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Anais da Academia Brasileira de Ciências

versão impressa ISSN 0001-3765versão On-line ISSN 1678-2690

An. Acad. Bras. Ciênc. vol.90 no.3 Rio de Janeiro jul./set. 2018

http://dx.doi.org/10.1590/0001-3765201820170829 

Mathematical Sciences

The phase portrait of the Hamiltonian system associated to a Pinchuk map

JOAN CARLES ARTÉS1 

FRANCISCO BRAUN2 

JAUME LLIBRE3 

1Departament de Matemàtiques, Universitat Autònoma de Barcelona, Edifici Cc, Campus de la UAB, Bellaterra, Cerdanyola del Vallès, 08193 Barcelona, Catalonia, Spain

2Departamento de Matemática, Universidade Federal de São Carlos, Rodovia Washington Luís, Km 235, Caixa Postal 676, 13565-905 São Carlos, SP, Brazil

3Departament de Matemàtiques, Universitat Autònoma de Barcelona, Edifici Cc, Campus de la UAB, Bellaterra, Cerdanyola del Vallès, 08193 Barcelona, Catalonia, Spain

Abstract

In this paper we describe the global phase portrait of the Hamiltonian system associated to a Pinchuk map in the Poincaré disc. In particular, we prove that this phase portrait has 15 separatrices, five of them singular points, and 7 canonical regions, six of them of type strip and one annular.

Key words center; global injectivity; real Jacobian conjecture; Pinchuk map

Introduction

As far as we know, the simplest class of non-injective polynomial local diffeomorphisms of 2 are the Pinchuk maps, constructed by Pinchuk (1994). The existence of these maps disproves the real Jacobian conjecture, that a polynomial local diffeomorphism of 2 is globally injective. One open problem is to know what exactly fails in this conjecture.

One of the most known conditions for a local diffeomorphism to be a global one is that it is proper. The asymptotic variety of a map of 2 is the set of points where the map is not proper (i.e., points that are limits of the map under sequences tending to infinity). In particular, a local diffeomorphism is a global diffeomorphism if and only if this set is empty. Gwoździewicz (2000) and Campbell (arXiv:math/9812032 in 1998, 2011) calculated the asymptotic variety of two Pinchuk maps in details. Our aim in this paper is to do a similar work, i.e., to describe a Pinchuk map, but now from a different point of view.

Let U2 be an open connected set. Let F=(p,q):U22 be a C2 local diffeomorphism. Let HF(x,y)=(p(x,y)2+q(x,y)2)/2 and consider the Hamiltonian system

ẋ=(HF)y(x,y),ẏ=(HF)x(x,y), (1)

where the dot denotes derivative with respect to the time t . The singular points of system (1) are characterized by the following result, that we shall prove below.

Lemma 1. The singular points of system (1) are the zeros of F , each of them is a center of system (1).

The following is a generalization of the characterization of global invertibility of polynomial maps given by Sabatini (1998). This version is due to Braun and Llibre (arXiv:1706.02643 in 2017).

Theorem 2. Let z0U such that F(z0)=(0,0) . The center z0 of system (1) is global if and only if (i) F is globally injective and (ii) F(U)=2 or F(U) is an open disc centered at the origin.

In the special case that F is a polynomial map and U=2 , it follows that F(2)=2 provided F is injective (Białynicki-Birula and Rosenlicht 1962). Hence, in this case, z0 is a global center of (1) if and only if F is globally injective. An application of this result was given by Braun et al. (2016).

Since the phase portrait on the Poincaré sphere of a Hamiltonian polynomial vector field having a global center is simple, i.e., at the infinite either it does not have singular points, or the infinite singular points are formed by two degenerate hyperbolic sectors (for Hamiltonian vector fields, the infinity contains only isolated singular points), it is interesting to know how complex can be the phase portrait of a non-global center of a Hamiltonian system (1).

In this paper we provide the qualitative global phase portrait of the Hamiltonian system (1) when F is given by the Pinchuk map considered by Campbell (1998, 2011), after a translation in the target in order to have only a point z0 such that F(z0)=(0,0) . More precisely, we prove the following result.

Theorem 3. Let F=(p,q):22 , where (p,q+208):22 is the Pinchuk map considered by Campbell (1998, 2011) (see the definition below). Then the phase portrait of the Hamiltonian system (1) in the Poincaré disc is topologically equivalent to the phase portrait given in Fig. 1.

Figure 1 The qualitative global phase portrait of system (1) in the Poincaré disc. 

To prove Theorem 3, we first study the infinite singular points of system (1). These infinite singular points are very degenerate, and we apply homogeneous and quasi-homogeneous blow ups to study them. Then, we complete the proof of Theorem 3 by proving that the separatrix configuration of system (1) is qualitatively the one presented in Fig. 1.

We think that a good understanding of what fails in the real Jacobian conjecture could be interesting to investigate a related problem, the Jacobian conjecture in 2 , that a polynomial local diffeomorphism whose Jacobian determinant is constant is globally injective. This conjecture remains unsolved until now. For the Jacobian conjecture we address the reader to the works of Bass et al. (1982) and Van Den Essen (2000).

Injectivity, centers and a Pinchuk map

We begin with the proof of Lemma 1.

Proof of Lemma 1. Let z0 be a singular point of the Hamiltonian system (1). We have

(py(z0)qy(z0)px(z0)qx(z0))(p(z0)q(z0))=(00),

which is true if and only in F(z0)=(p,q)(z0)=(0,0) because the Jacobian determinant of F is nowhere zero.

The point z0 is a center of the Hamiltonian system (1) because it is an isolated minimum of HF .

Now, we select the map F that we are going to work in this paper. Let t=xy1 h=t(xt+1) and f=(xt+1)2(t2+y) . A Pinchuk map is a non-injective polynomial map with nowhere zero Jacobian determinant of the form (P,Q):22 such that P=h+f and Q=t26th(h+1)u(h,f) , where u is chosen so that detD(P,Q)(x,y)=t2+(t+f(13+15h))2+f2 . The following is the Pinchuk map studied by Campbell (1998, 2011):

p¯=h+f,q¯=t26th(h+1)170fh91h2195fh269h375fh375h44.

According to Campbell (2011), the points (1,163/4) and (0,0)2 have no inverse image under (p¯,q¯) , all the other points of the curve

γ(s)=(s21,75s5+345s4429s3+117s221634),s,

which is a parametrization of the asymptotic variety of (p¯,q¯) , have exactly one inverse image under this map, and the points of 2\γ() have two inverse images. Hence, in particular, the point (0,208) has precisely one inverse image under (p¯,q¯) .

We consider the map F=(p,q):22 given by the translation

p(x,y)=p¯(x,y),q(x,y)=q¯(x,y)208. (2)

Observe that F is a Pinchuk map according to our above-definition. Moreover, now there exists exactly one point z02 such that F(z0)=(0,0) . From Lemma 1 the point z0 is the only finite singular point of system (1), corresponding to a non-global center of this system according to Theorem 2. Further, the curve

β(s)=γ(s)(0,208)=(s21,75s5+345s4429s3+117s229954), (3)

s , is the asymptotic variety of F , whose points have exactly one inverse image over F , but the points (1,995/4) and (0,208) , which have none.

From now on, we restrict our attention to the specific Pinchuk map (2).

We first calculate the coordinates of the point z0 . Observe that xt+1=x2yx+1 is a factor of p . If this factor annihilates, then h=0 and q=t2208<0 . The other factor of p is

g(x,y)=x+(12x+3x2)yx2(2+3x)y2+x4y3.

We observe that g(0,y)=y and q(0,y)=50y799/4 do not annihilate at the same time, thus the first coordinate of the point z0 is not 0 . Moreover, since the leading coefficient of q(x,y) as a polynomial in y is 75x15 , it follows that the first coordinate of z0 will be a point where the resultant in y between g(x,y) and q(x,y) is zero. This resultant is the cubic c(x)=3100839111757152x155580672x2+2239078400x3 multiplied by x36/64 . The discriminant of c(x) is negative, so it has only one real root, which will be the first coordinate of the point z0 .

Repeating a similar reasoning now looking g and q as polynomials in x , we calculate their resultant and obtain that its zero is the only real root of the cubic c(y)=1789023641600+100675956992y+26252413280y2+1506138481y3 , which will be the second coordinate of the point z0 .

Hence z0=(0,22568337...,17,491214...) approximately. Since z0 is a center, the only finite singular point of system (1), near z0 the phase portrait of this system is simple. Indeed, since z0 is the minimum point of HF , it follows that the gradient of HF points outward of each closed orbit of the center, and so each closed orbit of the center rotates in counterclockwise around z0 .

In the following section we shall investigate the infinite of system (1).

The infinite of system

In this section, we will use results and notations on the Poincaré compactification of polynomial vector fields of 2 . In particular Ui Vi i=1,2,3 , are the canonical local charts of the Poincaré sphere 𝕊2 .

For details on this technique we refer the reader to Chapter 5 of (Dumortier et al. 2006) or to (González Velasco 1969).

We call a singular point of a vector field linearly zero when the linear part of the vector field at this point is identically zero.

We begin by proving a general fact about the infinite singular points of Hamiltonian systems of the form (1). Writing H=H0+H1++Hd+1 , where Hi is the homogeneous part of degree i of the polynomial H , it is simple to conclude that the infinite singular points (u,0) of system (1) in the local charts U1 and U2 are the points satisfying Hd+1(1,u)=0 and Hd+1(u,1)=0 , respectively. Let (u,0) be an infinite singular point of system (1) and assume it is in the chart U1 . The linear part of the vector field at (u,0) is

((d+1)(Hd+1)y(1,u)dHd(1,u)0(Hd+1)y(1,u)).

Assuming m=degpdegq , we have d=2m1 and Hd=pmpm1+qmqm1 and Hd+1=pm2+qm2 . Since Hd+1(1,u)=0 , it follows that pm(1,u)=qm(1,u)=0 , and hence (Hd+1)y(1,u)=Hd(1,u)=0 . Therefore, (u,0) is a linearly zero singular point. This proves the following result.

Lemma 4. The infinite singular points of the Hamiltonian system (1) are linearly zero.

Now, we return to the Pinchuk map F defined by (2). Observe that the highest homogeneous part of HF(x,y) is 5625x30y20/2 . Thus, the origins of the charts U1,V1 and U2,V2 are the infinite singular points of the Hamiltonian system (1), each of them linearly zero from Lemma 4.

We will use the quasi-homogeneous directional blow up technique to desingularize each of these infinite singular points. An exposition about blow-ups can be found in (Álvarez et al. 2011), see also Chapter 3 of (Dumortier et al. 2006). We now recall the directional blow up transformations.

By the quasi-homogeneous blow up in the positive (resp. negative) x -direction with weights α and β , or simply (α,β) -blow up in the positive (resp. negative) x -direction, we mean the transformation which carries the variables (x1,y1) to the variables (x2,y2) according to the formulas

(x1,y1)=(x2α,x2βy2),(x1,y1)=(x2α,x2βy2),

respectively. Similarly, by the quasi-homogeneous blow up in the positive (resp. negative) y -direction with weights α and β , or simply (α,β) -blow up in the positive (resp. negative) y -direction, we mean the transformations

(x1,y1)=(x2y2α,y2β),(x1,y1)=(x2y2α,y2β),

respectively.

Clearly if α (resp. β ) is odd, then, the blow up in the positive x -direction (respec. y -direction) provides the information of the respectively negative blow ups. Also, if β is odd, the x -directional blow ups swap the second and third quadrants, while the y -directional blow ups swap the third and the fourth quadrants if α is odd. After the (α,β) -blow up in the x -direction, a system x1̇=P(x1,y1) y1̇=Q(x1,y1) is transformed into

x2̇=±Pαx2α1,y2̇=αx2α1Qβx2β1y2Pαx2α+β1,

with P=P(±x2α,x2βy2) and Q=Q(±x2α,x2βy2) , in the positive and negative directions according to ± . Similarly, the (α,β) -blow up in the y -direction transforms x1̇=P(x1,y1) y1̇=Q(x1,y1) into

x2̇=βy2β1Pαx2y2α1Qβy2α+β1,y2̇=±Qβy2β1,

with P=P(x2y2α,±y2β) and Q=Q(x2y2α,±y2β) , in the positive and in the negative directions according to ± .

After the blow up in the x -direction (resp. y -direction) we cancel a common appearing factor x2k ( y2k ) for a suitable k . So, if k is odd, the direction of the orbits are reversed in x2<0 ( y2<0 ).

The weights α and β are chosen analyzing the Newton polygon of (P,Q) , see the construction in (Álvarez et al. 2011).

The application of (α,β) -blow ups with αβ1 usually reduces the number of blow ups necessary for studying the local phase portrait of a linearly zero singular point.

To make the exposition clearer, we shall apply the most part of the blow ups in the x -direction. So, sometimes, we will first apply a xy -change (x1,y1)(y1,x1)=(x2,y2) , before making the blow-up.

In the next two subsections we will desingularize the origin of the charts U1 and U2 , respectively. We will denote the coordinates of the system in the step i of the algorithm as the variables (wi,zi) , so that after either a wz -change, a translation or a blow up, the new obtained system will be written in the variables (wi+1,zi+1) . In each step, we will denote the system wi̇=Pi(wi,zi) zi̇=Qi(wi,zi) simply as (Pi,Qi) .

Since the Hamiltonian system (1) with the polynomials p and q given by (2) has degree 49 , it follows that for the calculations in each step of the algorithm we have to deal with polynomials of very high degree. So, we persuade these calculations with the algebraic manipulator Mathematica. We do not show in each step the whole expressions of the systems (Pi,Qi) because this would be impractical.

The origin of the chart U1

We write the compactification of system (1) in the chart U1 in the variables (w0,z0) , as (P0,Q0) . From Lemma 4, the singular point (0,0) is linearly zero.

We first apply a wz -change and write the new system in the variables (w1,z1) as (P1,Q1) .

The Newton polygon of system (P1,Q1) has only one compact edge contained in the straight line x+2y=38 . We apply (1,2) -blow ups in the positive w -direction and in the positive and negative z -directions obtaining systems (P2,Q2) and (P2±,Q2±) , in the variables (w2,z2) and (w2±,z2±) , after canceling the common factors w238 and (w2±)38 , respectively. The first terms of these systems have the following expressions:

P2=w2(56250+11252(447w2+1900z2)+),Q2=2812511254(387w2+2000z2)+754(1967w22+24138w2z2+57000z22)+,

and

P2±=w2±(281252+281250(w2±)2+),Q2±=z2±(±14062521350000(w2±)2+).

The only singular point of (P2,Q2) over the line w2=0 is the linearly zero singular point (0,1) . The origin of the systems (P2±,Q2±) are saddles as depicted in the planes w2+z2+ and w2z2 of Fig. 2.

Figure 2 The sequence of blow downs in the study of the origin of the chart U1

The reader can follow a schema of each step of the calculations in Fig. 2. We just need to analyze the origin of the systems (P2±,Q2±) , because the other singularities over the lines z2±=0 will correspond to the singularity (0,1) of (P2,Q2) .

We now analyze this linearly zero singularity. We first do a translation bringing this point to the origin, obtaining the new system (P3,Q3) in the variables (w3,z3) . We also apply a wz -change obtaining the system (P4,Q4) in the variables (w4,z4) . The Newton polygon of this system has two compact edges. We choose the one contained in the straight line x+y=11 . This compact edge has the point of negative abscissa (1,12) , thus, concerning (1,1) -blow ups, it follows from Proposition 3.2 of (Álvarez et al. 2011) that w4 is not a characteristic direction, and so we only need to apply a w -directional (1,1) -blow up, obtaining the system (P5,Q5) in the variables (w5,z5) , after canceling the common factor w511 . The first terms of (P5,Q5) are:

P5=w5(14(w5+z5)(w5+2z5)(442125w55z5+824250w54z52+699990w53z53=+320532w52z54+215904w5z55+112500w56+217160z56)+),Q5=z5(14(w5+z5)(w5+2z5)(442125w55z5+824250w54z52+699990w53z53=+320532w52z54+215904w5z55+112500w56+217160z56)+). (4)

Over the line w5=0 , the singular points of (P5,Q5) are (0,0) and two points of the form (0,z5) , with z5 the two real solutions of

0=z54+70726z53+252941z52+290380z5+108580.

The discriminant of this quartic equation is negative, thus, it has two real solutions. Those are approximately z5=70722.424... and z5=1.6611121... . The singular point (0,0) is linearly zero and the other two singular points are saddles, as represented in the w5z5 -plane of Fig. 2.

Now, we study the linearly zero point (0,0) of (P5,Q5) . It is clear from (4) that the characteristic equation of (P5,Q5) is identically zero, so (0,0) is a dicritical singular point. We apply (1,1) -blow ups in both the w - and z -directions obtaining systems (P6,Q6) and (P6y,Q6y) in the variables (w6,z6) and (w6y,z6y) , after canceling the factors w69 and (z6y)9 , respectively. System (P6y,Q6y) does not have (0,0) as a singular point, so we just need to consider system (P6,Q6) over the line w6=0 . We have

P6(0,z6)=14(z6+1)(2z6+1)(217160z66+215904z65+320532z64+699990z63=+824250z62+442125z6+112500),Q6(0,z6)=14z62(z6+1)(2z6+1)(290380z66+260416z65+421348z64+904140z63=+1032225z62+542250z6+135000).

By using Sturm’s theorem (see for instance (Isaacson and Keller 1994); in the software Mathematica, the Sturm theorem is programed by the instruction CountRoots) we see that the polynomial of degree 6 multiplying (z6+1)(2z6+1)/4 in P6(0,z6) has no real roots, so, the only singular points are (0,1/2) and (0,1) . The first one is a weak focus and the second one is a saddle, as depicted in the plane w6z6 of Fig. 2. Since the origin of (P5,Q5) is dicritical, it follows that each orbit crossing the line w6=0 will correspond to two orbits tending to (0,0) in positive or negative directions.

We now begin the process of blowing down.

It is simple to conclude that the phase portrait of the system (P5,Q5) close to the origin is qualitatively the one depicted in (a) of Fig. 3.

Figure 3 The origin of system (P5,Q5) in ( a ) and the origin of system (P3,Q3) in ( b ).  

Consequently, by considering also the information close to the other two singular points in the line w5=0 (see the plane w5z5 of Fig. 2), we can understand the behavior near the origin of system (P4,Q4) . We then apply a wz -change and conclude that the behavior of system (P3,Q3) near the origin is the one presented qualitatively in (b) of Fig. 3.

By translating (0,0) to (0,1) and by using the information provided by the saddles of planes w2±z2± , we make the blow downs with α=1 and β=2 , obtaining the origin of system (P1,Q1) . We then finally apply a wz -change and conclude that the origin of system (P0,Q0) is qualitatively as drawn in Fig. 4.

Figure 4 The origin of the chart U1 .  

The origin of chart U2

As in the calculations made above, we write the compactified vector field in the chart U2 as (w0̇,z0̇)=(P0,Q0) . The Newton polygon of (P0,Q0) has two compact edges: one of them contained in the straight line 3x+2y=87 . We apply a (3,2) -blow up in the w -direction, obtaining the system (w1̇,z1̇)=(P1,Q1) after canceling the factor w187 . The first terms of P1 and Q1 are:

P1=w1(46875+11250z12(80w147z1)+),Q1=z1(9375+56250z12(2z13w1)+).

The polynomials P1 and Q1 have degree 61 .

It is clear that at (0,0) we have a saddle. The other singular point of (P1,Q1) in the line w1=0 is (0,1) , and it is a linearly zero point. See the w1z1 -plane of Fig. 5. The reader can follow the steps of the calculations in the schema shown in this figure. We just warn that, differently of Fig. 2, we already draw the final phase portrait of each step, including the behavior close to the linearly zero points (information that we will know only after persuading all the blow ups).

Figure 5 The sequence of blow downs in the study of the origin of the chart U2

We also apply (3,2) -blow ups in the positive and negative z -directions, obtaining the systems w1±̇=P1± z1±̇=Q1± , respectively, with linearly zero singular points at (w1±,z1±)=(0,0) . The polynomials P1± and Q1± have degree 30 and Q1± is a factor of z1± .

We do not need to analyze the other singular points over the lines z1±=0 , as the information provided by them is already contained in the w -directional blow up. We desingularize these points applying (1,1) -blow ups in the w -direction. Here, we do not need to apply blow ups in the z -directions because the characteristic equations of the systems are

0=z1±(4500(w1±)5z1±+1650(w1±)4(z1±)27800(w1±)3(z1±)3=+3025(w1±)2(z1±)4500w1±(z1±)5+5625(w1±)6+2501(z1±)6),

and so w1±=0 are not characteristic directions. We obtain the systems (P2±,Q2±) after canceling a factor (w2±)5 . The polynomials P2± and Q2± have degree 45 , and up to order 2 they have the same expressions:

P2±=w2±(281252+10125z2±+),Q2±=z2±(56254500z2±+).

Thus, at (0,0) the systems have a saddle, as depicted in the planes w2±z2± of Fig. 5. Moreover, any other singularity of the form (0,z2±) must satisfy

2501(z2±)6500(z2±)5+3025(z2±)47800(z2±)3+1650(z2±)24500z2±+5625=0.

By using Sturm’s theorem, we conclude that this equation has no real solution.

Now, we desingularize the point (0,1) of system (P1,Q1) . First, we apply a translation to bring this point to the origin, obtaining the system (P2,Q2) in the variables (w2,z2) . We also apply a wz -change obtaining the system (P3,Q3) in the variables (w3,z3) . The Newton polygon of this system has only one compact edge contained in the line x+2y=11 , and this edge has points of negative abscissa, so, concerning (1,2) -blow ups we just need to apply them in the w -direction, according to Proposition 3.2 of (Álvarez et al. 2011). Hence, we apply a (1,2) -blow up in the positive w -direction, obtaining the system (P4,Q4) in the variables (w4,z4) , after canceling a factor of w411 . These polynomials have degree 90 , and their first terms are:

P4=w4(4982259375+9964518754(260w4+z4)+),Q4=z4(3985807500110716875(639w4+2z4)+).

Clearly, (0,0) is a singularity corresponding to a saddle. The other singular point in the line w4=0 is (0,ξ) , where ξ is the only real root of the cubic

c(x)=4x3+216x2+6075x218700,

which is approximately ξ=18.8848.... . This cubic has only one real root because its discriminant is negative. A calculation shows that (0,ξ) is linearly zero. See the plane w4z4 in Fig. 5.

Now, we apply a translation to bring the point (0,ξ) to the origin, obtaining system (P5,Q5) written in the variables (w5,z5) . Since ξ is not a rational number, we do this translation with a parameter x , and thus P5 and Q5 are polynomials in w5 z5 and x . We simplify these polynomials substituting them by the remainder of the division of each of them by c(x) , obtaining so polynomials of degree 2 in x , and hence when we substitute x by ξ , we obtain the same expressions. We keep the notation (P5,Q5) .

The Newton polygon of this system has just one compact edge contained in the line x+y=1 . So, the blow ups here will be homogeneous ones. The characteristic equation of system (P5,Q5) is a multiple of

0=w5(729(23090824x2+532204875x18375684300)w52=216(149x2+1828125x32221800)w5z54(404x2+8325x54675)z52),

with x=ξ . It would thus be enough to apply a (1,1) -blow up in the z -direction, and to study the singularities of the new system in z6=0 (this could evidently also be concluded by observing that the compact edge of the Newton polygon of (P5,Q5) has a point of negative ordinate). We prefer though to apply (1,1) -blow ups in the w and z -directions and to study the singularities of the new systems either in the line w6=0 and in the origin, respectively. The reason why we do this is that the singularities other than the origin are linearly zero and we have to apply new blow ups after persuading a translation. The matter here is that the blow up in the z -direction produces a vector field of degree 158 , while the blow up in the w -direction produces a vector field of degree 109 . Thus, it is simpler to do a translation and after to apply the polynomial remainder in the vector field with smaller degree.

Then, after applying (1,1) -blow ups in either the positive w - and z -directions, we obtain the systems (P6,Q6) and (P6y,Q6y) in the variables (w6,z6) and (w6y,z6y) , after canceling factors w6 and z6y , respectively. The first terms of these systems are:

P6=w6(590494(149x2+1828125x32221800)+),Q6=478296916(23090824x2+532204875x18375684300)=+17714764(5928191012x29644385686625x+179165168144100)w6=+1771472(149x2+1828125x32221800)z6+,

with x=ξ

and

P6y=w6y(65614(404x2+8325x54675)+),Q6y=z6y(2187(404x2+8325x54675)+),

with x=ξ .

The origin of system (P6y,Q6y) is a saddle (see the plane w6yz6y in Fig. 5). On the other hand, the singularities of (P6,Q6) over the line w6=0 are the points (0,z6) , with z6 the real solutions of

0=4(404x2+8325x54675)z62+216(149x2+1828125x32221800)z6=729(23090824x2+532204875x18375684300), (5)

with x=ξ . The discriminant of this quadratic equation is a polynomial in x whose division by c(x) has remainder equal to 0 . This means that the only real solution of (5) is r1=b/(2a) , where a and b are the coefficients of z62 and z6 in (5), respectively. Substituting x by ξ after applying the polynomial remainder again we have

r1=3(2380ξ2+21ξ334440)5989

The point (0,r1) is linearly zero, so, we translate it to the origin obtaining the system (P7,Q7) in the variables (w7,z7) . We again persuade this translation considering r1=r1(x) as a polynomial of x . Again P7 and Q7 will be polynomials in w7 z7 and x . As before we substitute these polynomials by the remainder of the division of them by c(x) , obtaining polynomials of degree 2 in x . We keep the notation P7 and Q7 for them.

The Newton polygon of this system has only one compact edge contained in the straight line x+y=1 . The characteristic equation of this system has w7=0 as a solution.

As above, we apply (1,1) -blow ups in either the positive w - and z -directions, obtaining the systems (P8,Q8) and (P8y,Q8y) in the variables (w8,z8) and (w8y,z8y) , respectively. We then study the origin of (P8y,Q8y) and the singularities of (P8,Q8) over the line w8=0 . The reason is again computational, as the degree of (P8y,Q8y) is 196 and the degree of (P8,Q8) is 128 . The first terms of these systems are:

P8=w8(6561(610023097091x27154910819000x72219849901200)95824+),Q8=590494591119488(866106385697199684752x263678825997496319079125x=+894244583851567110026100)+,

with x=ξ , and

P8y=w8(21872(404x2+8325x54675)+),Q8y=z8(65614(404x2+8325x54675)+),

with x=ξ .

System (P8y,Q8y) has a saddle at the origin (see the plane w8yz8y in Fig. 5), while the singular points of (P8,Q8) over the line w8=0 are the points (0,z8) , with z8 the real roots of

0=2295559744(404x2+8325x54675)z82574944(610023097091x2=7154910819000x72219849901200)z8+27(866106385697199684752x2=63678825997496319079125x+894244583851567110026100),

with x=ξ . The discriminant of this equation is a polynomial in x whose division by c(x) has remainder 0 . Thus, the only real solution is r2=b/(2a) , where a and b are the coefficients of z82 and z8 of the equation, respectively. After applying the polynomial remainder, we substitute x by ξ obtaining

r2=38570325688ξ21361034154573ξ+41691943772820430417452.

The point (0,r2) is linearly zero, so, we translate it to the origin obtaining the system (P9,Q9) in the variables (w9,z9) . As before, we make this translation with the parameter x , so that P9 and Q9 are polynomials in x . Keeping the notation we substitute these polynomials by the remainder of the division of them by c(x) .

As before the Newton polygon of this system has only one compact edge contained in the straight line x+y=1 . Moreover, the characteristic equation does not have z9=0 as a solution. Here, we just apply a (1,1) -blow up in the positive z -direction, obtaining system (P10,Q10) in the variables (w10,z10) , after canceling the factor z10 (here we do not use the superscript y as this is the only system in this step). The degree of this new system is 234 , but as we are going to see, just the origin is a singular point in the line z10=0 . The first terms of P10 and Q10 are:

P10=w10(21874(404x2+8325x54675)+),Q10=z10(21872(404x2+8325x54675)+),

with x=ξ .

Now, over the line z10=0 , the singular points of (P10,Q10) are (0,0) and the points (w10,0) , with w10 the real roots of

0=27(224799605593831132981000196646508x2=+11060763183198622719418769173796625x=289048399074933337876985160926408100)w102=1721669808(375535867201456283x2+10785776535503894250x=338993606077717260600)w10+41168707330260512(404x2+8325x54675),

with x=ξ . The discriminant of this equation after applying the polynomial remainder is

Δ(x)=42795139080321190650757595864867731660278486158784x2=+2546081344010178238089386481604981090589087283168000x=63345629158853164845226783632142224359340633182668800.

It is simple to conclude that Δ(ξ)<0 , thus, only (0,0) is a singular point of (P10,Q10) in z10=0 . This singular point is the saddle depicted in the plane w10z10 of Fig. 5.

Since the behavior near each appearing singular points in each step above is very simple, the blow down of each step is also very simple: following the arrays in Fig. 5, it is easy to conclude that the origin of U2 has a degenerate hyperbolic sector as shown in the w0z0 -plane of Fig. 5.

The global phase portrait

We begin with a background on separatrices and canonical regions of the Poincaré compactification p(𝒳) in the Poincaré disc 𝔻 of a polynomial system ẋ=𝒳(x) . Let φ be the flow of p(𝒳) defined in 𝔻 . As usual we denote by (U,φ) the flow of p(𝒳) on an invariant subset U𝔻 . Two flows (U,φ) and (V,ψ) are said to be topologically equivalent if there exists a homeomorphism h:UV sending orbits of (U,φ) onto orbits of (V,ψ) preserving or reversing the orientation of all the orbits.

Following Markus (1954), we say that the flow (U,φ) is parallel if it is topologically equivalent to one of the following flows: (i) the flow defined in 2 by the system ẋ=1 ẏ=0 ; (ii) the flow defined in 2\{(0,0)} by the system in polar coordinates ṙ=0 θ̇=1 ; and (iii) the flow defined in 2\{(0,0)} by the system in polar coordinates ṙ=r θ̇=0 . Parallel flows topologically equivalent to (i), (ii) and (iii) are called stripannular and spiral (or radial), respectively.

We denote by γx the orbit of p(𝒳) passing through x when t=0 with maximal interval Ix , and the positive (resp. negative) orbit of γx by γx+={γx(t)|tIxandt0} (resp. γx-={γx(t)|tIxandt0} ). Then we set a±(x)=γx±¯\γx± , here as usual γx±¯ denotes the closure of γx± . Observe that a(x) differs from α(x) in the case of periodic orbits and singular points: indeed, a(x)= and α(x)=γx in this case (similarly for a+(x) and ω(x) ). An orbit γx of p(𝒳) is called a separatrix of p(𝒳) if it is not contained in an open neighborhood U such that (U,φ) is parallel and such that both a±(x)=a±(y) for all yU and U¯\U consists of a+(x) a(x) and exactly two orbits γy and γz such that a±(x)=a±(y)=a±(z) .

If 𝒳 is a polynomial vector field it is known that the separatrices of p(𝒳) are (i) the finite and infinite singular points of p(𝒳) ; (ii) the orbits of p(𝒳) contained in the boundary 𝕊1 of 𝔻 ; (iii) the limit cycles of p(𝒳) ; and (iv) the separatrices of the hyperbolic sectors of the finite and infinite singular points of p(𝒳) . Moreover, if p(𝒳) has finitely many finite and infinite singular points and finitely many limit cycles, then p(𝒳) has finitely many separatrices. We call each connected component of the complement of the union of separatrices a canonical region of p(𝒳) . Neumann (1975) proved that each canonical region of a vector field p(𝒳) is parallel.

To the union of the separatrices of p(𝒳) together with an orbit belonging to each canonical region of p(𝒳) we call a separatrix configuration of p(𝒳) . We say that the separatrix configurations S1 and S2 of p(𝒳1) and p(𝒳2) are topologically equivalent if there exists an orientation preserving homeomorphism from 𝔻 to 𝔻 which transforms orbits of S1 onto orbits of S2 . The following is the Markus-Neumann-Peixoto classification theorem (Markus 1954, Neumann 1975, Neumann and O’Brien 1976, Peixoto 1973, Dumortier et al. 2006) for the Poincaré compactification in the Poincaré disc of polynomial systems.

Theorem 5 (Markus-Neumann-Peixoto). Let p(𝒳1) and p(𝒳2) be the Poincaré compactification of two polynomial systems ẋ=𝒳1(x) and ẋ=𝒳2(x) , respectively. The flows of p(𝒳1) and p(𝒳2) on the Poincaré disc are topological equivalent if and only if the separatrix configurations of p(𝒳1) and p(𝒳2) are topological equivalent.

Hence, in order to qualitatively describe the phase portrait on the Poincaré disc of system (1) it is enough to qualitatively describe its separatrix configuration. This was done in Fig. 1, where we have drawn the separatrices other than singular points with bold lines. The other lines are orbits contained in its respective canonical regions. We observe from Fig. 1 that system (1) has 15 separatrices, five of them singular points, and 7 canonical regions, six of them of type strip and the one formed by the closed orbits surrounding z0 , annular.

Below, we prove Theorem 3 by proving that Fig. 1 is a separatrix configuration of system (1).

From the previous sections we conclude that close enough to the singular points, the phase portrait of system (1) is qualitatively the one presented in Fig. 6.

Figure 6 The phase portrait of system (1) near the singular points.  

For further references we label the hyperbolic, parabolic and elliptic sectors presenting in the origins of the charts U1 and V1 in Fig. 6 as h1 h2 h3 h4 p1 p2 and e1 e2 , respectively.

From the definition of system (1), each of its orbits is a connected component of a level set of HF=(p2+q2)/2 (because the only singular point of this system is the center z0 ), which in turn is the inverse image under F=(p,q) of circles surrounding the point (0,0) . Since F preserves orientation (because the Jacobian determinant of F is positive), each orbit of (1) is carried onto a curve contained in a circle with counterclockwise orientation. As we have seen, the curve β(s) defined in (3) is the asymptotic variety of F . Moreover, the points β(0)=(1,995/4) and β(1)=(0,208) of this curve have no inverse image under F , all the other points of this curve have exactly one inverse image and the other points of 2 have precisely two inverse images. Acting as Campbell (arXiv:math/9812032 1998), we delete from the curve β(s) the points β(0) and β(1) , obtaining three curves: C1=β(,0) C2=β(0,1) and C3=β(1,) . According to Campbell (1998), the inverse image under F of each Ci is a curve that divides the plane into two connected components. We call Di the inverse image of Ci i=1,2,3 . The set D1D2D3 is called by Campbell (1998) the asymptotic flower of F . It follows that 2\(D1D2D3) is formed by 4 connected components, each of them mapped twice onto each of the two connected components of 2\{β(s)} . Each curve Ci has a natural orientation, given by its parametrization (it is the opposite orientation used by Campbell (1998)). So, each curve Di also has a natural orientation (recall that F preserves orientation). The graphics of Ci and Di i=1,2,3 , are given in (a) and (b) of Fig. 7, respectively. As in (Campbell 1998, 2011) the axes in (a) have different scales. Following Campbell (1998), we label the regions as R (right) and L (left) of the curves Ci and Di .

Figure 7 The asymptotic variety of F in ( a ) and the asymptotic flower of F in ( b ).  

Since for each s

β(s)β(s)=14(1s)s(s+1)(112500s6232875s5+301125s4425760s3=+432312s286565s+116423),

and this polynomial of degree 6 multiplying (1s)s(s+1) has no real zeros by Sturm’s theorem, it follows that the curves C1 C2 and C3 are transversal to the circles centered at (0,0)=β(1) . As a consequence the curves D1 D2 and D3 are transversal to the non-singular orbits of system (1). In particular, the image of a non-periodic orbit of system (1) has α - and ω -limits contained in the curve β(s) . Below, we will say that the image of an orbit starts or finishes at β(s0) meaning that its α - or ω -limit is β(s0) , respectively. Moreover, through each point in the intersection of C1C2C3 with a circle, it crosses exactly one image of an orbit of system (1).

We call S1 and S2 the circles centered at (0,0) and containing the points β(1) and β(0) , respectively.

The point z0 , being the inverse image under F of (0,0)=β(1) , is contained in the curve D1 . The images under F of the closed orbits surrounding z0 are circles surrounding (0,0) contained in the bounded region defined by S1 . Thus the boundary of the period annulus of the center z0 corresponds to the arc of circle contained in S1 , starting and finishing at the point β(1) . This means that the boundary of the period annulus is an orbit that goes to infinity through the region labeled by L in (b) of Fig. 7. In particular, in the Poincaré disc, this orbit tends to the origin of the chart V1 . Then, analyzing the possibilities in Fig. 6, we see that this orbit contains the two separatrices of the hyperbolic sector h2 . This period annulus is an annular canonical region.

Now we analyze the parabolic sectors p1 and p2 .

Close to the two points of D1 cut by the orbit giving the boundary of the period annulus of the center (i.e., the orbit connecting the two separatrices of the hyperbolic sector h2 ), and outside the period annulus, there must exist orbits cutting D1 . Analyzing the images of these orbits, they are contained in circles surrounding the circle S1 . So, there are two possibilities for the images of these orbits: either they are arcs starting and finishing at a point of the curve C2 , or they are arcs starting at the curve C3 and finishing at the curve C2 or C3 . At a first glance both of these possibilities are compatible with the parabolic sectors p1 and p2 in Fig. 6. We claim that the correct possibility is the first one. Indeed, we can increase the radii of these circles containing the images of the orbits of p1 and p2 until we achieve the circle S2 . If we are in the second possibility, the orbit whose image is contained in S2 and starts at a point of C3 will contain the separatrice of the end of the parabolic sector p2 . But, this orbit will not contain the separatrice of the end of the parabolic sector p1 , because we can continue drawing arcs starting at C3 with radii bigger than the radius of S2 . Thus, the parabolic sector p1 will not finish, a contradiction with the nature of the vector field at the origin of the chart V1 , as shown in Fig. 6. This proves the claim.

So, the image of the orbits of the parabolic sectors p1 and p2 are arcs starting and finishing at a point of the curve C2 . And since we can continue drawing these arcs until we arrive at circle S2 , this means that the parabolic sectors p1 and p2 are connected, and the image of the orbit containing the separatrices that separate p1 from h1 and p2 from h3 is contained in the arc of S2 starting and finishing at the point β(0) . The region connecting p1 to p2 is a strip canonical region, see Fig. 1.

Now, since the image of this last orbit cuts the curves C3 and C1 , there must exist orbits near it whose images cross C3 and C1 . The only possibility is that those images are arcs of circles starting at the curve C1 , rotating a complete turn crossing C3 and C1 and continue up to finishing in the curve C3 . We call these orbits the big orbits. A big orbit whose image is contained in a circle close enough to S2 enters both the hyperbolic sectors h1 and h3 . We have to see where the big orbits start and finish.

The orbits whose images are arcs of the circles with radii smaller than the radius of S2 , starting and finishing at C1 and contained in the region R correspond to an elliptic sector with boundary formed by an orbit having image contained in the arc of S2 starting at C1 an finishing at β(0) . Close to this boundary and out of the elliptic sector there must exist orbits whose images start at C1 . These orbits are the big orbits. Hence, it follows that this elliptic sector is e1 and that the big orbits start at the origin of V1 , see Fig. 1.

The orbits whose images are arcs starting and finishing at C1 and contained in the region L form the elliptic sector e2 . Clearly its boundary is formed by the two orbits containing the separatrices of the hyperbolic sector h4 . The image of these orbits are the arcs starting at C1 and finishing at β(1) and starting at β(1) and finishing at C1 , respectively.

In particular, this means that the big orbits must finish at the origin of the chart U1 , below the hyperbolic sector h4 . Since their images are contained in the circles bigger than S2 , there exist orbits whose images are arcs contained in the circles between S1 and S2 , starting at C1 , crossing C2 and finishing at C3 . These orbits produce a parabolic sector between h3 and e2 , and give rise to a strip canonical region as presented in Fig. 1.

The big orbits also produce a strip canonical region.

The orbits of the strip canonical region placed above the hyperbolic sector h4 have their images contained in arcs of circles with radii bigger than the radius of S1 , starting at C3 and finishing at C1 .

The elliptic sectors e1 and e2 form another two strip canonical regions.

Hence we have 7 canonical regions, six of them are strip and one is annular. Analyzing Fig. 1, we see there are 6 finite orbits that are separatrices. The infinite has another 4 orbits. Hence, since there are 5 singular points, we have 15 separatrices in the separatrix configuration of system (1) in the Poincaré disc.

Acknowledgments

The first and third authors are partially supported by the Ministerio de Economia y Competitividad (MINECO) grant number MTM2013-40998-P, the Agència de Gestió d’Ajuts Universitaris i de Recerca (AGAUR) grant number 2014SGR 568 and the grants from European Commission (FP7-PEOPLE-2012-IRSES) numbers 316338 and 318999. The second author is partially supported by Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP), grant number 2014/ 26149-3. The second and third authors are also partially supported by Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES), grant 88881. 030454/ 2013-01 from the program CSF-PVE.

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Received: October 22, 2017; Accepted: December 13, 2017

Correspondence to: Francisco Braun E-mail: franciscobraun@dm.ufscar.br

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