Nonsingular Quadratic Differential Equations in the Plane

Camacho, M. I. T.; Palmeira, C. F. B.

doi:10.2307/2000673

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“…In this section we will state the main results on the classification of CS due to Cima and Llibre [3] .…”

Section: Classification Of Cubic Systemsmentioning

confidence: 99%

“…Actually, a construction leading to examples with 2n -4 inseparable leaves for all n >_ 4 can be already found in [14] . In [3] it is claimed that the case n = 2 has at most 2 inseparable leaves. This claim is not true because in [7] the quadratic system x = 1 + xy, y = m-1y2 , with m G -1 has 3 inseparable leaves: y = 0 and the two branches of the hyperbola xy = -m/(m + 1).…”

Section: Introductionmentioning

confidence: 99%

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Chordal cubic systems

Carbonell¹,

Llibre²

1989

Publicacions Matemàtiques

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“…In this section we will state the main results on the classification of CS due to Cima and Llibre [3] .…”

Section: Classification Of Cubic Systemsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Chordal cubic systems

Carbonell¹,

Llibre²

1989

Publicacions Matemàtiques

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“…The analysis of some of the cases in [1] is incomplete. In particular, in a case called (b2.2) in [1], the equation was reduced to (ey2 + ny +p)dx + [y(c'x + e'y)+m'x + n'y+p')dy = 0. It was claimed that for this to be nonsingular, one should have n2 -4pe < 0.…”

Section: Errata To "Nonsingular Quadratic Differential Equations In Tmentioning

confidence: 99%

“…As it was pointed out by Gasull and Llibre in [2], the correct value is 3. They also give the correct formulation of the theorem in [1] and its corollary, which should read as follows:…”

mentioning

confidence: 99%

Errata to: “Nonsingular quadratic differential equations in the plane” [Trans. Amer. Math. Soc. 301 (1987), no. 2, 845–859; MR0882718 (88b:58109)]

Camacho¹,

Palmeira²

1988

Trans. Amer. Math. Soc.

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It was claimed in the above-mentioned paper that the number of inseparable leaves of a foliation in the plane defined by a nonsingular quadratic differential equation is at most 2. As it was pointed out by Gasull and Llibre in [2], the correct value is 3. They also give the correct formulation of the theorem in [1] and its corollary, which should read as follows:THEOREM. Let P and Q be polynomials of degree at most 2 in two real variables, such that P2 + Q2 is never zero. The foliation defined by Pdx + Qdy = 0 is then conjugate to one of the following three: dx = 0; xdx + (1 -x2)dy = 0; y2dx +2(1+ xy)dy = 0.We recall that a conjugacy is a homeomorphism taking leaves of one foliation into leaves of the other.COROLLARY. There are at most 2 Reeb components for a nonsingular quadratic vector field in the plane.The proof of the theorem consists of a case by case analysis involving blow up of singularities. The analysis of some of the cases in [1] is incomplete. In particular, in a case called (b2.2) in [1], the equation was reduced to (ey2 + ny +p)dx + [y(c'x + e'y)+m'x + n'y+p')dy = 0. It was claimed that for this to be nonsingular, one should have n2 -4pe < 0. This condition although sufficient is not necessary. In particular one does not need to have p nonzero, which was assumed in the sequence. If p = 0, one can still have a nonsingular equation provided m! = n = 0 and this can lead to a configuration with three inseparable leaves (take for example e = 1, c' = 2, e' = 0, n' = 0, p' = 2).

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“…In [CP2] it is studied the case n = 3. In [CP1] it is claimed that the case n = 2 has at most 2 inseparable leaves. This claim is not true because the easily integrable quadratic system x = 1 + xy, y = y2 /m with m < -1 studied in [GLL] has 3 inseparable leaves given by y = 0 and the two branches of the hyperbola xy = -m/{m + 1).…”

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On the nonsingular quadratic differential equations in the plane

Gasull¹,

Llibre²

1988

Proc. Amer. Math. Soc.

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ABSTRACT. We point out that the maximum number of inseparable leaves of nonsingular polynomial differential equations of degree two is 3, and we present an example with exactly 3 inseparable leaves.A nonsingular differential equation in two real variables defines a foliation of the plane. It is well known that the topological classification of such foliations depends only on the number of inseparable leaves and the way they are distributed in the plane (see [Kl, K2 or HR]). Two leaves (or trajectories) Li and 7,2 are said to be inseparable if for any arcs Ti and T2 respectively transversal to 7,i and L2 there are infinitely many leaves which intersect both Ti and T2.For polynomial foliations of degree n, i.e. defined by a polynomial vector field (P, Q) where max{deg(7J), deg(Q)} = n, it is known that the number of inseparable leaves is at most 2n (see [M or SS]). A construction leading to examples with 2n -4 inseparable leaves for all n > 4 can be found in [P]. In [CP2] it is studied the case n = 3. In [CP1] it is claimed that the case n = 2 has at most 2 inseparable leaves. This claim is not true because the easily integrable quadratic system x = 1 + xy, y = y2 /m with m < -1 studied in [GLL] has 3 inseparable leaves given by y = 0 and the two branches of the hyperbola xy = -m/{m + 1).The main result of [GLL] shows that for nonsingular quadratic differential equations there are 23 different topological phase portraits in the Poincaré sphere. As a consequence there follows the next theorem.THEOREM. Consider the foliation defined by the differential equation Pdx + Qdy -0, where P and Q are polynomials of degree at most 2 in the two real variables and such that P2 + Q2 never vanishes. Then there Í3 a homeomorphism taking leaves of Pdx + Qdy = 0 to leaves of one of the three following foliations: dx = 0, xdx + (1 -x2)dy = 0, iy2/2)dx + (1 + xy)dy = 0.From this result we obtain immediately the following corollary, which answers a question raised in [CT] about the number of Reeb components that a quadratic system in the plane may admit.COROLLARY. There are at most two Reeb components for a nonsingular quadratic vector field in the plane.

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Nonsingular Quadratic Differential Equations in the Plane

Abstract: ABSTRACT. We consider the problem of determining the number of inseparable leaves of nonsingular polynomial differential equations of degree two. As a corollary of a classification theorem for the foliation defined by these equations, we prove that this number is at most 2.

Cited by 3 publications

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Chordal cubic systems

Chordal cubic systems

Errata to: “Nonsingular quadratic differential equations in the plane” [Trans. Amer. Math. Soc. 301 (1987), no. 2, 845–859; MR0882718 (88b:58109)]

On the nonsingular quadratic differential equations in the plane

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