o. Notice that if there is a periodic orbit Y = (71, 7~), it must be contained in {x > o} or in {x < o}. Also, from the expression for X, we have 7~ = "~a/Yl. Calculating the characteristic exponent of a periodic orbit of period T we obtain: 122 QUADRATIC VECTOR FIELDS I23 div X(v(t)) dt ---- by,(t) dt + (2~ -11- I) v2(t) dt Y; f; f: : b Tl(t) dt + (2ff + I) ~'t(t) dt f; I; = -b v (t) dr. So, if b ~e o, all possible periodic orbits in the same half-plane as well as the singularities must be of the same type: all repellors or all attractors. Since this is impossible, there is no periodic orbit at all. Finally, for b = o, X has the form X= = xy = Nx + ~x ~ + c-y~ and we easily show that AoX=--X for A(x,y) = (x,--y). The flow of Xisthen given in Figure 3. When ~2 _ 4~< 0 there are no singularities on ! and we get the graph of Figure 4 with hyperbolic singularities. By II'ya~enko's Theorem we know that these graphs are finite. Let us now consider the case k-=n=o and ~>o. In this case there are no singularities on the invariant line r = o, and there are two singularities lying on different sides of t. Both of them are center-type singularities. If there is a periodic orbit "r = ('rl, T2) of period T we calculate its characteristic exponent and obtain the number b [: yl(t) dt. As before, we see that there is no periodic orbit if b ~e o. For -----o the vector field verifies A. X =- X for A: (x,y)~ (x,--y) and so we obtain the graph of Figure 6. This proves Theorem B in the case (I. i). 2.2(I.2) b--a:~--b:O and ~+- O. Let X e ~ be given by (I) with c : o and verify the relations above. In this case p---- (o, I, o) and --p are the unique singularities at infinity. Let X 2 be the expression of X in coordinates (u2, v~) (see 1.2). Since X2: /fie----nv~+ ... ( i~ 2 = -- bv~ + ..., the point p is not hyperbolic for X restricted to infinity (us-axis) and it is semi-hyperbolic if and only if b+ o. Lemma 2.2. -- Let b ~ o. If X has an unbounded graph F, then: (i) p and -- p are saddles and they belong to F. An arc at infinity joining p and -- p is contained in F. 123 x24 RODRIGO BAMON (ii) The line g : x = -- n/b is invariant and contained in F. (iii) There are coordinates in which =~+~x+~y+/ix 2+y2. Proof. -- Since p is semi-hyperbolic and is a node along the center manifold (the infinite line), it is either a node or a saddle. If it is a node, no unbounded graph is possible. The rest of the proof goes as for Lemma 2.I. [] Lemma 2.3. -- Let b = o. If X has a periodic orbit then there are coordinates in which {~=mx--y+ax' X: r 4: o. = x + my + ~x 2 + ayx Proof. -- We first note that the conditions b--a=o, b=i=o and a-#:o are invariant under affine change of coordinates that keep p = (o, I, o) fixed. The lemma then follows by the periodic orbit property. [] Notice that we are only interested in quadratic vector fields which have periodic orbits. Therefore, we will frequently use the coordinates given by Lemma 2-3. If a~e o, using (x,y)~(~tx,~2y), r x, i= 1,2 if necessary, we can suppose ff>o and a>o. Lemma 2.4. ~ Let X e ff be given by {;=mx--y+ax' ~>o = x + my + ~x z + axy a > o. Then the singularity p = (o, I, o) at infinity has the following topological type: Proof. -- By blowing-up the singularity at the origin for the vector field / ~ = - v~ - ~u~ - u~ v, X, b, = -- au z v 2 - m~ - ~u~ v, - u 2 ~, we recognize the above topological type. [] 124 QUADRATIC VECTOR FIELDS t25 We will now prove the following scheme: fib > o " there is no unbounded graph b4=o! I I ff 4= o : graph as in Fig. 7 ab < o (coordinates as in Lemma ~. ~) I r~ = o : graph as in Fig, 8 ' i m = o : X is Hamiltonian ia~o b o (coordinates ~ m 4= o : since div X(x,y) = 2m, there is no ----- ~ ' periodic orbit as inLemma z.3) la> i2am--~=~ , 2am -- ~ ~= o : there is no unbounded graph. Fro. 7.- The characteristic exponent is nonzero, therefore there is no periodic orbit FIo. 8. -- There i~ ~yrametry, hence graphs are finite Fro. 9-- There is a Liapunov function imide the ~aph. Thu.,, there is no periodic orbit Let us consider the different cases: When b 4= o and ~b -'> o, p is a node and therefore there is no unbounded graph. For b + o and db < o, p is a saddle. Let us consider coordinates as in Lemma 2.2. If there exists a periodic orbit of period T, we calculate its characteristic exponent obtai- ning ~T. Thus, as before, there is no periodic orbit if ~ + o. When ff = o, the vector field is symmetric with respect to (x,y) ~ (x, --y), and 1here are no limit cycle~. I25 x26 RODRIGO BAM6N We now suppose b = o. Take coordinates as in Lemma s.3 with ~> o and a> o. Let y =y(x) be the parabola y =y(x) = (a/s) x ~ -- mx -- (x + m2)/2a and let ~r~(x) be its normal vector ~(x) = (ax- m,- x). Easy calculations show that (X(x,y(x)), ~s(x) ) = (sam -- ~) x 2. Fix a > o, a > o and consider m as a parameter. Let m0 be given by sam o -- ~ = o and let y =y0(x) be the parabola y =yo(x) = (a/2) x 2 -- moX -- (x + m~)/sa. For m = m o the parabola y =yo(x) is invariant under the flow of X. Let b(x) = -- m ox - (x + m~)/a. By straightforward calculation we obtain: (i) b(x)<yo(x) for all x~ll. (ii) The function f(x,y) = (y --yo(X))/(y -- b(x)) 2 has the origin as a maximum and in the region f~ = {(x,y)/y>yo(x)} this is the only critical point. That is, f has the following level curves in f~: I.- (e7 p +Of I y--y.(x) ~< o for each (x,y) el2 and (iii) Xf(x,y)= ~x Ox O~](x,y) = -- s~ (Y _ b(x))So x+o. In this way, if 2am -- ~ = o, the origin is a repellor and there are no periodic orbits. For m 4: m o (i.e. 2am --d 4: o), the parabola y =y0(x) is transversal to X. Since X =X,,= X,,.+ (m--mo) R where X,, is the vector field in Lemma 2. 3 and R(x,y) = (x,y) is the radial vector field, we observe that the separatrices at p move to different sides of y =y0(x) when m changes. Thus there is no unbounded graph when m 4: m0. The proof of Theorem B in case (I) is complete. 2.2(II.z) (/~--a) ~ - 4~(?-. b) = 0 and ~-- b:t= O. In this case there are two pairs of singularities at infinity. The one different from {p = (o, I, o), --p} is a pair of saddle-nodes for X restricted to infinity. This is clear from the equation of X restricted to infinity: ~l=a + (b--a) ul + (~-- b) u, ~. lg6 QUADRATIC VECTOR FIELDS x27 By rotating coordinates we can carry this pair of singularities to {p, -- p }, leading us to the next case: 2.2(II.2) b--g=0 and b--a+- O. Besides {p, --p} there is another pair of singularities at infinity. When we res- trict X to infinity, p and --p are non-hyperbolic while the other pair is hyperbolic. As before, let X, be the expression of X in coordinates (u2, v2). The linear part ~ at the ~ is (: --b)" So p is semi-hyperbolic if and only if b+-o. If b4=o and X has an unbounded graph F, then F must contain two adjacent singularities at infinity and the corresponding arc between them. Also, for b + o, if X has an unbounded graph without singularities in the plane, it must be of one of the two following types: By Dulac's Proposition these graphs are finite. Now, if b 4= o and X has an unbounded graph which contains singularities in the plane and which has a return map, then by the contact property it is proved that the separatrices of the graph are contained in invariant lines. By changing coordinates we put these invariant lines in the axes and so the vector field takes the form l; =x(m-k-ax+by)_ =y(ff + bx § by). By the two invariant lines property this graph is finite. Suppose now b = o. Lemma 2.5. -- If X has a periodic orbit then there exist coordinates in which i i~ = mx --y + ax 2 a -- b :> o X: l.~ : x + my + ~x~ -+- 3xy ff >>. o. Proof. --- The same as in Lemma 2.3 and the remark following it. [] Lemma 2.6. -- Let X be given as in the lemma above. Then the topological type of the singularity p -~ (o, I, o) at infinity is one of the following: (i) , ~ /fo<h< a; 127 RODRIGO BAMON t~8 (ii) 9 __~__~_ /f2a<b< a< o; (iii) (iv) if b = o then the topological types are as in (i), (ii) or as in the following figures I r Proof. -- As for Lemma 2.4. [] From the possible topological types for p, we obtain that unbounded graphs may have i or 3 singularities at infinity. For example the following graphs can exist: Unbounded graphs with two singularities at infinity cannot exist because graphs with a return map must enclose a convex region. Suppose that X e X * as in Lemma 2.5 has a graph with a return map and with three singularities at infinity. Then, one of them is p (or --p) and the others are the adjacent ones which are themselves symmetric. This pair of symmetric singularities are in the direction y/x ---~[(a- b). Since they are contained in a graph they must be saddles and, by the contact property, the separatrix in the plane must be an invariant straight line g of the form y=y(x) = _x+N. a--b From the equation ~2 (X(x,y(x)), (~, b -- a) ) = d(a -- b) x' + (bN(b -- a) _+~-a) x a--b --~N+mN(b--a) =o, where (d,b--a) is avector normal to?,it follows that d=m=o and N= -- z/b. 128 QUADRATIC VECTOR FIELDS x29 Thus, X can be expressed in the form {;=--y+ax' X: = x + bxy. Since this vector field has the symmetry A. X = -- X for A(x,y) = (-- x,y), we see that the origin is a center, and so there is no limit cycle. We will now prove the assertions in the following scheme for the case b = o. --bm--~----o/~=~ : graph as in Fig. IO 3am h-> o : graph as in Fig. I I o<b<a 3am -- bm -- h 4= o : there is no unbounded graph ~< 2a, b < o : there is no unbounded graph with return map {~ = o : graph as in Fig. 12 I 3am -- bm -- ~ = o > o : graph as in Fig. x 3 2a< b< a< o [ 3am -- bm -- ~ 4: o 9 there is no unbounded graph (~ = o : graph as in Fig. 14 3am -- ~ = o > o : there is no unbounded graph b=o 3am- ~ 4= o : there is no unbounded graph. FIo. xo. -- There is a first integral inside the graph, therefore it is finite Fxo. i x. -- There is a Liapunov function inside the graph. Thin, there are no limit cycle$ Flo. 12. -- Same as in Fig. lo 17 13o RODRIGO BAMON FIo. 13. -- Same as in Fig. xx Fro. t 4. -- The graph is symmetric, therefore it is fimte To prove the above assertions, consider the parabola 2a -- b I + m ~ y=y(x) = -- x 9 - mx 2 2a and its normal vector ~N(x) = ((2a- b)x- m,- x). Easy calculations give (X(x,y(x)), ~s(x) ) = (3am -- bm -- ~) x'. Let m o be given by the relation 3amo--bmo--a=o and let y =y0(x) be the parabola 2a- x y=Yo(X) -- -- x ~-m ox 2 2a Fix a, -b and S satisfying O < b < a and ~ >>. O. Notice that m o = o for ~i ---- o. Let us consider m as a parameter. If m = m 0 the parabola y =y0(x) is invariant and forms an unbounded graph. Take r=b/a and let b(x) = -- moX-- (i -bm~)/b. Then: (i) b(x) <yo(X) for all x ~R. (ii) The function f(x,y)= (y--yo(X))'/(y- b(x)) ~ has the origin as a maximum and in the region f~ = {(x,y)/y >yo(x)} this is the only critical point. The level curves off in f~ are /~.~ = yoCx) 2, 130 QUADRATIC VECTOR FIELDS (Of P + Of Q) (x,Y) = -- 4(2a -- b) m o (y -- Y~ xZ< o (iii) Xf(x, y) = Ox (y -- b(x) ) s for all (x,y) ef~ with x4= o. Therefore, for ~ = o the origin is a center (Fig. 1o) and if ~ > o (so that m 0 # o) there are no periodic orbits inside the graph (Fig. IX). For m# m 0 (i.e. 3am--brn--ff# o), we have as before that the relation: X=X,,=X,,,+ (m--m0) R is satisfied, and so there is no unbounded graph. This ends the proof in case o < b < a. Now fix a, b and ~ satisfying 2a < b< a< 0 and ~ >10. Recall that if ~ = o then m 0 = o. Let us again consider m as a parameter. First let m = m 0. In this case the parabola y =y0(x) is invariant and forms an unbounded graph. Take r and b(x) as before. Then: (i) y0(x)< b(x) for all xzR. (ii) The function f(x,y)= (yo(x)--y)'l(b(x)-y)~ has the origin as a maximum and in the region ~ -----{(x,y)/y <yo(x)} this is the only critical point. The level curves off in f~ are (y0(x) -y)' < o (iii) Xf(x,y) = 4(2a -- [~) rao (b(x) -- y)' for all (x,y) ef~ with x# o. Therefore, if ~ = o there is a first integral and the origin is a center (Fig. 12) and if ~> o (so that m 0 4= o) there is a Liapunov function and there is no periodic orbit inside the graph (Fig. i3). To conclude the case above, we now let m 4: m 0 (i.e. 3am -- bm -- ~i 4= o). The same arguments as in the previous case prove that there is no unbounded graph. If b <~ 2a and -b < O, then from the topological type ofp we conclude that no unbounded graph is possible. Fix a, -b and ~ satisfying b=O< a and if> 0. Consider m as a parameter. If m = m 0 the parabola y =y0(x) is invariant and it is easily shown that there is a saddle on the parabola. Hence there is no unbounded graph. When m 4= m o the parabola y =y0(x) is transversal to X. Suppose that the topological type ofp is the one in (i) of Lemma 2.6. (This is the only possibility when b = o for the existence of an unbounded graph.) Since the separatrices at infinity bound hyperbolic sectors, the 131 RODRIGO BAM6N transversal parabola y =y0(x) must leave the separatrices at different sides. The following picture illustrates the situation: Thus, there is no unbounded graph. Finally fix a, b and ~ satisfying -b = 0 < a and h- = O, so that m 0 = o. When m = m 0 = o the vector field verifies A. X = -- X for A(x,y) = (-- x,y), the origin is a center and there are no limit cycles. If m # mo, as before there is no unbounded graph. Thus, all assertions concerning the case b = o are proved and case (II) is settled. 2.2 (III) (~ - a)" - 4a(~- b) > 0 and ~- b, O. We now come to the most difficult part of the proof of our main result. For X e Z 2 given by (I) (see Chapter i) with c = o and satisfying the relation above, there are three pairs of symmetric singularities at infinity. Lemma 2.7. -- If X ~ X ~ has three pairs of symmetric singularities at infinity and if two of them are hyperbolic, then X has a finite number of limit cycles. Proof. ~ When a quadratic vector field X has three pairs of symmetric singularities at infinity, all of them are hyperbolic for the restriction of X to infinity (see Lemma i. I). So the only possible unbounded graphs with a return map are of the following types: (double arrows indicate hyperbolicity) The first one is finite by II'ya]enko's Theorem. The middle one is not accumulated by periodic orbits (Dulac's Proposition). The last one, with singularities in the plane, must have separatrices contained in invariant lines, and as explained before, in this case there are no limit cycles. [] 132 QUADRATIC VECTOR FIELDS I33 Lemma 2.8. -- If X E Z ~ has three pairs of symmetric singularities at infinity and two of them are not hyperbolic, then: (i) the two pairs of non,hyperbolic singularities are semi-hyperbolic; (ii) the third pair of symmetric singularities consists of hyperbolic nodes; (iii) there exist coordinates in which the hyperbolic pair of singularities is {, = ,12 i, ,, o), - r} and the semi-hyperbolic pairs of singularities are {p = (o, x,o);--p} and {q= (x, o, o),--q}. In these coordinates the vector field has an expression as in (x) with a ----- o, c = o, ~ = o, ~=o, b+b=o and b#o. Proof. -- We first observe that given any order in the pairs of singularities, there are coordinates in R 2 such that the first pair is {p, --p}, the second pair is {q, -- q} and the third one is { r, -- r}. In fact, with a rotation of coordinates we carry the first pair to {p, -- p}; with a linear change of coordinates of the form A(x,y) = (x, ~x -k-y) (which fixes p) the second pair is taken to {q, -- q}; and finally with a change of coor- dinates (x,y) ~ (x, Xy), X # o, the third one is taken to {r, -- r}. If X is expressed as in (I), then in the coordinates above the following relations are true c:o, d=o, ~--b+a--b=o and b--gar o (this follows from (2) and (3) in I. 2, by imposing the conditions P~(o, o) = Pl(o, o) = PI(-- x, o) = o). Moreover we have the following table singularities eigenvalues p = (o, o) --~, b--~# o q= (~, o, o) --a,b--a=g--b# o r= I/2 Vr2(- I, I, O) -b--c,c--b # o. If we suppose that p and q are not hyperbolic then a = r = o and b = -- b # o. The lemma now follows directly. [] By the two lemmas above we can restrict ourselves to quadratic vector fields X with expression {;=o~+mx-Fnyq-bxy X: b#o. =-~ + Tnx + ~y--bxy Moreover, by translating the coordinates we can suppose n = ~ = o. Also, if necessary, the change of coordinates (x,y) ~ (y, x) makes b > o. 133 RODRIGO BAMON z34 Lemma 2.9. ~ Let X ~ x 2 be given by {;=oc+mx+bxy X: = ~t + ~y -- bxy with b > o. Then: (i) p = (o, x, o) and q = (o, -- I, o) are semi-hyperbolic singularities of X at infinity (hyperbolic for the restriction of X to infinity) and r = i/2 ~/~(-- I, x, o) is a hyperbolic node; (ii) if X has an unbounded graph then it must contain p and q (or -- p and -- q) and the cor- responding arc between them; moreover p and q (-- p and -- q) must be saddles or saddles- nodes; (iii) if X has an unbounded graph that contains singularities in the plane and has a return map, then the separatrices are contained in invariant lines. In this case the vector field does not have limit cycles. Proof. -- Parts (i) and (ii) are clear from Lemmas 2.7 and 2.8. If X satisfies the hypothesis in (iii) then by the contact property it follows that there exist two invariant lines that must contain the separatrices. We know that in this case there is no limit cycle. [] In what follows we will consider X e )~* to be given by {y=OC+mx+bxy X: = ~ + ~y -- bxy with b > o, and we will study, in terms of the coefficients, when X can have unbounded graphs without singularities in the plane. We will prove the following assertions: (i) if ~0c = o there is no graph without singularities in the plane; (ii) if either ~<o, e>o or mE>o there is no unbounded graph. Notice that when m > o and ~< o, we can change coordinates (x,y) ~ (--y, -- x) so that we may suppose m< o and K>o. Let ~>o, m~< o, 0c< o and ~>1 o. Then: (iii) if 0c+~ =m+ff=o the only possible graph is as follows (iv) if (e+~)(m +~) =o but ~+~+m +n~e o, there is no unbounded graph; (v) if e + ~ + o and m -t- n ~e o, then any graph without singularities in the plane is finite. 134 QUADRATIC VECTOR FIELDS t35 To prove (i) suppose 8 = o. The line y = 0 is invariant and contains the center manifold of q. So, there is no graph, as required. The same happens if ~ = o. To prove the other assertions let us consider X expressed in the coordinates at infinity: Xt v t =--bu tv,-m~-~, Xi ~ = bu. v~ -- ~ -- ~. In both systems, the origin is a semi-hyperbolic singularity. The center manifold for X t has the form ,,~ = h,(Vl) = (~./b) ~, + 0(4) and the flow along it is given by i t = -- m~ -- (~ + 7,) ~, + 0(~,~). Similarly, the center manifold for X~ has the form ~, = ~(~,) = - (s/b) ~ + 0(~) and the flow is given by ~ = - ~4 - (~ + ~) d + o(~). We can now prove (ii). Suppose if< o. Since b> o, the center manifold of X t is locally contained in the half plane u t ~< o. That is: center manifold "~....._..1 ~ center manifold From the contact property it follows that all graphs with return map must enclose a convex region. On the other hand, by Lemma 2.9 any such graph must contain the adjacent singularities p and q (or --p and -- q). But this is impossible because of the location of the center manifold (see figure above). The same happens when ,t > o. 138 I36 RODRIGO BAMON If m~ > o then p or q is a node, and thus there is no unbounded graph. Now suppose ~ > o, m <~ o, ot < o and fi >i o. Notice that from the expressions for the center manifolds of p and q we have the following situations: Q) (double arrows indicate hyperbolicity) Besides proving (iii) to (v) we will see that these are the only cases where we can have unbounded graphs without singularities in the plane. In fact, consider the hyperbola y =y(x)=- ~/(bx) and its normal vector ~s(x) = (--cx, bx~). Easy calculations show that (X(x,y(x)), ~(x)) = x(b(o, + ~) x -- ~(m + ~)). To prove (iii), observe that, since e -{- ~ = m q- fi = o, the hyperbola is invariant, the vector field verifies A. X = -- X for A(x,y) = (y, x) and we obtain the graph indicated above. To prove (iv), i.e. when (~-+-~)(m+~) =o but ~-b~+mq-fi:~ o, notice that the hyperbola y =y(x) is transversal to X and no unbounded graph can exist (the separatrices ofp and q must be on different sides of y =y(x)). Let us now prove (v). We suppose ~q-~+ o and m-Fh-~e o. We recall that ~>o, m~< o, o~<o and fi>_-o. The relations m=o and 0cq-~>o or n-= o and 0c + ~ < o imply respectively that q or p are nodes (see the expres- sions for the center manifolds), and thus in these cases there are no unbounded graphs. Now we arrive at the hardest part of the proof of Theorem B. There are three casesto consider: mK#o; m=o, fi>o, 0t-k-~< o; and m<o, K=o, ~+~>o. As shown in the figures above in the three cases there can exist an unbounded graph without singularities in the plane. We will prove now that if such a graph exists, then it is finite. For that purpose we will analyse return maps (Poincard maps) associated to these graphs and show that these maps have isolated fixed points. To help clarify our arguments let us consider the following figure and dia- grams: 136 QUADRATIC VECTOR FIELDS t37 ~ X Y t ,.~l I_ "l 1~ ~,1.. 1 1 'b % 9 to "" '' 1" 1 Let us explain the notation. For o< r let Y~, = {(8,,,)/I,,I < ~}, Y~ = {(~, ~)/Izl < ~}, 21 ={(u, 8)llul < ~}, ~ = {(w, 8)/Iwl < ~}, n, = {(8, "~)I1", I < ~}, o3 ={(8, ~',)/1'~,1 < ~}, be transversal sections as indicated. Let Pl denote the change of coordinates from the (%, v,)-plane to the (ux, v,)-plane. Since vx = l /x and v, = I [y we have v2/v, = x]y = u I and so pt(u,, v,) = ( I ]u2, v2/u2). Moreover, O, (f~2) = { (I/8, vx)/1 vl [ < r }. 18 ~3 8 RODRIGO BAMON Let P~:0x(Dz) ~Dx; ~0:E 2~y~t and d/: Zt -+ Z* be the Poincar6 maps naturally defined by X. Let ta, s~, t~, s~ be local changes of coordinates given as follows. First take ua = ux -- h,(va) = ua -- (~]b) ~ -- 0(~) (i) t~ : __ V 1 ~ UI.. In these new coordinates, X x has the form u~ = -- b~a -- b~ + ~, ~, gt(a~, v~) v a ----- -- bff x b, -- mO~ -- (~ + ~) ~ + ~fx(~Tx). Now, to avoid the term -- bgx ~i in the component of the gx-axis we take sl: { u=~l = ~/(x + ~1), obtaining = -- bu -- bu ~ + uvg"x(u, v) xl/: = -- mv ~ -- (o~ + ~) v 3 + v4f(u, v) + uv" h'x(u, v). Now take = ~, - h,O,,) = ~ + (~,/b) ~ - o(~) (ii) t2 : _ /31 U 2 . In these coordinates X~ is expressed as u~ = bu2 ~ = b~ ~, - ~ - (~ + ~) ~7 + ~'A(~;). Again, to avoid the term b~a g~ in the component of the 17s-axis we take $~ : Z = v-~/(~ + U~) obtaining (0 = bw -q- bw 2 -q- wz~(w, z) Finally, let u = u(v) be the Poincar6 map from Y~+ ={(~, v) ~Zl/v> o} to ~1 defined by X1 and let z = z(w) be the Poincar6 map from ~.+ = {(w, 8) ~ ~Jw > o} to Z 2 defined by X2. To prove the finiteness of the graphs we will compare the Poincar~ maps above to other ones defined by auxiliary vector fields. To do this consider the following vector fields X 1 : and X~ : ~' z 2 138 QUADRATIC VECTOR FIELDS t39 with m'< o and K'>o. Let ~':Yt + -+~.l and ~':~+ ~X 2 be the Poincard maps associated to X't and X~ respectively. We will use the following expressions det(Xx, X'l) = uv*[b(m ' -- ra) + Fj.(u, v)], det(X,, X~) = wz'[b(~ -- n') + F,(w, z)], where Fa(o ,o) =o and F2(o ,o) =o, to compare the flows of XiandX~, i= i or2. Notice for example that if ]m] > I m'[ then det(X,, X[) > o for u> o, and so ~(v) < u(v) for small enough v > o. Let us first calculate u = ~'(v), z = ~'(w) and show that ?'(o) ~- I. We have l u = ~'(v) = ~e-o/t,: ~1 eb/Im'~l = k~ e b/C''o) (4) ~ ~ S z = = = x + (~' S/b) In S -- (~' S/b) In w k2 -- (~' ~lb) in w Now, to calculate ~'(o) we use the following lemma. Lemma 2. to. -- We have p~(o) = 8. Proof. -- We calculate O~(o) by the following formula (see [A]) p;(o) -- I Xl(I/8' ~ exp div Xx(T(t)) dt, ; x,(8, o) t The vector where 7(t) is the orbit of Xt that goes from (l/~, o) to (S, o) in time 1". field Xt, when restricted to v t = o, has the equation fil = -- bus -- buy. Integrating we obtain "f(t) = I + S -- e -hI' 0 9 From ~-(T) :8 we obtain T= -- I/blnS. Since div Xl(ul, O) ---- -- b -- 3bul we calculate ff be- bt div Xt(7(t)) dt = -- bT -- 3 x + ~ -- e -~ dt f =ln3--31n(t +~--e -~) ] ----- In M. Finally, since [Xt(8 , o)t ---- bS(i + 8) and ]X1(I/8 , o)] = b(i + 8)[M we obtain p~(o) = 8, proving the lemma. 139 ,4o RODRIGO BAM6N Remark. -- If ~t and ~s are cross sections for X 1 with ~t tangent to fl t at (8, o) and ~s tangent to Ot(f~2) at (I[~, o) and they are parametrized by the projection to the vl-axis, then the corresponding Poincar6 map ~s : ~s ~ fit satisfies ~s'(o) = p~(o) = a. From the above remark and from the fact that 0t(8, v2) ---- (I/8, v,/8) we finally obtain 9'(0) = I. We can now give the expression of our modified Poincar~ map. I.emma 2.t 3. ~ Let u = ~(o) and z = ~(w) be as in (4)- Let v = ~(z) = xz and w = ~(u) = ~tu. Then (~o T o ~ o'~) (u) = ku ~'ll-'lx where k is a positive real number. Proof. -- The formula is obtained by composing the maps. [] Let us now compare the return map of X with the modified return map and prove that if there exists a graph then it is finite (we will prove slightly more: they are not accumulated by periodic orbits). Suppose o< (h-/lrat)< I or else h= o. Chose m', h-', X and ~ such that: m' < o, I m'l < m, ~' > ~, x < x, (~'/1 m' I x) < I and o < V.u < ~b(u) for all u > o small enough. Then for ~, ~', ~ and ~" defined in (4) and in Lemma 2.13 we have since, for u > o, det(Xt, X~) > o near (o, o), ~(o) < uCv) 7(w) < z(w) since, for w > o, det(X2, X~) < o near (o, o), ~(u) < +(,,) for u > o small enough, for z > o small enough. T(~) < ,p(z) Thus (~o ~ o 7 o ~) (u) = ku ~'/''~'l ~ < (u o ~ o z o +) (~), and since (K'/[ m'] X) < x we conclude that u < (u o ~ o z o +) (u) for all u small enough. With this it is proved that if there is an unbounded graph then it is a repellor (i.e. it is the ~-limit set of some orbit). In the same way if (ff/[m D > I or m = o we prove that if there is an unbounded graph it is an attractor (i.e. it is the t0-1imit set of some orbit). This ends the proof of Case (III). 140 QUADRATIC VECTOR FIELDS t4 t a.2 (IV) ~-b=b--a=h=O. Let XeZ 2 be given by (I) (see Section I) with c=E--b=b--a=d=o. In this case, X ix transversal to infinity with the exception of two symmetric points. Since the relations above are invariant under any affine change of coordinates, the periodic orbit property allows us to restrict ourselves to vector fields of the form ; = mx -- y + ax ~ + bxy = x + my + axy + by 2. For these vector fields it is not hard to prove that with a rotatiol~ of coordinates we can make b = o. We then obtain (~ = mx --y + ax" = x +my + axy. With this last expression we see that the origin is the only singularity and therefore, there is a finite number of limit cycles. The proof of Theorem B is now complete. REFERENCES A. A~vRoNov et al., Od~alitative Theory of Second Order Dynamical Systems, John Wiley & Sore, New York, [A] W. Corrst, A survey of Quadratic Systems, Journal of Di.fferential Equations, 2 (1966), 293-3o4 9 [c] J. ~, Applications of Center Manifold Theory, Applied Math. Sciences, BS, Springer-Verlag, 1981. [Ca] C. Cmco~rs and S. Scan, Separatrix and Limit Cycles of Quadratic Systems avd Dulac's Theorem, [Ch-S] Transactwns Amer. Math. Sor 9.78 (t983) , 585-6x2. M. H. I)ULAC, Sur les cycles limltes, Bull. Soc. Math. France, 51 (1923), 45-188. [D] F. Douo~trmR, Singularities of Vector Fields, Journal of Differential Equations, 28, I (~977), 53 -t~ [Du] M. Hw.-~H, C. PuGs and M. Smm, Invariant Manifolds, Springer Lecture Notes in Math., 588 (i977). [H-P-S] [[] Yu. S. tl'ya.~enko, Limit cycles of polynomial vector fields with non degenerate singular points in the real plane (in Russian), Functional Analysis and its applications, 18 (3) (x984), 32-34 (ha translation : 18 (3) (x985), ,99-~o9)- I. G. P~rt~ovsxu and E. U. LA~DXS, On the number of limit cycles of the equation dy[d.~ = P(x,y)[Q.(x,y) [P-L l] where P and Q are polynomials of the second degree, Amer. Math. Soc. Transl. (2), t6 (t95fl), t77-22t. I. G. Pv taov~Ktz and E. ~'. LA.,~DIS, On the number of llmit cycles of the equation dy/dx = P(x,y)lQ,(x,y) [P-Ld where P and Q.. are polynomials, Amer. Math. Soo. Transl. (2), 14 (t96o), t8~-2oo. 1. G. P~Taovs~tt and E. U. LANDIS, Corrections to the articles : " On the number of limit cycles of the [P-L s] equation d3/d.x = P(x,.y)[Q.(x,y) where P and Q. are polynomials ", Math. Sb.N.S., 48 (9 o) (1959). a53-~55- 14/ RODRIGO BAMON ~42 j. PALLS and W. de MELO, Geometric Theory of Dynamical Systems; An Introduction, New York, Springer- [P-M] Verlag, x982. J. SOTOUAYOR, Curvas definidas por equa~oes diferenciais no plano, f3 o Col6quio Bros. de Mal., IMpA, [S] I~I, [Sh,] Sm SONOLINO, A concrete example of the existence of four limit cycles for plane quadratic systems, Sci., Sinica, Ser. A, 28 (x98o), z53-158. [$h~ Sm SONOLINO, A method for constructing cycles without contact around a weak focus, Journal of Differential Equa/~s, 41 (~98~), 3o~-3xa. Universidad de Chile Departamento de Matematicas Casilla 653 Santiago, Chile. Manuscrit refu le t6 septembre z985. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Publications mathématiques de l'IHÉS Springer Journals

Quadratic vector fields in the plane have a finite number of limit cycles

Publications mathématiques de l'IHÉS , Volume 64 (1) – Aug 31, 2007

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Springer Journals
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Copyright © 1986 by Publications Mathématiques de L’I.É.E.S.
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Mathematics; Mathematics, general; Algebra; Analysis; Geometry; Number Theory
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10.1007/BF02699193
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Abstract

QUADRATIC VECTOR FIELDS IN THE PLANE HAVE A FINITE NUMBER OF LIMIT CYCLES by Ronmoo BAMON INTRODUCTION An isolated periodic orbit of a vector field in R z is called a limit cycle. Part of Hilbert's x6-th problem is to find an upper bound for the number of limit cycles of polynomial vector fields of a given degree. Still today, very little is known about these upper bounds. Moreover it is not known if an arbitrary polynomial vector field has a finite number of limit cycles. In x923, Dulac [D] claimed that all graphs (see definitions in Chapter I) of analytic vector fields in the plane arefinite (i.e. they are not accumulated by limit cycles). From this result follows the finiteness of limit cycles for polynomial vector fields. Recently, II'y~enko [I] gave a strikingly simple counterexample to one of Dulac's main assertions, and gave a correct proof for the fact that all hyperbolic graphs (see Chapter x) of analytic vector fields are finite. This represents a major step and is essential for the result in this paper. Around I956 Petrovskii and Landis [P-Lt, P-L2] claimed that quadratic vector fields in the plane have at most 3 limit cycles. In I959 they withdrew their proof [P-Ls]. Later in I979, the chinese mathematician Shi Song Ling [Shi, Sh~] produced examples of quadratic vector fields with 4 limit cycles, disproving the estimate of Petrovskii and Landis. For our work we start from the fact that a polynomial vector field with infinitely many limit cycles must contain a graph (bounded or unbounded, see Chapter i) which is accumulated by limit cycles. This follows from the Poincar&Bendixon Theorem. Taking into account very special qualitative properties of quadratic systems, all of which are recalled in Chapter I, we prove the following theorem The author acknowledges the very kind hospitality and a fine mathematical atmosphere provided by IMPA/CNPq during the preparation of this paper. This work was partially supported by CNPq (Brasil), CONICYT (Chile) and PNUD-UNESCO. 111 I12 RODRIGO BAM6N Theorem B. -- All graphs (bounded or unbounded) of quadratic vector fields in R ~ are finite. From the fact we mentioned above follows immediately Theorem A. -- Every quadratic vector field in R e has a finite number of limit cycles. In I983, Chicone and Shafer [Ch-S] proved the finiteness of bounded graphs. Here we give an alternative proof. In Chapter i we give definitions and general information. We also recall pro- perties of quadratic systems. In Chapter 2 we prove Theorem B. We will denote by Z e the space of quadratic vector fields endowed with the topology of the coefficients. Acknowledgements. -- I am indebted to a number of people for helping me in several ways to achieve the results in this paper, among them A. Lins Neto, R. Moussu, J. PalLs, J. Sotomayor. I also acknowledge N. Yus for his earlier stimulus in the area of Dyna- mical Systems. x. Pre *hmlnarles I.I. General definitions Let X be a differentiable vector field in R e. Definition I. -- An orbit q~(t) = (x(t),y(t)) of X is called a separatrix of X if its t0-1imit set (or its a-limit set) is a singular point p = (xo,yo) of X and ~im x(t)-x~ (or lim x(t)--~o ) x) y(t) --Yo t~'-~Y(t) ~ exists and belongs to R w {+ oo}; 2) there exists r >~ and T>o (T<o) with 19(t)--P[<e 1 for all t>t T (t~< T) such that for all e>o there exist an orbit d/(t) of X and T>T (T<T) such that ]~b(t) -- q~(t) l < ~ for all t e [o, T] (t ~ IT, o]) and ]+(~) -- p] > r Definition 2. -- A closed curve F in R e is called a graph of X if it can be parametrized by a: [o, i] ~R * of class C 1 with ct(o) = g(I) satisfying: x) if a'(t) = o then X(ct(t)) = o; 2) if ~'(t) ~: o then a(t) belongs to a separatrix of X and there exists ?~ > o such that We will say that a graph I" of X has a return map if for all cross sections Y~ of X intersecting F, there exists p e Z such that to(p) n Y. :~ q) or ~(p) n Y. ~: ~. I . 2. Poincart' s compactification Let S 2 ---- { (x,y, z) ~ R 3 I xe + ye + z e = I } and let H + and H- be the hemispheres { (x, y, z) e S" ] z > o] and { (x,y, z) e S e [ z < o }, respectively. 112 QUADRATIC VECTOR FIELDS xx 3 To a differentiable vector field X in R 2 we can associate vector fields X + and X- in H + and H- in the following way: first we define in {(x,y, z)[ z = i } the vector field X(x,y, z) ---- (X(x,y), o) and then we project it by central projection onto H + and H-. When X is a polynomial vector field of degree n, it happens that multiplying X + and X- by z"- a we obtain a vector field in H + u H- that can be continuously extended to S x={(x,y,z) ~S *[z=o} in such a way that the resulting vector field on S * is analytic. This is the so-called Poincarg compactification of X and is denoted by ~(X). This construction allows us to study the flow of a polynomial vector field far away from the origin. The equator S 1 of S ~ represents the " points at infinity " and it happens (by construction) that points at infinity remain at infinity under the flow of :~(X) (i.e. S x is invariant for t~(X)). A graph of X containing separatrices (and thus singularities) at infinity will be called an unbounded graph. To study a polynomial vector field at infinity we consider the following coordinates u 1 = y/x and v 1 = I/x v~ = Ily. Geometrically they can be represented as follows: ?)lt If X~(2 is given by {y=P(x,Y)=o~+mx+ny+ax2+bxy+cy ' (~) x : Q(x,y) = ~ + ~x + ~y + ~x, + ~xy + ~y, then in the coordinates ul, vt it is expressed in the form (2) X, : [ '~' = Pl(ul, vl) ~t = Q.,(ut, ~t) with PI(ua,vl) =d+ (b--a) u I +Nv I+ (~--b) u~+ (ff--m) u tv 1 Qx(ut, vx) = -- avt -- but v x - m~ - cu~ v, - nut ~ - e~ 15 it4 RODRIGO BAMON and in the coordinates u2, v s it is expressed in the form u2 = Ps(us, vs) (~) xs : is = Q2(us, v,) with Ps(u2, vz) =c+ (b--c) u s +nv 2 + (a--b) u~+ (m--g) u sv s Lemma x.x. ~ Every X ~ X z has one, two or three pairs of symmetric singularities at infinity or the whole infinity is filled up with singularities. Proof. -- If X e X s is given by (i), using the expressions of X 1 and X z we have that the singularities at infinity are obtained by solving Px(u,,o) =a + (-b--a) u x + (i--b) u]--cuat = o and Ps(us, o) = c + (b -- ~) u s + (a -- b) u~ -- ~u~ ----- o. From these expressions the lemma follows immediately. Note that u x 4: o and u s = u~-t represent the same point at infinity. [] Remark. ~ By a rotation of coordinates we can always carry one of the pairs of symmetric singularities at infinity to the pair of points p : (o, l,o) and -- p = (o, -- I, o). Hence we can always suppose that the origin in the (us, vs)-plane is a singularity of X s or, in other words, we can always suppose c --- o. The following corollary is now clear. Corollary x.2. -- Let X e x s be given by (x) with c = o. Then: I) X has one pair of symmetric singularities at infinity if and only if [(5 -- a) ~ -- 4a(c -- b) < o] or [b--a =~--b =o and ~4: o]; II) x has two pairs of symmetric singularities at infinity if and only if [(b-- a) s-4n(c- b) ---- o and c-- b4: o] or [g-- b =o and b -- a 4= o]; III) X has three pairs of symmetric singularities at infinity if and on(~ if (b -- a) ~ -- 4d(i -- b) > o and i -- b = o; IV) the whole infinity is filled with singularities of X if and only if ~ -- b = b -- a = ~ = o. The proof of Theorem B in the next chapter will consider separately each one of the cases arising from this corollary. I. 3. General properties of quadratic vector field in R s Among the properties of planar quadratic vector fields there is a simple but basic one that we will use throughout the paper. It will be refered to as the periodic orbit property and says the following: 114 QUADRATIC VECTOR FIELDS ss5 If a planar quadratic vector field has a periodic orbit Y, then in the compact region bounded by Y there is a unique singularity. Moreover, the linear part of the vector field at this singularity has conjugate complex eigenvalues. This property can be used to find simple expressions for the vector field. In fact, if a quadratic vector field X has a periodic orbit, it is clear that there exists an afine change of coordinates such that X: {y=mX--y+P2(x,y) = x _ my + Q2(x,y), where P~ and Q.2 are homogeneous polynomial vector fields of degree two. In some cases we will use the fact that this form of X is invariant under rotations. Let us recall another basic property of planar quadratic vector fields. Contact property. -- Let X s )~2 be a quadratic vector field in R ~. Then every straight line t in R ~ is either invariant or has at most two contacts with X (i.e. points where t is not transversal to X). This property enables us to find geometric properties for bounded or unbounded graphs of quadratic systems. For example we can prove: (i) all graphs with return map enclose a convex region; (ii) a graph with return map and with at least two singularities must contain the straight line segment joining two adjacent singularities. In this case the quadratic vector field has an invariant line. With these properties we obtain simple expressions for the vector field. In fact, if a quadratic vector field X has an invariant line, then there exists an afine change of coordinates such that {~x(m+ax+by) x: = Q(x,y) and if X has two transversal invariant lines then {y~X(m-k-ax+by) x : + + Whenever necessary we will use these forms of X. Let us recall two more properties of quadratic systems. Invariant line property. ~ If a quadratic system has an invariant line then it has at most one limit cycle. Two invariant lines property. ~ The quadratic system k ---- x(m -~- ax q- by) .} -----y(K -k bx "k- cy) has no limit cycle. 115 xl6 RODRIGO BAM6N Although the invariant line property may substantially simplify the proofs, we are not going to use it, because we do not know of a good reference for its proof. However, we shall make use of the two invariant lines property, first proved by Bautin (see [C]). I .4. Singularities and periodic orbits Let p be a singular point of a vector field X in R * (i.e. X(p) = o) and let )'1, )', be the eigenvalues of DX(p). We say that p is hyperbolic if Re ),i ue o, i = z, 2, that it is semi-hyperbolic if X1 X2 ---- o but )~1 + )'~ + o and that it is a degenerate singularity if k 1 ---- ),, = o. Moreover, we say thatp is a center type singularity ifX 1 and ),~ are complex conjugate with zero real part. For the moment we are interested in the description of the flow of X in a neigh~ borhood of p (the topological type of p). First we define some types of singularities by their geometrical features: \t/ Jr< %1{ repellor saddle-node saddle attractor If the orbits of an attractor (repellor) spiral around the singularity we speak of a focus; if not we speak of a node. Let us now relate hyperbolicity with the above topological types. Ifp is a hyperbolic singularity of X and X1, )', are the eigenvalues of DX(p), then p is a saddle if )~1 )'2 < o, an attractor if Re ~ < o, i = I, 2, and a repellor if Re )~ > o, i = I, 2. Moreover, p is a focus if and only if X1 and ),, are complex conjugate numbers. These are simple basic facts about dynamical systems and can be seen for example in [P-M]. Ifp is a semi-hyperbolic singularity of X there are two local invariant differentiable curves intersecting transversally at p, such that the behavior of X along these curves determines the topological type of p. The tangent lines to these curves at p are the lines generated by the eigenvectors of DX(p). The invariant curve whose tangent line at p is generated by the eigenvector associated with the vanishing eigenvalue is called the center manifold ofp. The flow of X on the center manifold ofp is given by the first nonzero derivative fCk/(o) of an associate one-dimensional differential equation ~? =f(x) for which f(o) =f'(o) = o. The flow on the other invariant curve is determined by the sign of ),, the nonzero eigenvalue. With this, the topological type ofp is a saddle- node if k is even, a saddle if k is odd and )~.f(k)(o) < o and a node if k is odd and k./(k)(o) > o (attractor if ~. < o and repellor if ), > o). The center manifold theory needed here can be found in [H-P-S] or [Ca]. 116 QUADRATIC VECTOR FIELDS II7 The topological type in the degenerate case can be obtained by means of the blowing-up method. This method consists in " opening " (blowing-up) the singularity into a circle using for example the map ~): R X S 1 -+R 2 (r, 0) ~ (r cos 0, r sin 0). (We suppose that the vector field X is defined in an open set ofR z and has the degenerate singularity p at the origin.) An important fact is that there exists a vector field X in R x S x (the blow-up of X) verifying Dq~cr, 0)(X(r , 0)) = X(q~(r, 0)) for (r, 0) ~ U and leaving { o } � S 1 invariant. The set U is open in R � S 1 and contains { o } � S t. If we know the flow of .X in a neighborhood of { o ) � S x (for example if all the singu- larities of X in {o} � S 1 are hyperbolic or semi-hyperbolic), then " blowing down " X we obtain the topological type ofp. IfX has degenerate singularities along {o} � S 1, we blow-up again each one of these singularities and observe if we can determine the corresponding flows. If not, we blow-up again and again. Fortunately this process ends; in fact (X being analytic) we know that after a finite number of blowings-up we only get hyperbolic and semi-hyperbolic singularities. This allows us to describe the topological type of p. In sections 2. i. i, 2.2 (1.2) and 2.2 (II.2), we will give the topological types of all degenerate singularities which will be needed. The blowing-up method can be seen in detail in [A], [Du] and [T]. Finally, we recall that the topological type of a center type singularity p of an analytic vector field is either a focus or a center (all orbits in a neighborhood of p are periodic). A periodic orbit y is called an attractor (repellor) if it is the o-limit set (,t-limit set) of all points in a neighborhood of y. Let X be a vector field in R ~ and let ? be a periodic orbit of X of period T. The number c = div X(y(t)) dt is called the characteristic exponent of y. It is a well known fact (see [A], [S]) that for c > o the orbit "~ is a repellor and for c< o it is an attractor. I. 5. ll'yMenko's Theorem ([I]) Definition. -- A graph I ~ of a vector field X in R ~ is called a hyperbolic graph of X if all singularities of X contained in 1 ~ are hyperbolic. Theorem (II'ya~enko). -- Every hyperbolic graph of an analytic vector field in R 2 is finite (i.e. not accumulated by limit cycles). 117 x~8 RODRIGO BAMON This theorem is crucial for our result because it allows us to consider only the non-hyperbolic graphs. i. 6. Dulac' s Proposition Definition. -- A semi-hyperbolic singularity p of a vector field X in R 2 is called contractive if div X(p)< o and expansive if div X(p)> o. Notice that in this case div X(p) is the nonzero eigenvalue of DX(p). Proposition (Dulac [D]). -- Every graph of an analytic vector field in R ~ which contains only hyperbolic or contractive (expansive) semi-hyperbolic singularities is finite. The proof of this fact is straightforward. 2. Proof of Theorem B 2.z. We will first prove that all bounded graphs of quadratic vector fields are finite. To do this we observe that a quadratic vector field has at most four singularities in the plane. Since we are interested in periodic orbits we may suppose that one of the singularities is a focus or a center (see I. 3 and 1.4). Hence bounded graphs contain one, two or three singularities (this is true for every quadratic vector field; see Berlinskii's Theorem in [C]). 2.x.x. Let us first consider graphs with one singularity. If the singularity is either hyperbolic or semi-hyperbolic the graph is finite. This follows from II'ya~enko's Theorem and Dulac's Proposition, respectively. Suppose now that the singularity is at the origin and that both eigenvalues are zero. If the linear part of the vector field at (o, o) is identically zero then the vector field is homogeneous and there is no limit cycle. We may then suppose that after a linear change of coordinates the vector field has the form i k =y + ax ~ + bxy + cy z Following the blowing-up method we observe that if ~ # o the topological type of the origin is ~>o /~<o Since the line y = o is transversal except in (o, o) this singularity does not belong to any graph. Thus, necessarily ff ---- o. 118 QUADRATIC VECTOR FIELDS If9 In this case (~ ---- o) the line y ---- o is invariant and the existence of a periodic orbit implies both b 4= o and the existence of another singularity which must be a focus or a center. Changing coordinates by (x,y) ~-~ ([~x + ~y, ~,y) with an appro- priate ~,, we obtain the following form for X: i fc=ny+ax 2 +bxy--ny 2 X: . y = xy with (o, o) and (o, x) as singularities. For (o, I) to be a focus or a center it is necessary that b ~<4 n. We also need a4= o. Suppose b = o. Then the vector field verifies A, X = -- X for A(x,y) = (--y, x). It follows that (o, i) is a center and that there is no limit cycle. Now denote by X b the vector field Xb:l; ~yg. Then, X b = X 0 +b (xy) and det (X0, Xb) = -- bx'yL It follows that the orbits of X b are topologically transverse to the ones of X 0 and since X 0 has a center and is symmetric with respect to the y-axis we see that Xb, b + o, does not have any periodic orbit. Moreover, calculating the topological type of (o, o) (by the blowing-up method) we conclude that X 0 has the following bounded graphs according to the values of the coefficient a. , @ a<o o<a<l/2 Also, the vector fields X 0 for a/> I/2 and X n for b 4= o do not have any bounded graph. This settles the case of bounded graphs with one singularity. 2.x.2. Let us now consider bounded graphs with two singularities. Since we are interested in graphs with return map we may suppose (by the contact property) that both singularities are points of an invariant line for the vector field. By a linear change of coordinates we can carry these points to (o, o) and (o, I). The vector field then takes the form i k~- x(m q- ax + by) X: !,~ ---- k-y(y -- i) + -mx + ax' + bxy Roe o. The eigenvalues of the linear part of X are {m, -- ~} at (o, o) and {m + b, ~} at (o, x). Since ~ 4 o each singularity is either hyperbolic or semi-hyperbolic. If =x.y.ny~axqt-[-bxy 120 RODRIGO BAMON at least one of them is hyperbolic then the graph is finite (by II'ya~enko's Theorem or by Dulac's Proposition). If both singularities are semi-hyperbolic then m = b = o. But in this case J = ax ~ and there is no bounded graph. 2. x.3. Finally, if there is a graph with three singularities and with return map then, by the contact property, there are three invariant lines and by the two invariant lines property we know that there is no limit cycle. This conclude the proof for bounded graphs. 2.2. Now we prove that all unbounded graphs are finite. We proceed by consi- dering separately each one of the relations in Corollary 1.2. 2.2(I.I) (b--a) 2-4~(~-b)< 0 Let X e Z ~ be given by (I) with c = o and verify the relation above. In this case p = (o, I, o) and -- p are the unique singularities at infinity. Since X is expressed in coordinates (u~, v~) (see x.2) as ~= (b--~) us + nv~ + ... X2 b2 = -- ~v~ + ... the point p is hyperbolic for X restricted to infinity (u,-axis) and it is hyperbolic if and only if ~4: o. Lemma 2. I. -- If X has an unbounded graph I', then: (i) p and -- p are saddles (and so b * o) and they belong to r. An arc at infinity joining p and --p must be contained in F. (ii) The line t : x = -- n/b is invariant and contained in I'. (iii) There exist coordinates in which we can write i ic = xy with b 2 -- 4a(C- I) < O. X: t.Y = Q.(x,y) Proof. -- Part (i) is clear. To prove (ii) we notice that (X(-- n/b,y), (I, o)) = o~ -- mn/b q- an~/b ~ proving that ,e is invariant or transversal to the flow. If it is transversal, the separatrices of the saddlesp and -- p must be on different sides of/(see the figure below). Therefore, there is no unbounded graph 120 QUADRATIC VECTOR FIELDS We now prove (iii). By translating coordinates we carry t to the line x = o. The vector field is now given by k = x(m + ax + by) t-} = Q(x,y). Since b + o (p is a saddle), we obtain the desired form for X by changing coordinates: (x,y) ~ (x, m + ax -t- by) [] Let X e ff be given by k = xy b~ -- 4ti(E- x) < o. x: t) = Q(x,y) Since in this case ,~,=,,,[I -- i-- b,, -- ~, -- i 4-~u,v2-~41 X2 : ~,- = Qa(u,-, v~) = - iv, + ... we conclude that the v~-axis is invariant and that the flow along this line is given by = Q,-(o, v,) = - - - Thus, the origin (u2, v2) = (o, o) (and hence p) is a saddle if and only if (I -- i) E> o (hyperbolic case) or C---- ~i =- o and ~ > o (non-hyperbolic ease). For X e ff as above we will prove the following scheme: : K S- 4~> o : there is no unbounded graph i IK+ o : graph as in Fig. x ~(I --E)>o ti,---4~E=ol i ~ + o : graph as in Fig. 2 I n=~ tb=o : graph as in Fig. 3 Kz 4~ < o : graph as in Fig. 4 =/i=o and i>o! ~ + o : graph as in Fig. 5 = o : graph as in Fig. 6 -,.,, ~*"~ Fro. L -- The graph does not have a return map and so it is finite Fro. 2. -- Calculating the characteristic exponent it follows that three is no periodic orbit 16 RODRIGO BAMON Flo. 3. -- There is syrrunetry, therefore the graph is finite Flo. 4. -- By II'ya~enko's theorem, the graph is finite Fro. 5. -- Same as in Fig. 2 lhG. 6. -- Same as in Fig. 3 Let us consider the case ~(1 -- c) > 0. We first look at the singularities of X on the invariant line t : x = o. These singularities are given by the roots of Q(o,y) = ~ + n) + ~y~ = o. If K S -- 4fi~ > 0, there are two singularities on t which are hyperbolic fi~r X restricted to t. In this case there is no unbounded graph. For K S -- 4fii = 0, there is a unique singularity on t, namely P0 --- (o, -- n/~i), which has eigenvalues -- K]2c and o. Thus, for ff + o, P0 is a saddle-node and we obtain the graph of Figure ]. For K := o, we necessarily have ~ = o. Notice that if there is a periodic orbit Y = (71, 7~), it must be contained in {x > o} or in {x < o}. Also, from the expression for X, we have 7~ = "~a/Yl. Calculating the characteristic exponent of a periodic orbit of period T we obtain: 122 QUADRATIC VECTOR FIELDS I23 div X(v(t)) dt ---- by,(t) dt + (2~ -11- I) v2(t) dt Y; f; f: : b Tl(t) dt + (2ff + I) ~'t(t) dt f; I; = -b v (t) dr. So, if b ~e o, all possible periodic orbits in the same half-plane as well as the singularities must be of the same type: all repellors or all attractors. Since this is impossible, there is no periodic orbit at all. Finally, for b = o, X has the form X= = xy = Nx + ~x ~ + c-y~ and we easily show that AoX=--X for A(x,y) = (x,--y). The flow of Xisthen given in Figure 3. When ~2 _ 4~< 0 there are no singularities on ! and we get the graph of Figure 4 with hyperbolic singularities. By II'ya~enko's Theorem we know that these graphs are finite. Let us now consider the case k-=n=o and ~>o. In this case there are no singularities on the invariant line r = o, and there are two singularities lying on different sides of t. Both of them are center-type singularities. If there is a periodic orbit "r = ('rl, T2) of period T we calculate its characteristic exponent and obtain the number b [: yl(t) dt. As before, we see that there is no periodic orbit if b ~e o. For -----o the vector field verifies A. X =- X for A: (x,y)~ (x,--y) and so we obtain the graph of Figure 6. This proves Theorem B in the case (I. i). 2.2(I.2) b--a:~--b:O and ~+- O. Let X e ~ be given by (I) with c : o and verify the relations above. In this case p---- (o, I, o) and --p are the unique singularities at infinity. Let X 2 be the expression of X in coordinates (u2, v~) (see 1.2). Since X2: /fie----nv~+ ... ( i~ 2 = -- bv~ + ..., the point p is not hyperbolic for X restricted to infinity (us-axis) and it is semi-hyperbolic if and only if b+ o. Lemma 2.2. -- Let b ~ o. If X has an unbounded graph F, then: (i) p and -- p are saddles and they belong to F. An arc at infinity joining p and -- p is contained in F. 123 x24 RODRIGO BAMON (ii) The line g : x = -- n/b is invariant and contained in F. (iii) There are coordinates in which =~+~x+~y+/ix 2+y2. Proof. -- Since p is semi-hyperbolic and is a node along the center manifold (the infinite line), it is either a node or a saddle. If it is a node, no unbounded graph is possible. The rest of the proof goes as for Lemma 2.I. [] Lemma 2.3. -- Let b = o. If X has a periodic orbit then there are coordinates in which {~=mx--y+ax' X: r 4: o. = x + my + ~x 2 + ayx Proof. -- We first note that the conditions b--a=o, b=i=o and a-#:o are invariant under affine change of coordinates that keep p = (o, I, o) fixed. The lemma then follows by the periodic orbit property. [] Notice that we are only interested in quadratic vector fields which have periodic orbits. Therefore, we will frequently use the coordinates given by Lemma 2-3. If a~e o, using (x,y)~(~tx,~2y), r x, i= 1,2 if necessary, we can suppose ff>o and a>o. Lemma 2.4. ~ Let X e ff be given by {;=mx--y+ax' ~>o = x + my + ~x z + axy a > o. Then the singularity p = (o, I, o) at infinity has the following topological type: Proof. -- By blowing-up the singularity at the origin for the vector field / ~ = - v~ - ~u~ - u~ v, X, b, = -- au z v 2 - m~ - ~u~ v, - u 2 ~, we recognize the above topological type. [] 124 QUADRATIC VECTOR FIELDS t25 We will now prove the following scheme: fib > o " there is no unbounded graph b4=o! I I ff 4= o : graph as in Fig. 7 ab < o (coordinates as in Lemma ~. ~) I r~ = o : graph as in Fig, 8 ' i m = o : X is Hamiltonian ia~o b o (coordinates ~ m 4= o : since div X(x,y) = 2m, there is no ----- ~ ' periodic orbit as inLemma z.3) la> i2am--~=~ , 2am -- ~ ~= o : there is no unbounded graph. Fro. 7.- The characteristic exponent is nonzero, therefore there is no periodic orbit FIo. 8. -- There i~ ~yrametry, hence graphs are finite Fro. 9-- There is a Liapunov function imide the ~aph. Thu.,, there is no periodic orbit Let us consider the different cases: When b 4= o and ~b -'> o, p is a node and therefore there is no unbounded graph. For b + o and db < o, p is a saddle. Let us consider coordinates as in Lemma 2.2. If there exists a periodic orbit of period T, we calculate its characteristic exponent obtai- ning ~T. Thus, as before, there is no periodic orbit if ~ + o. When ff = o, the vector field is symmetric with respect to (x,y) ~ (x, --y), and 1here are no limit cycle~. I25 x26 RODRIGO BAM6N We now suppose b = o. Take coordinates as in Lemma s.3 with ~> o and a> o. Let y =y(x) be the parabola y =y(x) = (a/s) x ~ -- mx -- (x + m2)/2a and let ~r~(x) be its normal vector ~(x) = (ax- m,- x). Easy calculations show that (X(x,y(x)), ~s(x) ) = (sam -- ~) x 2. Fix a > o, a > o and consider m as a parameter. Let m0 be given by sam o -- ~ = o and let y =y0(x) be the parabola y =yo(x) = (a/2) x 2 -- moX -- (x + m~)/sa. For m = m o the parabola y =yo(x) is invariant under the flow of X. Let b(x) = -- m ox - (x + m~)/a. By straightforward calculation we obtain: (i) b(x)<yo(x) for all x~ll. (ii) The function f(x,y) = (y --yo(X))/(y -- b(x)) 2 has the origin as a maximum and in the region f~ = {(x,y)/y>yo(x)} this is the only critical point. That is, f has the following level curves in f~: I.- (e7 p +Of I y--y.(x) ~< o for each (x,y) el2 and (iii) Xf(x,y)= ~x Ox O~](x,y) = -- s~ (Y _ b(x))So x+o. In this way, if 2am -- ~ = o, the origin is a repellor and there are no periodic orbits. For m 4: m o (i.e. 2am --d 4: o), the parabola y =y0(x) is transversal to X. Since X =X,,= X,,.+ (m--mo) R where X,, is the vector field in Lemma 2. 3 and R(x,y) = (x,y) is the radial vector field, we observe that the separatrices at p move to different sides of y =y0(x) when m changes. Thus there is no unbounded graph when m 4: m0. The proof of Theorem B in case (I) is complete. 2.2(II.z) (/~--a) ~ - 4~(?-. b) = 0 and ~-- b:t= O. In this case there are two pairs of singularities at infinity. The one different from {p = (o, I, o), --p} is a pair of saddle-nodes for X restricted to infinity. This is clear from the equation of X restricted to infinity: ~l=a + (b--a) ul + (~-- b) u, ~. lg6 QUADRATIC VECTOR FIELDS x27 By rotating coordinates we can carry this pair of singularities to {p, -- p }, leading us to the next case: 2.2(II.2) b--g=0 and b--a+- O. Besides {p, --p} there is another pair of singularities at infinity. When we res- trict X to infinity, p and --p are non-hyperbolic while the other pair is hyperbolic. As before, let X, be the expression of X in coordinates (u2, v2). The linear part ~ at the ~ is (: --b)" So p is semi-hyperbolic if and only if b+-o. If b4=o and X has an unbounded graph F, then F must contain two adjacent singularities at infinity and the corresponding arc between them. Also, for b + o, if X has an unbounded graph without singularities in the plane, it must be of one of the two following types: By Dulac's Proposition these graphs are finite. Now, if b 4= o and X has an unbounded graph which contains singularities in the plane and which has a return map, then by the contact property it is proved that the separatrices of the graph are contained in invariant lines. By changing coordinates we put these invariant lines in the axes and so the vector field takes the form l; =x(m-k-ax+by)_ =y(ff + bx § by). By the two invariant lines property this graph is finite. Suppose now b = o. Lemma 2.5. -- If X has a periodic orbit then there exist coordinates in which i i~ = mx --y + ax 2 a -- b :> o X: l.~ : x + my + ~x~ -+- 3xy ff >>. o. Proof. --- The same as in Lemma 2.3 and the remark following it. [] Lemma 2.6. -- Let X be given as in the lemma above. Then the topological type of the singularity p -~ (o, I, o) at infinity is one of the following: (i) , ~ /fo<h< a; 127 RODRIGO BAMON t~8 (ii) 9 __~__~_ /f2a<b< a< o; (iii) (iv) if b = o then the topological types are as in (i), (ii) or as in the following figures I r Proof. -- As for Lemma 2.4. [] From the possible topological types for p, we obtain that unbounded graphs may have i or 3 singularities at infinity. For example the following graphs can exist: Unbounded graphs with two singularities at infinity cannot exist because graphs with a return map must enclose a convex region. Suppose that X e X * as in Lemma 2.5 has a graph with a return map and with three singularities at infinity. Then, one of them is p (or --p) and the others are the adjacent ones which are themselves symmetric. This pair of symmetric singularities are in the direction y/x ---~[(a- b). Since they are contained in a graph they must be saddles and, by the contact property, the separatrix in the plane must be an invariant straight line g of the form y=y(x) = _x+N. a--b From the equation ~2 (X(x,y(x)), (~, b -- a) ) = d(a -- b) x' + (bN(b -- a) _+~-a) x a--b --~N+mN(b--a) =o, where (d,b--a) is avector normal to?,it follows that d=m=o and N= -- z/b. 128 QUADRATIC VECTOR FIELDS x29 Thus, X can be expressed in the form {;=--y+ax' X: = x + bxy. Since this vector field has the symmetry A. X = -- X for A(x,y) = (-- x,y), we see that the origin is a center, and so there is no limit cycle. We will now prove the assertions in the following scheme for the case b = o. --bm--~----o/~=~ : graph as in Fig. IO 3am h-> o : graph as in Fig. I I o<b<a 3am -- bm -- h 4= o : there is no unbounded graph ~< 2a, b < o : there is no unbounded graph with return map {~ = o : graph as in Fig. 12 I 3am -- bm -- ~ = o > o : graph as in Fig. x 3 2a< b< a< o [ 3am -- bm -- ~ 4: o 9 there is no unbounded graph (~ = o : graph as in Fig. 14 3am -- ~ = o > o : there is no unbounded graph b=o 3am- ~ 4= o : there is no unbounded graph. FIo. xo. -- There is a first integral inside the graph, therefore it is finite Fxo. i x. -- There is a Liapunov function inside the graph. Thin, there are no limit cycle$ Flo. 12. -- Same as in Fig. lo 17 13o RODRIGO BAMON FIo. 13. -- Same as in Fig. xx Fro. t 4. -- The graph is symmetric, therefore it is fimte To prove the above assertions, consider the parabola 2a -- b I + m ~ y=y(x) = -- x 9 - mx 2 2a and its normal vector ~N(x) = ((2a- b)x- m,- x). Easy calculations give (X(x,y(x)), ~s(x) ) = (3am -- bm -- ~) x'. Let m o be given by the relation 3amo--bmo--a=o and let y =y0(x) be the parabola 2a- x y=Yo(X) -- -- x ~-m ox 2 2a Fix a, -b and S satisfying O < b < a and ~ >>. O. Notice that m o = o for ~i ---- o. Let us consider m as a parameter. If m = m 0 the parabola y =y0(x) is invariant and forms an unbounded graph. Take r=b/a and let b(x) = -- moX-- (i -bm~)/b. Then: (i) b(x) <yo(X) for all x ~R. (ii) The function f(x,y)= (y--yo(X))'/(y- b(x)) ~ has the origin as a maximum and in the region f~ = {(x,y)/y >yo(x)} this is the only critical point. The level curves off in f~ are /~.~ = yoCx) 2, 130 QUADRATIC VECTOR FIELDS (Of P + Of Q) (x,Y) = -- 4(2a -- b) m o (y -- Y~ xZ< o (iii) Xf(x, y) = Ox (y -- b(x) ) s for all (x,y) ef~ with x4= o. Therefore, for ~ = o the origin is a center (Fig. 1o) and if ~ > o (so that m 0 # o) there are no periodic orbits inside the graph (Fig. IX). For m# m 0 (i.e. 3am--brn--ff# o), we have as before that the relation: X=X,,=X,,,+ (m--m0) R is satisfied, and so there is no unbounded graph. This ends the proof in case o < b < a. Now fix a, b and ~ satisfying 2a < b< a< 0 and ~ >10. Recall that if ~ = o then m 0 = o. Let us again consider m as a parameter. First let m = m 0. In this case the parabola y =y0(x) is invariant and forms an unbounded graph. Take r and b(x) as before. Then: (i) y0(x)< b(x) for all xzR. (ii) The function f(x,y)= (yo(x)--y)'l(b(x)-y)~ has the origin as a maximum and in the region ~ -----{(x,y)/y <yo(x)} this is the only critical point. The level curves off in f~ are (y0(x) -y)' < o (iii) Xf(x,y) = 4(2a -- [~) rao (b(x) -- y)' for all (x,y) ef~ with x# o. Therefore, if ~ = o there is a first integral and the origin is a center (Fig. 12) and if ~> o (so that m 0 4= o) there is a Liapunov function and there is no periodic orbit inside the graph (Fig. i3). To conclude the case above, we now let m 4: m 0 (i.e. 3am -- bm -- ~i 4= o). The same arguments as in the previous case prove that there is no unbounded graph. If b <~ 2a and -b < O, then from the topological type ofp we conclude that no unbounded graph is possible. Fix a, -b and ~ satisfying b=O< a and if> 0. Consider m as a parameter. If m = m 0 the parabola y =y0(x) is invariant and it is easily shown that there is a saddle on the parabola. Hence there is no unbounded graph. When m 4= m o the parabola y =y0(x) is transversal to X. Suppose that the topological type ofp is the one in (i) of Lemma 2.6. (This is the only possibility when b = o for the existence of an unbounded graph.) Since the separatrices at infinity bound hyperbolic sectors, the 131 RODRIGO BAM6N transversal parabola y =y0(x) must leave the separatrices at different sides. The following picture illustrates the situation: Thus, there is no unbounded graph. Finally fix a, b and ~ satisfying -b = 0 < a and h- = O, so that m 0 = o. When m = m 0 = o the vector field verifies A. X = -- X for A(x,y) = (-- x,y), the origin is a center and there are no limit cycles. If m # mo, as before there is no unbounded graph. Thus, all assertions concerning the case b = o are proved and case (II) is settled. 2.2 (III) (~ - a)" - 4a(~- b) > 0 and ~- b, O. We now come to the most difficult part of the proof of our main result. For X e Z 2 given by (I) (see Chapter i) with c = o and satisfying the relation above, there are three pairs of symmetric singularities at infinity. Lemma 2.7. -- If X ~ X ~ has three pairs of symmetric singularities at infinity and if two of them are hyperbolic, then X has a finite number of limit cycles. Proof. ~ When a quadratic vector field X has three pairs of symmetric singularities at infinity, all of them are hyperbolic for the restriction of X to infinity (see Lemma i. I). So the only possible unbounded graphs with a return map are of the following types: (double arrows indicate hyperbolicity) The first one is finite by II'ya]enko's Theorem. The middle one is not accumulated by periodic orbits (Dulac's Proposition). The last one, with singularities in the plane, must have separatrices contained in invariant lines, and as explained before, in this case there are no limit cycles. [] 132 QUADRATIC VECTOR FIELDS I33 Lemma 2.8. -- If X E Z ~ has three pairs of symmetric singularities at infinity and two of them are not hyperbolic, then: (i) the two pairs of non,hyperbolic singularities are semi-hyperbolic; (ii) the third pair of symmetric singularities consists of hyperbolic nodes; (iii) there exist coordinates in which the hyperbolic pair of singularities is {, = ,12 i, ,, o), - r} and the semi-hyperbolic pairs of singularities are {p = (o, x,o);--p} and {q= (x, o, o),--q}. In these coordinates the vector field has an expression as in (x) with a ----- o, c = o, ~ = o, ~=o, b+b=o and b#o. Proof. -- We first observe that given any order in the pairs of singularities, there are coordinates in R 2 such that the first pair is {p, --p}, the second pair is {q, -- q} and the third one is { r, -- r}. In fact, with a rotation of coordinates we carry the first pair to {p, -- p}; with a linear change of coordinates of the form A(x,y) = (x, ~x -k-y) (which fixes p) the second pair is taken to {q, -- q}; and finally with a change of coor- dinates (x,y) ~ (x, Xy), X # o, the third one is taken to {r, -- r}. If X is expressed as in (I), then in the coordinates above the following relations are true c:o, d=o, ~--b+a--b=o and b--gar o (this follows from (2) and (3) in I. 2, by imposing the conditions P~(o, o) = Pl(o, o) = PI(-- x, o) = o). Moreover we have the following table singularities eigenvalues p = (o, o) --~, b--~# o q= (~, o, o) --a,b--a=g--b# o r= I/2 Vr2(- I, I, O) -b--c,c--b # o. If we suppose that p and q are not hyperbolic then a = r = o and b = -- b # o. The lemma now follows directly. [] By the two lemmas above we can restrict ourselves to quadratic vector fields X with expression {;=o~+mx-Fnyq-bxy X: b#o. =-~ + Tnx + ~y--bxy Moreover, by translating the coordinates we can suppose n = ~ = o. Also, if necessary, the change of coordinates (x,y) ~ (y, x) makes b > o. 133 RODRIGO BAMON z34 Lemma 2.9. ~ Let X ~ x 2 be given by {;=oc+mx+bxy X: = ~t + ~y -- bxy with b > o. Then: (i) p = (o, x, o) and q = (o, -- I, o) are semi-hyperbolic singularities of X at infinity (hyperbolic for the restriction of X to infinity) and r = i/2 ~/~(-- I, x, o) is a hyperbolic node; (ii) if X has an unbounded graph then it must contain p and q (or -- p and -- q) and the cor- responding arc between them; moreover p and q (-- p and -- q) must be saddles or saddles- nodes; (iii) if X has an unbounded graph that contains singularities in the plane and has a return map, then the separatrices are contained in invariant lines. In this case the vector field does not have limit cycles. Proof. -- Parts (i) and (ii) are clear from Lemmas 2.7 and 2.8. If X satisfies the hypothesis in (iii) then by the contact property it follows that there exist two invariant lines that must contain the separatrices. We know that in this case there is no limit cycle. [] In what follows we will consider X e )~* to be given by {y=OC+mx+bxy X: = ~ + ~y -- bxy with b > o, and we will study, in terms of the coefficients, when X can have unbounded graphs without singularities in the plane. We will prove the following assertions: (i) if ~0c = o there is no graph without singularities in the plane; (ii) if either ~<o, e>o or mE>o there is no unbounded graph. Notice that when m > o and ~< o, we can change coordinates (x,y) ~ (--y, -- x) so that we may suppose m< o and K>o. Let ~>o, m~< o, 0c< o and ~>1 o. Then: (iii) if 0c+~ =m+ff=o the only possible graph is as follows (iv) if (e+~)(m +~) =o but ~+~+m +n~e o, there is no unbounded graph; (v) if e + ~ + o and m -t- n ~e o, then any graph without singularities in the plane is finite. 134 QUADRATIC VECTOR FIELDS t35 To prove (i) suppose 8 = o. The line y = 0 is invariant and contains the center manifold of q. So, there is no graph, as required. The same happens if ~ = o. To prove the other assertions let us consider X expressed in the coordinates at infinity: Xt v t =--bu tv,-m~-~, Xi ~ = bu. v~ -- ~ -- ~. In both systems, the origin is a semi-hyperbolic singularity. The center manifold for X t has the form ,,~ = h,(Vl) = (~./b) ~, + 0(4) and the flow along it is given by i t = -- m~ -- (~ + 7,) ~, + 0(~,~). Similarly, the center manifold for X~ has the form ~, = ~(~,) = - (s/b) ~ + 0(~) and the flow is given by ~ = - ~4 - (~ + ~) d + o(~). We can now prove (ii). Suppose if< o. Since b> o, the center manifold of X t is locally contained in the half plane u t ~< o. That is: center manifold "~....._..1 ~ center manifold From the contact property it follows that all graphs with return map must enclose a convex region. On the other hand, by Lemma 2.9 any such graph must contain the adjacent singularities p and q (or --p and -- q). But this is impossible because of the location of the center manifold (see figure above). The same happens when ,t > o. 138 I36 RODRIGO BAMON If m~ > o then p or q is a node, and thus there is no unbounded graph. Now suppose ~ > o, m <~ o, ot < o and fi >i o. Notice that from the expressions for the center manifolds of p and q we have the following situations: Q) (double arrows indicate hyperbolicity) Besides proving (iii) to (v) we will see that these are the only cases where we can have unbounded graphs without singularities in the plane. In fact, consider the hyperbola y =y(x)=- ~/(bx) and its normal vector ~s(x) = (--cx, bx~). Easy calculations show that (X(x,y(x)), ~(x)) = x(b(o, + ~) x -- ~(m + ~)). To prove (iii), observe that, since e -{- ~ = m q- fi = o, the hyperbola is invariant, the vector field verifies A. X = -- X for A(x,y) = (y, x) and we obtain the graph indicated above. To prove (iv), i.e. when (~-+-~)(m+~) =o but ~-b~+mq-fi:~ o, notice that the hyperbola y =y(x) is transversal to X and no unbounded graph can exist (the separatrices ofp and q must be on different sides of y =y(x)). Let us now prove (v). We suppose ~q-~+ o and m-Fh-~e o. We recall that ~>o, m~< o, o~<o and fi>_-o. The relations m=o and 0cq-~>o or n-= o and 0c + ~ < o imply respectively that q or p are nodes (see the expres- sions for the center manifolds), and thus in these cases there are no unbounded graphs. Now we arrive at the hardest part of the proof of Theorem B. There are three casesto consider: mK#o; m=o, fi>o, 0t-k-~< o; and m<o, K=o, ~+~>o. As shown in the figures above in the three cases there can exist an unbounded graph without singularities in the plane. We will prove now that if such a graph exists, then it is finite. For that purpose we will analyse return maps (Poincard maps) associated to these graphs and show that these maps have isolated fixed points. To help clarify our arguments let us consider the following figure and dia- grams: 136 QUADRATIC VECTOR FIELDS t37 ~ X Y t ,.~l I_ "l 1~ ~,1.. 1 1 'b % 9 to "" '' 1" 1 Let us explain the notation. For o< r let Y~, = {(8,,,)/I,,I < ~}, Y~ = {(~, ~)/Izl < ~}, 21 ={(u, 8)llul < ~}, ~ = {(w, 8)/Iwl < ~}, n, = {(8, "~)I1", I < ~}, o3 ={(8, ~',)/1'~,1 < ~}, be transversal sections as indicated. Let Pl denote the change of coordinates from the (%, v,)-plane to the (ux, v,)-plane. Since vx = l /x and v, = I [y we have v2/v, = x]y = u I and so pt(u,, v,) = ( I ]u2, v2/u2). Moreover, O, (f~2) = { (I/8, vx)/1 vl [ < r }. 18 ~3 8 RODRIGO BAMON Let P~:0x(Dz) ~Dx; ~0:E 2~y~t and d/: Zt -+ Z* be the Poincar6 maps naturally defined by X. Let ta, s~, t~, s~ be local changes of coordinates given as follows. First take ua = ux -- h,(va) = ua -- (~]b) ~ -- 0(~) (i) t~ : __ V 1 ~ UI.. In these new coordinates, X x has the form u~ = -- b~a -- b~ + ~, ~, gt(a~, v~) v a ----- -- bff x b, -- mO~ -- (~ + ~) ~ + ~fx(~Tx). Now, to avoid the term -- bgx ~i in the component of the gx-axis we take sl: { u=~l = ~/(x + ~1), obtaining = -- bu -- bu ~ + uvg"x(u, v) xl/: = -- mv ~ -- (o~ + ~) v 3 + v4f(u, v) + uv" h'x(u, v). Now take = ~, - h,O,,) = ~ + (~,/b) ~ - o(~) (ii) t2 : _ /31 U 2 . In these coordinates X~ is expressed as u~ = bu2 ~ = b~ ~, - ~ - (~ + ~) ~7 + ~'A(~;). Again, to avoid the term b~a g~ in the component of the 17s-axis we take $~ : Z = v-~/(~ + U~) obtaining (0 = bw -q- bw 2 -q- wz~(w, z) Finally, let u = u(v) be the Poincar6 map from Y~+ ={(~, v) ~Zl/v> o} to ~1 defined by X1 and let z = z(w) be the Poincar6 map from ~.+ = {(w, 8) ~ ~Jw > o} to Z 2 defined by X2. To prove the finiteness of the graphs we will compare the Poincar~ maps above to other ones defined by auxiliary vector fields. To do this consider the following vector fields X 1 : and X~ : ~' z 2 138 QUADRATIC VECTOR FIELDS t39 with m'< o and K'>o. Let ~':Yt + -+~.l and ~':~+ ~X 2 be the Poincard maps associated to X't and X~ respectively. We will use the following expressions det(Xx, X'l) = uv*[b(m ' -- ra) + Fj.(u, v)], det(X,, X~) = wz'[b(~ -- n') + F,(w, z)], where Fa(o ,o) =o and F2(o ,o) =o, to compare the flows of XiandX~, i= i or2. Notice for example that if ]m] > I m'[ then det(X,, X[) > o for u> o, and so ~(v) < u(v) for small enough v > o. Let us first calculate u = ~'(v), z = ~'(w) and show that ?'(o) ~- I. We have l u = ~'(v) = ~e-o/t,: ~1 eb/Im'~l = k~ e b/C''o) (4) ~ ~ S z = = = x + (~' S/b) In S -- (~' S/b) In w k2 -- (~' ~lb) in w Now, to calculate ~'(o) we use the following lemma. Lemma 2. to. -- We have p~(o) = 8. Proof. -- We calculate O~(o) by the following formula (see [A]) p;(o) -- I Xl(I/8' ~ exp div Xx(T(t)) dt, ; x,(8, o) t The vector where 7(t) is the orbit of Xt that goes from (l/~, o) to (S, o) in time 1". field Xt, when restricted to v t = o, has the equation fil = -- bus -- buy. Integrating we obtain "f(t) = I + S -- e -hI' 0 9 From ~-(T) :8 we obtain T= -- I/blnS. Since div Xl(ul, O) ---- -- b -- 3bul we calculate ff be- bt div Xt(7(t)) dt = -- bT -- 3 x + ~ -- e -~ dt f =ln3--31n(t +~--e -~) ] ----- In M. Finally, since [Xt(8 , o)t ---- bS(i + 8) and ]X1(I/8 , o)] = b(i + 8)[M we obtain p~(o) = 8, proving the lemma. 139 ,4o RODRIGO BAM6N Remark. -- If ~t and ~s are cross sections for X 1 with ~t tangent to fl t at (8, o) and ~s tangent to Ot(f~2) at (I[~, o) and they are parametrized by the projection to the vl-axis, then the corresponding Poincar6 map ~s : ~s ~ fit satisfies ~s'(o) = p~(o) = a. From the above remark and from the fact that 0t(8, v2) ---- (I/8, v,/8) we finally obtain 9'(0) = I. We can now give the expression of our modified Poincar~ map. I.emma 2.t 3. ~ Let u = ~(o) and z = ~(w) be as in (4)- Let v = ~(z) = xz and w = ~(u) = ~tu. Then (~o T o ~ o'~) (u) = ku ~'ll-'lx where k is a positive real number. Proof. -- The formula is obtained by composing the maps. [] Let us now compare the return map of X with the modified return map and prove that if there exists a graph then it is finite (we will prove slightly more: they are not accumulated by periodic orbits). Suppose o< (h-/lrat)< I or else h= o. Chose m', h-', X and ~ such that: m' < o, I m'l < m, ~' > ~, x < x, (~'/1 m' I x) < I and o < V.u < ~b(u) for all u > o small enough. Then for ~, ~', ~ and ~" defined in (4) and in Lemma 2.13 we have since, for u > o, det(Xt, X~) > o near (o, o), ~(o) < uCv) 7(w) < z(w) since, for w > o, det(X2, X~) < o near (o, o), ~(u) < +(,,) for u > o small enough, for z > o small enough. T(~) < ,p(z) Thus (~o ~ o 7 o ~) (u) = ku ~'/''~'l ~ < (u o ~ o z o +) (~), and since (K'/[ m'] X) < x we conclude that u < (u o ~ o z o +) (u) for all u small enough. With this it is proved that if there is an unbounded graph then it is a repellor (i.e. it is the ~-limit set of some orbit). In the same way if (ff/[m D > I or m = o we prove that if there is an unbounded graph it is an attractor (i.e. it is the t0-1imit set of some orbit). This ends the proof of Case (III). 140 QUADRATIC VECTOR FIELDS t4 t a.2 (IV) ~-b=b--a=h=O. Let XeZ 2 be given by (I) (see Section I) with c=E--b=b--a=d=o. In this case, X ix transversal to infinity with the exception of two symmetric points. Since the relations above are invariant under any affine change of coordinates, the periodic orbit property allows us to restrict ourselves to vector fields of the form ; = mx -- y + ax ~ + bxy = x + my + axy + by 2. For these vector fields it is not hard to prove that with a rotatiol~ of coordinates we can make b = o. We then obtain (~ = mx --y + ax" = x +my + axy. With this last expression we see that the origin is the only singularity and therefore, there is a finite number of limit cycles. The proof of Theorem B is now complete. REFERENCES A. A~vRoNov et al., Od~alitative Theory of Second Order Dynamical Systems, John Wiley & Sore, New York, [A] W. Corrst, A survey of Quadratic Systems, Journal of Di.fferential Equations, 2 (1966), 293-3o4 9 [c] J. ~, Applications of Center Manifold Theory, Applied Math. Sciences, BS, Springer-Verlag, 1981. [Ca] C. Cmco~rs and S. Scan, Separatrix and Limit Cycles of Quadratic Systems avd Dulac's Theorem, [Ch-S] Transactwns Amer. Math. Sor 9.78 (t983) , 585-6x2. M. H. I)ULAC, Sur les cycles limltes, Bull. Soc. Math. France, 51 (1923), 45-188. [D] F. Douo~trmR, Singularities of Vector Fields, Journal of Differential Equations, 28, I (~977), 53 -t~ [Du] M. Hw.-~H, C. PuGs and M. Smm, Invariant Manifolds, Springer Lecture Notes in Math., 588 (i977). [H-P-S] [[] Yu. S. tl'ya.~enko, Limit cycles of polynomial vector fields with non degenerate singular points in the real plane (in Russian), Functional Analysis and its applications, 18 (3) (x984), 32-34 (ha translation : 18 (3) (x985), ,99-~o9)- I. G. P~rt~ovsxu and E. U. LA~DXS, On the number of limit cycles of the equation dy[d.~ = P(x,y)[Q.(x,y) [P-L l] where P and Q are polynomials of the second degree, Amer. Math. Soc. Transl. (2), t6 (t95fl), t77-22t. I. G. Pv taov~Ktz and E. ~'. LA.,~DIS, On the number of llmit cycles of the equation dy/dx = P(x,y)lQ,(x,y) [P-Ld where P and Q.. are polynomials, Amer. Math. Soo. Transl. (2), 14 (t96o), t8~-2oo. 1. G. P~Taovs~tt and E. U. LANDIS, Corrections to the articles : " On the number of limit cycles of the [P-L s] equation d3/d.x = P(x,.y)[Q.(x,y) where P and Q. are polynomials ", Math. Sb.N.S., 48 (9 o) (1959). a53-~55- 14/ RODRIGO BAMON ~42 j. PALLS and W. de MELO, Geometric Theory of Dynamical Systems; An Introduction, New York, Springer- [P-M] Verlag, x982. J. SOTOUAYOR, Curvas definidas por equa~oes diferenciais no plano, f3 o Col6quio Bros. de Mal., IMpA, [S] I~I, [Sh,] Sm SONOLINO, A concrete example of the existence of four limit cycles for plane quadratic systems, Sci., Sinica, Ser. A, 28 (x98o), z53-158. [$h~ Sm SONOLINO, A method for constructing cycles without contact around a weak focus, Journal of Differential Equa/~s, 41 (~98~), 3o~-3xa. Universidad de Chile Departamento de Matematicas Casilla 653 Santiago, Chile. Manuscrit refu le t6 septembre z985.

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