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10.1007/s00010-024-01051-7
Article
Global centers of a family of cubic systems
Appis Raul Felipe 1
Llibre Jaume jaume.llibre@uab.cat

2
1 https://ror.org/00qdc6m37 grid.411247.5 0000 0001 2163 588X Departamento de Matemática, Universidade Federal de São Carlos, São Paulo, São Carlos 13565-905 Brazil
2 https://ror.org/052g8jq94 grid.7080.f 0000 0001 2296 0625 Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia Spain
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Consider the family of polynomial differential systems of degree 3, or simply cubic systems x′=y,y′=-x+a1x2+a2xy+a3y2+a4x3+a5x2y+a6xy2+a7y3,

in the plane R2. An equilibrium point (x0,y0) of a planar differential system is a center if there is a neighborhood U of (x0,y0) such that U\{(x0,y0)} is filled with periodic orbits. When R2\{(x0,y0)} is filled with periodic orbits, then the center is a global center. For this family of cubic systems Lloyd and Pearson characterized in Lloyd and Pearson (Comput Math Appl 60:2797–2805, 2010) when the origin of coordinates is a center. We classify which of these centers are global centers.

Keywords

Global center
Center
Cubic system
Mathematics Subject Classification

Primary 34A34
Secondary 34C25
37C37
14R15
Conselho Nacional de Desenvolvimento Científico e Tecnol´ogico200950/2022-3 Appis Raul Felipe Agencia Estatal de Investigación, SpainPID2019-104658GB-I00 Llibre Jaume H2020 European Research CouncilMSCA-RISE-2017-777911 Llibre Jaume AGAUR (Generalitat de Catalunya)2021SGR00113 Llibre Jaume Acadèmia de Ciències i Arts de Barcelona1 Llibre Jaume Universitat Autònoma de BarcelonaOpen Access Funding provided by Universitat Autonoma de Barcelona.

issue-copyright-statement© Springer Nature Switzerland AG 2024
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pmcIntroduction and statement of the main result

Let P,Q:R2⟶R be polynomials and consider the differential system1 x′=P(x,y),y′=Q(x,y).

Denote by X(x,y)=(P(x,y),Q(x,y)) the vector field associated to the differential system (1). The degree d of system (1) is the maximum of the degrees of the polynomials P and Q. Here the apostrophe denotes derivative with respect to the time t. A point (x0,y0) is an equilibrium point of system (1) if X(x0,y0)=(0,0).

An equilibrium point (x0,y0) of system (1) is a center if there is a simply connected open neighborhood W of (x0,y0) such that (x0,y0) is the only equilibrium point in W and all the trajectories contained in W\{(x0,y0)} are periodic. The largest simply connected open neighborhood P of (x0,y0) such that P\{(x0,y0)} is filled of periodic trajectories is called the period annulus. When P=R2 the point (x0,y0) is a global center.

Dulac [5] and Poincaré [13] were the first in studying the centers of the differential systems in the plane. While Conti [4] was the first in studying the global centers.

To classify the centers of polynomial differential systems as well as to determine the necessary and sufficient conditions to know whether a center is global are in general difficult problems.

Kapteyn [8] and Bautin [2] classified the centers of the polynomial differential systems of degree 2, the quadratic centers. For degrees higher than 2 the classification of all centers remain unsolved.

Galleotti and Villarini [7] proved that polynomial differential systems of even degree do not admit global centers, a shorter proof was given in [10].

The classification of the global centers for homogeneous polynomial differential systems is well known, see for instance [3]. The classification of the global centers of quasi-homogeneous polynomial differential systems is also well studied, see [9].

In this paper we classify the global centers of the following family of cubic polynomial differential systems2 x′=y,y′=-x+a1x2+a2xy+a3y2+a4x3+a5x2y+a6xy2+a7y3.

Lloyd and Pearson in [12] classified when the origin of coordinates of systems (2) is a center. This result is stated as follows.

Theorem 1

The origin of system (2) is a center if and only if one of the following five conditions holds: (i) a2=a5=a7=0;

(ii) a1=a3=a5=a7=0;

(iii) a4=a3(a1+a3), a5=-a2(a1+a3) and (a1+2a3)a6+a32(a1+a3)=a7=0;

(iv) a5+3a7+a2(a1+a3)=0, 9a6a22+2a24+27a7μ+9μ2=0, a4a22+a5μ=0, (3a7μ+μ2+a6a22)a5-3a7μ2-a6a22μ=0 where μ=3a7+a2a3;

(v) a5+3a7+a2(a1+a3)=0, 18a4a5-27a4a7+9a5a12+9a5a6+2a5a22=0, 27a4a1+4a5a2+9a13+2a1a22=0, 18a42+9a4a12+2a4a22+2a52=0, 18a4a2+9a5a1+9a5a3+9a12a2-27a1a7+9a6a2+2a23=0.

The next two results help for classifying the global centers of the cubic polynomial differential systems (2).

Proposition 2

If the origin of a differential system (2) is a global center, then a7=0.

Proposition 3

A differential system (2) has the unique equilibrium point (0, 0) if and only if either a1=a4=0, or a12+4a4<0.

In the following result we classify the global centers of system (2). In what follows from Proposition 2 we assume that a7=0, and from Proposition 3 that a4≤0.

Theorem 4

Assume that the unique equilibrium of the differential system (2) is the (0, 0). Under the condition (i) of Theorem 1 the cubic polynomial differential system (2) has a global center if and only if one the following two conditions holds: a6<0;

a3=a6=0 , a4<0.

Under the condition (ii) of Theorem 1, the cubic polynomial differential system (2) has a global center if and only if one the following three conditions holds: (c) a4=0 and a22+4a6<0;

(d) a4<0 and a6<0;

(e) a6=0 and a22+8a4<0.

Under the conditions (iii), (iv) and (v) of Theorem 1, the classification of the global center of system (2) is reduced to conditions (i) and (ii).

In Sect. 2 we present some tools that we need for proving Theorem 4. In Sect. 3 we initially prove Propositions 2 and 3 and after we prove Theorem 4.

Preliminary results

The Poincaré compactification

To determine conditions in order that a center of a polynomial differential system in R2 be global, we need to study the behavior of the flow at infinity, so we recall the Poincaré compactification of a polynomial differential system (1), essential for the study of the dynamics in a neigborhood of the infinity of the polynomial differential systems.

Let R2≡{(x1,x2,1);x1,x2∈R} and the sets H+={(x1,x2,x3)∈S2;x3>0}, H-={(x1,x2,x3)∈S2;x3<0} and S1≡{(x1,x2,x3)∈S2;x3=0}, where S2={(x1,x2,x3)∈R3;x12+x22+x32=1}. In order to study a vector field over S2 we consider six local charts that cover the whole sphere S2. So, for i=1,2,3, letUi={(x1,x2,x3)∈S2;xi>0}andVi={(x1,x2,x3)∈S2;xi<0}.

Consider the diffeomorphisms φi:Ui⟶R2 and ψi:Vi⟶R2 given byφi(x1,x2,x3)=ψi(x1,x2,x3)=xjxi,xkxi

with j,k≠i and j<k. The sets (Ui,φi) and (Vi,ψi) are called the local charts over S2.

Let f±:R2⟶H± be the central projections from R2 to S2 given byf±(x1,x2)=±x1Δ(x1,x2),x2Δ(x1,x2),1Δ(x1,x2)

where Δ(x1,x2)=x12+x22+1. In other words f±(x1,x2) is the intersection of the straight line through the points (0,0,0),(x1,x2,1)∈R3 with H±. Note that f+=φ3-1 and f-=ψ3-1. Moreover, the maps f± induces over H± vector fields analytically conjugate to system (1). Indeed, f+ induces on H+=U3 the vector field X1(y)=Df+(φ3(y))X(φ3(y)), and f- induces on H-=V3 the vector field X2(y)=Df-(ψ3(y))X(ψ3(y)). Thus we obtain a vector field on S2\S1 that admits an analytic extension p(X) on S2, see for more details [6, chapter 5]. The vector field p(X) is called the Poincaré compactification.

Denote (u,v)=φi(x1,x2,x3)=ψi(x1,x2,x3). The expression of p(X) in the chart U1 isu′=vdQ1v,uv-uP1v,uv,v′=-vd+1P1v,uv.

The expression of p(X) in U2 isu′=vdPuv,1v-uQuv,1v,v′=-vd+1Quv,1v.

The expression of p(X) in U3 isu′=P(u,v),v′=Q(u,v).

For i=1,2,3 the expression of p(X) in the chart Vi differs of the expression in Ui only by the factor (-1)d-1.

Note that we can identify the infinity of R2 with the set S1. Two points for each direction in R2 provide two antipodal points of S1. An equilibrium point of p(X) on S1 is called infinite equilibrium point and an equilibrium point on S2\S1 is called a finite equilibrium point. Observe that the infinite equilibrium points are in correspondence with the points (u, 0) on the charts U1,V1,U2 and V2. Thus, if (x1,x2,0)∈S1 is an infinite equilibrium point, then your antipode (-x1,-x2,0) is also a infinite equilibrium point.

Denote by Pi and Qi the homogeneous parts of degree i of the polynomials P and Q, respectively. Consider the polynomials3 F(s)=Qd(1,s)-sPd(1,s)andG(s)=Pd(s,1)-sQd(s,1).

So a point (s,0)∈S1∩(U1∪V1) is an infinite equilibrium point if and only if F(s)=0. Analogously (s,0)∈S1∩(U2∪V2) is an infinite equilibrium point if and only if G(s)=0. Note that, if (s,0)∈U1∪V1, then4 Dp(X)(s,0)=F′(s)Qd-1(1,s)-sPd-1(1,s)0-Pd(1,s),

and if (s,0)∈U2∪V2, then5 Dp(X)(s,0)=G′(s)Pd-1(s,1)-sQd-1(s,1)0-Qd(s,1).

The vertical homogeneous blow-up

Let (x0,y0) be an equilibrium point of system (1). Denote by λ1 and λ2 the eigenvalues of the Jacobian matrix DX(x0,y0). It is said that (x0,y0) is hyperbolic if λ1 and λ2 have no zero real part;

(x0,y0) is semi-hyperbolic if λ1λ2=0 and λ12+λ22≠0;

(x0,y0) is nilpotent if λ1=λ2=0 and the matrix DX(x0,y0) is not the zero matrix;

(x0,y0) is linearly zero if the matrix DX(x0,y0) is the zero matrix.

The hyperbolic and semi-hyperbolic equilibrium points are also called elementary equilibrium points

In the following we present an important technique for determining the local phase portrait around an equilibrium point when it is neither hyperbolic, nor semi-hyperbolic. This method determine the local phase portrait of an equilibrium point using changes of variables called vertical blow-ups. The idea of a blow-up is to turn a non-elementary equilibrium point into a vertical straight line and study the phase portrait in a neighborhood of this straight line, applying a new blow-up to the equilibrium points which appear on this straight line if necessary. In general, such equilibrium points are less degenerate. For more details see [6, chapter 3].

We considerP(x,y)=Pm(x,y)+⋯,Q(x,y)=Qn(x,y)+⋯

in system (1), where Pm and Qn are homogeneous polynomials of degree m≥1 and n≥1 respectively, and the dots mean higher order terms in x and y of m in the polynomial P and of n in the polynomial Q. Consider the polynomialF(x1,x2)=xQm(x1,x2)-yPm(x1,x2)ifm=n-yPm(x1,x2)ifm<nxQn(x1,x2)ifn<m.

The homogeneous polynomial F is called the characteristic polynomial of system (1) and the straight lines through the origin defined by the real linear factors of the polynomial F are called the characteristic directions at the origin, see for more details [1].

The vertical blow-up is the changes of variables (x1,x2)⟶(u1,u2) where (x1,x2)=(u1,u1u2). The new system in the variables u1 and u2 is given by6 u1′=x1′=P(u1,u1u2),u2′=x1x2′-x1′x2x12=Q(u1,u1u2)-u2P(u1,u1u2)u1.

Note that the vertical blow-up is a diffeomorphism of R2\{(0,x2)} to R2\{(0,u2)} that swaps the second quadrant for the third quadrant, and vice versa.

The following result establishes relationships between the equilibrium at the origin of system (1) and the equilibrium points on the line u1=0 of system (6), for more details see [1].

Theorem 5

Let φ be a trajectory of the differential system (1) tending to origin when t⟶+∞ (or t⟶-∞) tangent to one of the two directions θ determined by tanθ=w≠±∞. Assume that F≢0. Then (i) the straight line x1,wx1 is a characteristic direction;

(ii) the point (u1,u2)=0,w is a equilibrium point of system (6) and

(iii) a trajectory φ as in the hypothesis is in biunivocal correspondence with a trajectory of system (6) tending to an equilibrium point 0,w.

The next result provides necessary and sufficient conditions in order that a polynomial differential system in the plane R2 has a global center, for a proof see [11].

Proposition 6

A polynomial differential system in R2 without a line of equilibrium points at infinity has a global center if and only if it has a unique finite equilibrium point which is a center and all the local phase portraits of the infinite equilibrium points are formed by two hyperbolic sectors having all of them both separatrices on the infinite circle S1.

Proofs

Proof of Proposition 2

We have from (2) and (3) thatG(s)=-s(a4s3+a5s2+a6s+a7).

Thus the origin of the chart U2 is always an infinite equilibrium point and G′(0)=-a7. If a7≠0 then, from (5) and Theorem 2.15 of [6], the origin of the local chart U2 is a hyperbolic node with eigenvalues -a7 of multiplicity two. Therefore the origin of (2) cannot be global center because there are trajectories going or coming from the origin of the local chart U2. Therefore a7=0. □

Proof of Proposition 3

The equilibrium points of system (2) are of the form (x, 0) where x is a real root of the polynomialh(x)=x(-1+a1x+a4x2).

Therefore, the origin is the unique equilibrium point of system (2) if and only if the polynomial h(x) has no nonzero roots if and only if either a1=a4=0. or a12+4a4<0. □

Proof of Theorem 4

By Proposition 2 we can assume a7=0. From the result of Proposition 3 we divide the proof into two cases.

Case 1. a1=a4=0.

Suppose that statement (i) of Theorem 1 holds. Then a6≠0, otherwise the differential system (2) would have degree 2, and consequently cannot have a global center. System (2) in the chart U2 writesu′=u2v2-a6u2-a3uv+v2,v′=v(uv2-a6u-a3v).

Note that u=0 is not a characteristic direction at the origin of U2. Doing the vertical blow up (u,v)=(u1,u1v1) and eliminating with a rescaling of the time the common factor u1 between u1′ and v1′ we obtainu1′=P1(u1,v1)=u1(u12v12+v12-a3v1-a6),v1′=Q1(u1,v1)=-v13,

with the Jacobian matrixD(P1,Q1)(0,0)=-a6000.

As Q1(0,v1)=-v13 it follows, by Theorem 2.19 of [6] that, if a6>0, then (u1,v1)=(0,0) is a semi-hyperbolic node, and consequently system (2) cannot have a global center because there are trajectories of system (2) going or coming from the infinity. If a6<0, then (u1,v1)=(0,0) is a semi-hyperbolic saddle. Going back through the vertical blow up we conclude that the origin of U2 is formed by two hyperbolic sectors having both separatrices at infinity, see Fig. 1.Fig. 1 Blow up of the equilibrium point (0, 0) of the local chart U2 in case 1

Now system (2) in the chart U1 becomesu′=-u2v2+a3u2v+a6u2-v2,v′=-uv3.

Since u=0 is not a characteristic direction at the origin of U1, doing the vertical blow up (u,v)=(u1,u1v1) and eliminating the common factor u1 between u1′ and v1′ we obtainu1′=P1(u1,v1)=u1(-u12v12+a3u1v1-v12+a6),v1′=Q1(u1,v1)=v1(v12-a3u1v1-a6),

withD(P1,Q1)(0,0)=a600-a6.

If a6<0, then (u1,v1)=(0,0) is a hyperbolic saddle. Going back through the vertical blow up we obtain that the origin of U1 is formed by two hyperbolic sectors having both separatrices at infinity, see Fig. 2.Fig. 2 Blow up of the equilibrium point (0, 0) of the local chart U1 in case 1

In summary, by Proposition 6 system (2) has a global center, and statement (a) of Theorem 4 is proved.

Assume that statement (ii) of Theorem 1 holds. We have a6≠0, otherwise the differential system (2) would be quadratic. System (2) in the chart U2 isu′=u2v2-a2u2v-a6u2+v2,v′=uv(v2-a2v-a6).

Observe that v=0 is the only characteristic direction at the origin of U2. Doing the vertical blow up (u,v)=(u1,u1v1) and eliminating the factor u1 it followsu1′=P1(u1,v1)=u1(u12v12-a2u1v1+v12-a6),v1′=Q1(u1,v1)=-v13,

andD(P1,Q1)(0,0)=-a6000.

If a6>0, then (u1,v1)=(0,0) is a semi-hyperbolic node, and then system (2) cannot have a global center. If a6<0, then (u1,v1)=(0,0) is a semi-hyperbolic saddle, and going back through the vertical blow up we conclude that the origin of U2 is formed by two hyperbolic sectors, see Fig. 1.

System (2) in the chart U1 isu′=-u2v2+a6u2+a2uv-v2,v′=-uv3.

Since u=0 is not a characteristic direction at the origin of U1, doing the vertical blow up (u,v)=(u1,u1v1) and eliminating the factor u1 we get the systemu1′=P1(u1,v1)=u1(-u12v12-v12+a2v1+a6),v1′=Q1(u1,v1)=v1(v12-a2v1-a6).

This differential system on the straight line u1=0 has the equilibria (0, 0) and (0,(a2±a22+4a6)/2) if a22+4a6≥0. By Theorem 2.15 of [6] the equilibrium (0, 0) is always a hyperbolic saddle.

When a22+4a6>0 by Theorem 2.19 of [6] the two equilibria (0,(a2±a22+4a6)/2) are semi-hyperbolic saddle-nodes, so the differential system cannot have a global center.

If a22+4a6=0, then doing blow ups the local phase portrait of the equilibrium point (0,a2/2) is formed by two hyperbolic sectors separatec by two parabolic sectors, so again the differential system cannot have a global center.

Then going back through the vertical blow ups, we conclude that the origin of U1 is formed by two hyperbolic sectors when a22+4a6<0, see for instance Fig. 2. Therefore statement (c) of Theorem 4 is proved.

If statement (iii) of Theorem 1 holds, then a3=a5=0 and the study comes down to statement (ii).

Now, suppose that either statement (iv) or (v) of Theorem 1 holds. If a2=0, then a5=0 and consequently the study comes down to statement (i). If a2≠0, then a3=a5=0 and the study comes down to statement (ii).

Case 2. a12+4a4<0.

Assume that statement (i) of Theorem 1 holds. We have from (3) that F(s)=a4+a6s2. If a6>0 andp±=±-a4a6,

then (p±,0) are equilibrium points at infinity in the chart U1 with F′(p±)=2a6p±≠0. So from (4) we have that (p±,0) are semi-hyperbolic saddles, nodes or saddle-nodes. Then system (2) cannot have a global center. So we can suppose that a6≤0, and consequently there are no infinite equilibrium points in the chart U1. System (2) in the chart U2 becomesu′=-a4u4-a1u3v+u2v2-a6u2-a3uv+v2,v′=v(-a4u3-a1u2v+uv2-a6u-a3v).

Since u=0 is not a characteristic direction at the origin of U2, doing the vertical blow up (u,v)=(u1,u1v1) and eliminating the factor u1 we obtain7 u1′=P1(u1,v1)=u1(u12v12-a1u12v1-a4u12+v12-a3v1-a6),v1′=Q1(u1,v1)=-v13,

andD(P1,Q1)(0,0)=-a6000.

If a6<0, then (u1,v1)=(0,0) is a semi-hyperbolic saddle. Going back through the vertical blow up we conclude that the origin of U2 is formed by two hyperbolic sectors and consequently follows statement (a), see Fig. 1.

Now suppose a6=0. Then both coordinate axes u1=0 and v1=0 are characteristic directions. Doing the twist (u2,v2)=(u1+v1,v1), that translates the direction u1=0 to u2=v2, system (7) becomes8 u2′=u23v22-3u22v23+3u2v24-v25-a1u23v2+3a1u22v22-3a1u2v23+a1v24-a4u23+3a4u22v2+(1-3a4)u2v22+(a4-2)v23-a3u2v2+a3v22,v2′=-v23.

First suppose a3≠0. Then doing the vertical blow up (u2,v2)=(u3,u3v3) in (8) and eliminating the common factor u3 we obtain9 u3′=P3(u3,v3)=u3[-u33v35+3u33v34-3u33v33+a1u32v34+u33v32-3a1u32v33+3a1u32v32+(a4-2)u3v33-a1u32v3+(1-3a4)u3v32+3a4u3v3+a3v32-a4u3-a3v3],v3′=Q3(u3,v3)=-v3(v3-1)[-u33v34+2u33v33-u33v32+a1u32v33-2a1u32v32+a1u32v3+(a4-2)u3v32-2a4u3v3+a4u3+a3v3].

So the unique equilibrium points of system (9) in the v3-axis are (0, 0) and (0, 1). SinceD(P3,Q3)(0,1)=000-a3

the equilibrium (0, 1) is a semi-hyperbolic saddle-node. Consequently system (2) cannot have a global center.

Now assume that a3=0. Then we can eliminate another common factor u3 in system (9) and we have10 u3′=P3(u3,v3)=u3[-u32v35+3u32v34-3u32v33+a1u3v34+u32v32-3a1u3v33+3a1u3v32+(a4-2)v33-a1u3v3+(1-3a4)v32+3a4v3-a4],v3′=Q3(u3,v3)=-v3(v3-1)[-u32v34+2u32v33-u32v32+a1u3v33-2a1u3v32+a1u3v3+(a4-2)v32-2a4v3+a4].

Then the equilibrium points (0,v3) of system (10) are determined by the zeros of the polynomialv3(v3-1)p(v3)=0,

where p(v3)=(a4-2)v32-2a4v3+a4. Since a4<0 and the discriminant of p is 8a4 it follows that (0, 0) and (0, 1) are the unique equilibrium points of system (10), withD(P3,Q3)(0,0)=-a400a4andD(P3,Q3)(0,1)=-1002.

Therefore both equilibrium points are hyperbolic saddles. Going back through the changes of variables we obtain that the origin of the chart U2 is formed by two hyperbolic sectors. Therefore statement (b) is proved (Fig. 3).Fig. 3 Blow up’s of the equilibrium point (0, 0) of the local chart U2 under the case 2 with a3=a6=0

Suppose (ii) holds in Theorem 1. System (2) in the chart U2 writesu′=-a4u4+u2v2-a2u2v-a6u2+v2,v′=uv(-a4u2+v2-a2v-a6).

Now, doing the vertical blow up (u,v)=(u1,u1v1) and eliminating the common factor u1 we getu1′=P1(u1,v1)=u1(u12v12-a4u12-a2u1v1+v12-a6),v1′=Q1(u1,v1)=-v13,

andD(P1,Q1)(0,0)=-a6000.

Therefore if a6>0, then (u1,v1)=(0,0) is a semi-hyperbolic node, and consequently the center of system (2) cannot be global. So, since a4<0 and a6≤0, this implies that there are no infinite equilibrium points on the local chart U1.

If a6<0, then (u1,v1)=(0,0) is a semi-hyperbolic saddle and going back through the vertical blow up we have that the origin of U2 is formed by two hyperbolic sectors, see Fig. 1. Then statement (d) follows.

Assume a6=0. Doing the change of variables (u2,v2)=(u1+v1,v1) and then doing the vertical blow up (u2,v2)=(u3,u3v3) and eliminating the commun factor u32 we obtain11 u3′=P3(u3,v3)=-u3[u32v35-3u32v34+3u32v33-u32v32+(a2-a4+2)v33+(3a4-2a2-1)v32+(a2-3a4)v3+a4],v3′=Q3(u3,v3)=v3(v3-1)[u32v34-2u32v33+u32v32+(a2-a4+2)v32+(2a4-a2)v3-a4].

The equilibrium points of system (11) are (0, 0), (0, 1) and the points (0,v3) such that v3 be a real zero of the polynomialq(v3)=(a2-a4+2)v32+(2a4-a2)v3-a4=0.

The points (0, 0) and (0, 1) are hyperbolic saddles becauseD(P3,Q3)(0,0)=-a400a4andD(P3,Q3)(0,1)=-1002.

First we assume that a2=a4-2. Suppose a4≠-2. Thenp0=0,a4a4+2

is an equilibrium point of system (11) withD(P3,Q3)(p0)=-a42(a4+2)200-2a4a4+2.

If a4<-2, then p0 is a hyperbolic stable node and system (2) cannot have a global center. If -2<a4<0, then p0 is a hyperbolic saddle. However, going back through the change of variables there are trajectories that tend to the origin of U2, see Fig. 4. Hence the center of system (2) cannot be global.Fig. 4 Blow-up’s of the equilibrium point (0, 0) of the local chart U2 under the case 2 with a2=a4-2 and -2<a4<0

When a4=-2, the unique infinite equilibrium points of system (11) are (0, 0) and (0, 1). Going back through the changes of variables it follows that the origin of U2 has an elliptic sector, so the differential system cannot have a global center, see Fig. 4.

Suppose a4≠a2+2. The discriminant of the polynomial q is a22+8a4.

Assume a22+8a4=0. Then, since a4<0 we have that a2≠-4,0, andq0=0,a2a2+4

is an equilibrium point of system (11) withD(P3,Q3)(q0)=-a22(a2+4)2000.

Note that the vector field over the line u3=0 is given by(P3(0,v3),Q3(0,v3))=0,18v3(v3-1)[(a2+4)v3-a2]2.

Consequently, q0 is a semi-hyperbolic saddle-node and system (2) cannot have a global center, because going back through the changes of variables there are orbits which go or come from the origin of the local chart U2.

Now assume that a22+8a4>0. So a2≠0 because a4<0. Thenp±=0,a2-2a4±a22+8a42(a2-a4+2)

are equilibrium points of system (11). Denote a22+8a4=b2 with b>0. ThenD(P3,Q3)(p-)=-(a2-b)2(a2+b+4)23200b(a2-b)(a2+b+4)28,

where -(a2-b)2(a2+b+4)2/32<0.

If a2<0 it follows that b(a2-b)(a2+b+4)2/8<0, and then p- is a hyperbolic stable node and system (2) cannot have a global center.

If a2>0, then we haveD(P3,Q3)(p+)=-(a2+b)2(a2-b+4)23200-b(a2+b)(a2-b+4)28,

with -(a2+b)2(a2-b+4)2/32<0 and -b(a2+b)(a2-b+4)2/8<0. Thus p+ is a hyperbolic stable node and, consequently system (2) cannot have a global center.

Finally assume that a22+8a4<0. Then (0, 0) and (0, 1) are the unique equilibrium points, because the polynomial q is positive. Going back through the changes of variables we obtain that the origin of U2 is formed by two hyperbolic sectors. Therefore this completes the proof of statement (e).

Observe that statement (iii) of Theorem 1 cannot hold in Case 2, otherwise a12+4a4=(a1+2a3)2≥0, in contradiction with the fact that a12+4a4<0.

Assume that statement (iv) holds in Theorem 1. If a2≠0 we obtaina6=-2a22+9a329,a4=a3(a1+a3)anda1=-a3(9a32+4a22)2a22.

Therefore 81a36/(4a24)=a12+4a4<0, a contradiction.

Assume now that a2=0. Then the study boils down to studying global centers under condition (i).

Suppose that statement (v) in Theorem 1 holds. Assume a2≠0. Then a1=-a3, a5=a6=0, a4=-(2a22+9a32)/18 and a3(2a22+9a32)/2=0. Hence a3=0 and the only infinite equilibrium point is the origin of the chart U2. System (2) in the chart U2 isu′=a22u49+u2v2-a2u2v+v2,v′=uv(a22u2+9v2-9a2v)9.

Doing the vertical blow up (u,v)=(u1,u1v1) and eliminating the common factor u1 between u1′ and v1′ we getu1′=P1(u1,v1)=u1(9u12v12+a22u12-9a2u1v1+9v129,v1′=Q1(u1,v1)=-v13.

Doing the change of variables (u2,v2)=(u1+v1,v1), after doing the vertical blow up (u2,v2)=(u3,u3v3), and eliminating the commun factor u32, we obtain12 u3′=P3(u3,v3)=-u39[9u32v35-27u32v34+27u32v33-9u32v32+(a22+9a2+18)v33-(3a22+18a2+9)v32+(3a22+9a2)v3-a22],v3′=Q3(u3,v3)=v3(v3-1)9[9u32v34-18u32v33+9u32v32+(a22+9a2+18)v32-(2a22+9a2)v3+a22].

Thus the equilibrium points of system (12) are (0, 0), (0, 1) and the points (0,v3) such that v3 is a real zero of the polynomial(a2+6)(a2+3)v32-a2(2a2+9)v3+a22=0.

We haveD(P3,Q3)(0,0)=a22900-a229andD(P3,Q3)(0,1)=-1002,

i.e., (0, 0) and (0, 1) are hyperbolic saddles.

If a2=-6, then (0, 2) is a hyperbolic stable node, and consequently system (2) cannot have a global center.

If a2=-3, then (0,-1) is a hyperbolic saddle withD(P3,Q3)(0,-1)=-1002.

Going back through the changes of variables we obtain that there are trajectories that tend to the origin of U2, so system (2) cannot have a global center.

Now if (a2+6)(a2+3)≠0, then the point (0,a2/(a2+3)) is a hyperbolic stable node and system (2) cannot have a global center.

Finally, if a2=0 then a5=0 the study boils down to studying global centers under statement (i) of Theorem 1. □

Acknowledgements

We thank to Professor Luis Fernando Mello his comments which help us to improve this paper.

Author Contributions

The three authors have worked equally in this paper.

Funding

Open Access Funding provided by Universitat Autonoma de Barcelona. The first author is partially supported by the Conselho Nacional de Desenvolvimento Cient’ıfico e Tecnol’ogico - Brasil (CNPq) - by the Grants 200950/2022-3. The second author is partially supported by the Agencia Estatal de Investigación grant PID2019-104658GB-I00, the H2020 European Research Council grant MSCA-RISE-2017-777911, AGAUR (Generalitat de Catalunya) grant 2021SGR00113, and by the Acadèmia de Ciències i Arts de Barcelona.

Publisher's Note

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