J. Math. Anal. Appl. 335 (2007) 1335–1346

www.elsevier.com/locate/jmaa

Periodic orbits for a class of reversible quadratic vector
field on R

3 ✩

Claudio A. Buzzi a, Jaume Llibre b,∗, João C. Medrado c

a UNESP-IBILCE – São José do Rio Preto, SP, CEP 15054-000, Brazil
b Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain
c Instituto de Matemática e Estatística, Universidade Federal de Goiás, 74001-970 Goiânia, Goiás, Brazil

Received 26 September 2005

Available online 14 February 2007

Submitted by C.E. Wayne

Abstract

For a class of reversible quadratic vector fields on R
3 we study the periodic orbits that bifurcate from

a heteroclinic loop having two singular points at infinity connected by an invariant straight line in the finite
part and another straight line at infinity in the local chart U2. More specifically, we prove that for all n ∈ N,
there exists εn > 0 such that the reversible quadratic polynomial differential system

ẋ = a0 + a1y + a3y2 + a4y2 + ε
(
a2x2 + a3xz

)
,

ẏ = b1z + b3yz + εb2xy,

ż = c1y + c4z2 + εc2xz

in R3, with a0 < 0, b1c1 < 0, a2 < 0, b2 < a2, a4 > 0, c2 < a2 and b3 /∈ {c4,4c4}, for ε ∈ (0, εn) has at
least n periodic orbits near the heteroclinic loop.
© 2007 Elsevier Inc. All rights reserved.

Keywords: Periodic orbits; Quadratic vector fields; Reversibility

✩ The first and third authors are partially supported by CNPq grant number 472873/2004-0. The second author
is partially supported by a MCYT/FEDER grant number MTM2005-06098-C02-01 and by a CICYT grant number
2005SGR 00550. The third author is partially supported by a CNPq grant number 620029/2004-8. All authors are also
supported by the joint project CAPES-MECD grant 071/04 and HBP2003-0017, respectively.

* Corresponding author.
E-mail addresses: buzzi@ibilce.unesp.br (C.A. Buzzi), jllibre@mat.uab.es (J. Llibre), medrado@mat.ufg.br

(J.C. Medrado).
0022-247X/$ – see front matter © 2007 Elsevier Inc. All rights reserved.
doi:10.1016/j.jmaa.2007.02.011



1336 C.A. Buzzi et al. / J. Math. Anal. Appl. 335 (2007) 1335–1346
1. Introduction

A vector field X : R3 → R
3 of the form X = (P 1,P 2,P 3) is called a quadratic vector field

if P 1, P 2 and P 3 are polynomials of degree less or equal to two and at least one of them has
degree two.

The dynamics of linear vector fields in R
3 is simple and very well understand. In particular,

these vector fields only present periodic orbits when they have some invariant plane with a global
center; i.e. a plane with a unique singular point surrounded by periodic orbits and no other or-
bits. The quadratic vector fields are the easiest nonlinear vector fields which can present a rich
dynamics of periodic orbits. Here we shall show the existence of an easy mechanism around two
straight line solutions, already present in the quadratic vector fields, which generates as many
periodic orbits as we want. Of course, this mechanism is different from a center.

A diffeomorphism ϕ : R3 → R
3 is called an involution if ϕ ◦ ϕ = Id. Given an involution

ϕ : R3 → R
3 we say that a vector field X over R

3 is ϕ-reversible, if ϕ∗X = −X ◦ ϕ, i.e.,
dϕp(X(p)) = −X(ϕ(p)). Let S be the fixed point set of ϕ. An orbit γ is said symmetric if
ϕ(γ ) = γ . Hence, every singular point of X in S is symmetric of X. Some classical properties of
reversible systems are:

(i) The phase portrait of X is symmetric with respect to S.

(ii) A symmetric singular point or a symmetric periodic orbit cannot be attractor or repellor.

(iii) If X(p) = 0 and p /∈ S, then X(ϕ(p)) = 0.

(iv) If a regular orbit γ intersects S in two distinct points, then γ is a periodic orbit.

(v) If X(p) �= 0 and p ∈ S, then X(p) /∈ TpS.

(vi) Any periodic orbit γ of X not crossing S, has a symmetric one given by ϕ(γ ).

For more details about these properties see [3,11,13].
In this work we deal with ϕ-reversible quadratic vector fields Xε on R

3, depending on
one-parameter ε, where we assume that the dimension of the fixed point set S of the linear
involution ϕ is equal to 1. Our main goal is to establish sufficient conditions for the existence of
arbitrary number of periodic orbits which are not in a center.

A polynomial vector field X in R
3 can be extended to an analytic vector field on the closed ball

of radius one, the interior of this ball is diffeomorphic to R
3 and its boundary (a 2-dimensional

sphere S
2) plays the role of infinity. The technique for making such an extension is called

Poincaré compactification, see Section 2.
We consider the heteroclinic loop L formed by two singular points p and q at infinity con-

nected by two orbits γ1 in R
3 and γ2 in S

2 such that L satisfies: (1) γ1 intersects S in a point s;
(2) one endpoint of S on S

2 is the point r of γ2, see Fig. 1.
Let N be the set of positive integers. For all n ∈ N we shall prove that there exists εn > 0

sufficiently small such that the vector field Xε has at least n periodic orbits near the loop L for
all ε ∈ (0, εn).

We will take a small interval G ⊂ S having r as an endpoint (see Fig. 1), and we will follow
its image under the flow of Xε until its intersection with a cross section Σ of the orbit γ1 which
contains a neighborhood of S near the point s ∈ L, see Fig. 1. We denote by π :G → Σ the
Poincaré map. Then, we shall prove that π(G) is a spiral near the point s giving finitely many
turns for every ε > 0 sufficiently small. This number of turns tends to infinity as ε → 0. The



C.A. Buzzi et al. / J. Math. Anal. Appl. 335 (2007) 1335–1346 1337
Fig. 1. The vector field Xε on the Poincaré ball.

orbits through the points of π(G)∩S are periodic because, by construction, they have two points
on S. Using these ideas in Section 3 we shall prove our main results stated in the following three
theorems.

Theorem 1. Let Xε be the vector field associated to the quadratic polynomial differential system:

ẋ = a0 + a1y + a4y
2 + a5z

2 + ε
(
a2x

2 + a3xz
)
,

ẏ = b1z + b3yz + εb2xy,

ż = c1y + c4z
2 + εc2xz, (1)

where a0 < 0, b1c1 < 0, a2 < 0, b2 < a2, a4 > 0, c2 < a2 and b3 /∈ {c4,4c4}. Then, the vector
field Xε is ϕ-reversible, with ϕ(x, y, z) = (−x, y,−z). Moreover,

(a) For ε > 0 the vector field Xε satisfies:

(a1) the straight line {(x,0,0): x ∈ R} is an invariant line of the vector field without singular
points and the flow goes in the decreasing direction of the x-axis;

(a2) the set {(z1,0,0)} on the chart U2 on the Poincaré ball is an invariant straight line
without singular points and the flow goes in the increasing direction of z1-axis;

(a3) in the chart U1 the point p = (0,0,0) is a hyperbolic singular point, the plane
{(z1, z2,0)} is its local stable manifold and the line {(0,0, z3)} is its unstable mani-
fold;

(a4) Xε has the heteroclinic loop L formed by the straight line {(x,0,0)} in R
3, two singular

points {(0,0,0)} of the Poincaré ball in the chart U1 and V1, and the straight line
{(z1,0,0)} on the chart U2 of the Poincaré ball.

(b) For ε = 0 the vector field X0 has invariant cylinders surrounding the x-axis and the restric-
tion of X0 to x = 0 has a nonisochronous center at the origin.

(c) For all n ∈ N there exists εn > 0 sufficiently small such that the vector field Xε has at least
n periodic orbits near the heteroclinic loop L.



1338 C.A. Buzzi et al. / J. Math. Anal. Appl. 335 (2007) 1335–1346
Theorem 2. Let Xε be a quadratic vector field such that

(a) Xε is ϕ-reversible and has the heteroclinic loop L on the Poincaré ball as in Theorem 1 for
ε > 0;

(b) X0 has invariant cylinders surrounding the x-axis and the restriction of X0 to x = 0 has a
nonisochronous center at the origin.

Then the most general Xε satisfying (a) and (b) is the vector field (1).

The next theorem illustrate that we may have a similar result even if the heteroclinic loop L
is not formed by an invariant straight line in the chart U2.

Theorem 3. Let Xε be the vector field associated to the quadratic polynomial differential system:

ẋ = a0 + a1y + a4y
2 + a5z

2 + ε
(
a2x

2 + a3xz
)
,

ẏ = b1z + b3yz + εb2xy,

ż = c1y + c3y
2 + c4z

2 + εc2xz, (2)

with a0 < 0, b1c1 < 0, a2 < 0, b2 < a2, (a3 − c4)
2 − 4a5(a2 − c2) < 0, c2 < a2, a5(a2 − b2) −

(a3 − b3)
2/4 > 0 and a4(a5 + a2 − b2) > 0 (or a4 = 0). Then, statement (c) of Theorem 1 holds

for system (2).

The paper is organized as follows. In Section 2 we describe the Poincaré compactification for
a polynomial differential system in R

3. In Section 3 we deal with the perturbed vector field Xε

for ε > 0 and in Section 4 we deal with the unperturbed vector field X0. Finally, in Sections 5–7
we prove Theorems 2, 1 and 3, respectively.

The idea of using symmetries for finding periodic orbits is very old. Thus Poincaré used it
for finding periodic orbits in the restricted three-body problem, see [12]. After this technique
became popular for looking for periodic orbits in different problems of Celestial Mechanics see
for instance the book of Meyer [9] and the references quoted there. But this tool has been used
for studying periodic orbits in problems very far from Celestial Mechanics, see for instance the
paper of Devaney [3], the books of Poénaru [11] and of Sevryuk [13], the survey of Lamb and
Roberts [5], and again the references quoted there.

The idea that heteroclinic loops having some component at infinity can create large amplitude
periodic orbits appeared already in [6,10] for different kinds of heteroclinic loops; or again in
problems related with Celestial Mechanics as in [1]; or in bifurcations of periodic orbits from
infinity in planar polynomial vector fields [4] or in planar piecewise linear vector fields [7], etc.

2. The Poincaré compactification in R
3R
3

R
3

In R
3 we consider the polynomial differential system

ẋ = P 1(x, y, z),

ẏ = P 2(x, y, z),

ż = P 3(x, y, z),

or equivalently its associated polynomial vector field X = (P 1,P 2,P 3). The degree n of X is
defined as n = max{deg(P i): i = 1,2,3}.



C.A. Buzzi et al. / J. Math. Anal. Appl. 335 (2007) 1335–1346 1339
Let S
3 = {y = (y1, y2, y3, y4) ∈ R

4: ‖y‖ = 1} be the unit sphere in R
4, and

S+ = {
y ∈ S

3: y4 > 0
}

and S− = {
y ∈ S

3: y4 < 0
}

be the northern and southern hemispheres, respectively. The tangent space to S
3 at the point y is

denoted by TyS
3. Then, the tangent hyperplane

T(0,0,0,1)S
3 = {

(x1, x2, x3,1) ∈ R
4: (x1, x2, x3) ∈ R

3}
is identified with R

3.
We consider the central projections

f+ : R3 = T(0,0,0,1)S
3 → S+ and f− : R3 = T(0,0,0,1)S

3 → S−,

defined by

f+(x) = 1

�x
(x1, x2, x3,1) and f−(x) = − 1

�x
(x1, x2, x3,1),

where �x = (1 + ∑3
i=1 x2

i )1/2. Through these central projections, R
3 can be identified with the

northern and the southern hemispheres, respectively. The equator of S
3 is S

2 = {y ∈ S
3: y4 = 0}.

Clearly, S
2 can be identified with the infinity of R

3.
The maps f+ and f− define two copies of X, one Df+ ◦ X in the northern hemisphere and

the other Df− ◦X in the southern one. Denote by X the vector field on S3 \S2 = S+ ∪ S− which
restricted to S+ coincides with Df+ ◦ X and restricted to S− coincides with Df− ◦ X.

In what follows we shall work with the orthogonal projection of the closed northern hemi-
sphere to y4 = 0. Note that this projection is a closed ball B of radius one, whose interior is
diffeomorphic to R

3 and whose boundary S
2 corresponds to the infinity of R

3. We shall extend
analytically the polynomial vector field X to the boundary, in such a way that the flow on the
boundary is invariant. This new vector field on B will be called the Poincaré compactification
of X, and B will be called the Poincaré ball. Poincaré introduced this compactification for poly-
nomial vector fields in R

2, and its extension to R
m can be found in [2].

The expression for X(y) on S+ ∪ S− is

X(y) = y4

⎛
⎜⎜⎜⎝

1 − y2
1 −y2y1 −y3y1

−y1y2 1 − y2
2 −y3y2

−y1y3 −y2y3 1 − y2
3

−y1y4 −y2y4 −y3y4

⎞
⎟⎟⎟⎠

⎛
⎝P 1

P 2

P 3

⎞
⎠ ,

where P i = P i(y1/|y4|, y2/|y4|, y3/|y4|). Written in this way X(y) is a vector field in R
4 tangent

to the sphere S
3.

Now we can extend analytically the vector field X(y) to the whole sphere S
3 by

p(X)(y) = yn−1
4 X(y);

this extended vector field p(X) is called the Poincaré compactification of X.
As S

3 is a differentiable manifold, to compute the expression for p(X) we can consider the
eight local charts (Ui,Fi), (Vi,Gi) where Ui = {y ∈ S

3: yi > 0}, and Vi = {y ∈ S
3: yi < 0}

for i = 1,2,3,4; the diffeomorphisms Fi :Ui → R
3 and Gi :Vi → R

3 for i = 1,2,3,4, are the
inverses of the central projections from the origin to the tangent planes at the points (±1,0,0,0),
(0,±1,0,0), (0,0,±1,0) and (0,0,0,±1), respectively. We now do the computations on U1.



1340 C.A. Buzzi et al. / J. Math. Anal. Appl. 335 (2007) 1335–1346
Suppose that the origin (0,0,0,0), the point (y1, y2, y3, y4) ∈ S
3 and the point (1, z1, z2, z3) in

the tangent plane to S
3 at (1,0,0,0) are collinear, then we have

1

y1
= z1

y2
= z2

y3
= z3

y4
,

and consequently

F1(y) =
(

y2

y1
,
y3

y1
,
y4

y1

)
= (z1, z2, z3)

defines the coordinates on U1.
As

DF1(y) =

⎛
⎜⎜⎜⎝

− y2

y2
1

1
y1

0 0

− y3

y2
1

0 1
y1

0

− y4

y2
1

0 0 1
y1

⎞
⎟⎟⎟⎠

and yn−1
4 = (

z3
�z

)
n−1, the analytical field p(X) becomes

zn
3

(�z)n−1

(−z1P
1 + P 2,−z2P

1 + P 3,−z3P
1), (3)

where P i = P i(1/z3, z1/z3, z2/z3).
In a similar way we can deduce the expressions of p(X) in U2 and U3. These are

zn
3

(�z)n−1

(−z1P
2 + P 1,−z2P

2 + P 3,−z3P
2), (4)

where P i = P i(z1/z3,1/z3, z2/z3) in U2, and

zn
3

(�z)n−1

(−z1P
3 + P 1,−z2P

3 + P 2,−z3P
3), (5)

where P i = P i(z1/z3, z2/z3,1/z3) in U3.
The expression for p(X) in U4 is zn+1

3 (P 1,P 2,P 3) where the component P i = P i(z1, z2, z3).
The expression for p(X) in the local chart Vi is the same as in Ui multiplied by (−1)n−1.

When we shall work with the expression of the compactified vector field p(X) in the local
charts we shall omit the factor 1/(�z)n−1. We can do that through a rescaling of the time.

We remark that all the points on the sphere at infinity in the coordinates of any local chart
have z3 = 0.

3. The perturbed vector field Xε

We write the quadratic polynomial differential system in R
3 in the form

ẋ =
∑

0�i+j+k�2

aijkx
iyj zk,

ẏ =
∑

0�i+j+k�2

bijkx
iyj zk,

ż =
∑

cijkx
iyj zk. (6)
0�i+j+k�2



C.A. Buzzi et al. / J. Math. Anal. Appl. 335 (2007) 1335–1346 1341
We consider that its associated vector field Xε is ϕ-reversible with respect to the linear invo-
lution

ϕ(x, y, z) = (−x, y,−z),

which has the fix point set S = {(0, y,0)}. So, the system becomes

ẋ = a000 + a010y + a200x
2 + a101xz + a020y

2 + a002z
2,

ẏ = b100x + b001z + b110xy + b011yz,

ż = c000 + c010y + c200x
2 + c101xz + c020y

2 + c002z
2. (7)

We assume that the straight line {(x,0,0): x ∈ R} is invariant for system (7), and that the flow
along this line goes in the decreasing direction of the x-axis. Therefore, we get that system (7),
after a renaming of the parameters, has the form

ẋ = a0 + a1y + a2x
2 + a3xz + a4y

2 + a5z
2,

ẏ = b1z + b2xy + b3yz,

ż = c1y + c2xz + c3y
2 + c4z

2. (8)

Here, we have that a0 < 0 and a2 < 0.
The next step is to analyze system (8) at infinity.

Lemma 4. If the straight line {(z1,0,0)} on the chart U2 is formed by a unique orbit of system (8)
and the flow goes in the increasing direction of the z1-axis, then c3 = 0, b2 − a2 < 0 and a4 > 0.

Proof. According to (4), system (8) in the chart U2 has the expression

ż1 = a4 + a1z3 + (a2 − b2)z
2
1 + (a3 − b3)z1z2 + a5z

2
2 + a0z

2
3 − b1z1z2z3,

ż2 = c3 + c1z3 + (c4 − b3)z
2
2 + (c2 − b2)z1z2 − b1z

2
2z3,

ż3 = −b2z1z3 − b3z2z3 − b1z2z
2
3.

In order to have the invariant straight line {(z1,0,0)} with the flow going in the increasing
direction of the z1-axis we need the conditions c3 = 0, b2 − a2 < 0 and a4 > 0. �

Now, we analyze the vector field on the chart U1.

Lemma 5. If b2 − a2 < 0, c2 − a2 < 0 and a2 < 0, then on the chart U1 we have that (0,0,0)

is a hyperbolic singular point such that {(z1, z2,0)} and {(0,0, z3)} are the local stable and the
unstable manifold of (0,0,0), respectively.

Proof. According to (3), on the chart U1, the system has the expression

ż1 = (b2 − a2)z1 + (b3 − a3)z1z2 + b1z2z3 − a4z
3
1 − a5z1z

2
2 − a1z

2
1z3 − a0z1z

2
3,

ż2 = (c2 − a2)z2 + c3z
2
1 + (c4 − a3)z

2
2 + c1z1z3 − a4z

2
1z2 − a5z

3
2 − a1z1z2z3 − a0z2z

2
3,

ż3 = −a2z3 − a3z2z3 − a4z
2
1z3 − a5z

2
2z3 − a1z1z

2
3 − a0z

3
3.

So, the conditions to have (0,0,0) as an attractor node on S
2 and a repellor on R

3 are b2 −a2 < 0,
c2 − a2 < 0 and a2 < 0. �



1342 C.A. Buzzi et al. / J. Math. Anal. Appl. 335 (2007) 1335–1346
4. The unperturbed vector field X0

Until now we have that a system satisfying the hypotheses of Lemmas 4 and 5 can be written
in the form

ẋ = a0 + a1y + a2x
2 + a3xz + a4y

2 + a5z
2,

ẏ = b1z + b2xy + b3yz,

ż = c1y + c2xz + c4z
2, (9)

with a0 < 0, a2 < 0, a4 > 0, b2 − a2 < 0 and c2 − a2 < 0.
We choose the cross section Σ to the orbit {(x,0,0)} as the plane x = 0 in a neighborhood

of (0,0,0). We want that for ε = 0 there exist invariant cylinders surrounding the x-axis and that
the restriction of system (9) to Σ has a nonisochronous center at the origin. In order to obtain
this we need to take b2 = c2 = 0.

The restriction of system (9) on Σ becomes

ẏ = b1z + b3yz,

ż = c1y + c4z
2. (10)

The next result proved by Loud [8] is useful for classifying the nonisochronous centers of
quadratic differential systems in the plane.

Theorem 6. The origin is an isochronous center of the quadratic system

ẋ = −y + a20x
2 + a11xy + a02y

2,

ẏ = x + b20x
2 + b11xy + b02y

2,

if and only if the system can be brought to one of the following systems:

(a) ẋ = −y + x2 − y2, ẏ = x(1 + 2y);
(b) ẋ = −y + x2, ẏ = x(1 + y);
(c) ẋ = −y − 4

3
x2, ẏ = x

(
1 − 16

3
y

)
;

(d) ẋ = −y + 16

3
x2 − 4

3
y2, ẏ = x

(
1 + 8

3
y

)
through a linear change of coordinates and a rescaling of time.

Lemma 7. If a2 = 0, a3 = 0, b2 = 0, c2 = 0, b1c1 < 0, and b3 /∈ {c4,4c4}, then the origin is
a nonisochronous center of system (10) and for system (9) we have invariant cylinders surround-
ing the x-axis.

Proof. The vector field (10) is reversible with respect to the involution (y, z) �→ (y,−z).
If c1b1 < 0, then the origin is a center or a focus of (10). Using the fact that (10) is reversible

we conclude that it is a center.
According to Theorem 6 the center (10) is isochronous if and only if there is a linear change

of variables and a time rescaling such that the system (10) becomes one of the four systems given
in Theorem 6.



C.A. Buzzi et al. / J. Math. Anal. Appl. 335 (2007) 1335–1346 1343
First of all we apply a rescaling of time to system (10). By multiplying the system by the

factor
√

1
−b1c1

, we have

ẏ = −
√

−b1

c1
z + b3√−b1c1

yz,

ż =
√

− c1

b1
y + c4√−b1c1

z2. (11)

We call β =
√

− b1
c1

. We can prove that all linear transform of variables that change the linear
part of (11) in(

0 −1

1 0

)

are of the form

B =
( d

β
−bβ

b d

)
,

with d2 + b2β2 �= 0.
The nonlinear part of (11) is given by f (y, z) = (

b3√−b1c1
yz, c4√−b1c1

z2). We impose that

Bf (B−1(y, z)) is equal to one of the four for nonlinear parts of the systems given in Theo-
rem 6. Then, we obtain that the vector field can be brought to vector field (b) or (c), if b3 = c4
or b3 = 4c4, respectively.

In short we get that if b3 /∈ {c4,4c4}, then the origin is a nonisochronous center for (11).
Using the fact that the origin is a center for (11), for y and z small enough the periodic

orbit passing thought (y, z) can be given implicitly by h(y, z) = 0. Observe that the cylinder
{(x, y, z) ∈ R

3: h(y, z) = 0} is invariant by the flow of (10) and surrounds the x-axis. �
5. Proof of Theorem 2

According to Lemmas 4 and 5 the vector field Xε for ε > 0 must be

ẋ = a0 + a1y + a4y
2 + a5z

2 + a2x
2 + a3xz,

ẏ = b1z + b3yz + b2xy,

ż = c1y + c4z
2 + c2xz, (12)

where a0 < 0, a2 < 0, b2 − a2 < 0, a4 > 0 and c2 − a2 < 0.
According to Lemma 7, for ε = 0 the vector field must satisfy

ẋ = a0 + a1y + a4y
2 + a5z

2,

ẏ = b1z + b3yz,

ż = c1y + c4z
2, (13)

where b1c1 < 0 and b3 /∈ {c4,4c4}. So, we put in front of a2, a3, b2 and c2 the parameter ε and
we obtain the vector field of the statement of Theorem 2. �



1344 C.A. Buzzi et al. / J. Math. Anal. Appl. 335 (2007) 1335–1346
6. Proof of Theorem 1

The proof of statements (a) and (b) of Theorem 1 follows directly from Theorem 2. So, the
only thing that remains to prove in Theorem 1 is that for all n ∈ N there is εn > 0 such that for
ε ∈ (0, εn) the system Xε has at least n periodic orbits near the heteroclinic loop L.

In order to prove this, we construct a Poincaré map in the following way. Consider a cross
section Σ at the point r = (0,0,0) in the chart U2 of the Poincaré ball that contains the small
interval G. In a neighborhood U of p in the chart U1 we take the cross sections Σ1 and Σ2 of
the orbits γ1 and γ2 such that q1 ∈ γ1 ∩ Σ1 and q2 ∈ γ2 ∩ Σ2, respectively. Finally, we consider
the cross section Σ of the orbit γ1 which contains a neighborhood of S near the point s of L, see
Fig. 2. We denote by π the Poincaré map going from Σ to Σ . The loop L and the local phase
portrait of the hyperbolic singular point p of U1 guarantee the existence of the Poincaré map π .

We may consider diffeomorphisms induced by the flow of the vector field Xε: π2 :Σ → Σ2
and π0 :Σ1 → Σ . Since the orbits going from Σ to Σ2 and Σ1 to Σ spend only a bounded time,
π1 and π2 are well defined.

In the neighborhood U of the hyperbolic singular point p = (0,0,0), we get that the plane
{(z1, z2,0)} is the local stable manifold of p and the line {(0,0, z3)} is the unstable manifold
of p. So, we may consider a diffeomorphism π1 :Σ2 → Σ1 induced by the flow of Xε in the
neighborhood U . Thus, we consider the Poincaré map given by π = π0 ◦ π1 ◦ π2 :Σ → Σ .

The image of the segment G with endpoint r and contained in Σ by π2 is the arc π2(G) in Σ2
having the point q2 as an endpoint. And consequently, (π1 ◦ π2)(G) is an arc in Σ1 having q1 as
an endpoint.

We claim that for ε > 0 sufficiently small the arc (π1 ◦ π2)(G) spirals with infinitely many
turns around the point q1. This is due to two facts. First, the time going from the point p to the
point q1 is infinite, and for ε = 0 the flow of system (1) restricted to Σ has a nonisochronous
center.

Since π0 is a diffeomorphism, we have that the arc π(G) = (π0 ◦ π1 ◦ π2)(G) is an arc in
Σ which spirals to the point s giving infinitely many turns around s. The orbits through the
intersection points of π(G) with the y-axis are periodic because, by construction, they have two
points on the y-axis and Xε is ϕ-reversible. This completes the proof of Theorem 1.

Fig. 2. The half-heteroclinic loop L on x � 0.



C.A. Buzzi et al. / J. Math. Anal. Appl. 335 (2007) 1335–1346 1345
7. Proof of Theorem 3

We recall that system (8), with a0 < 0 and a2 < 0, is ϕ-reversible and it has {(x,0,0)} as an
invariant straight line with the flow going in the decreasing direction of x-axis.

We assume that c3 �= 0, then by Lemma 4 we have that the straight line {(z1,0,0)} on the chart
U2 is no more an invariant set. We will find conditions for system (8) to have only the singular
points (0,0,0) on charts U1 and V1 and no more singular points on the Poincaré sphere S

2. We
will also find conditions for these points to be an attractor on U1 and repellor on V1, restrict
to S

2. This fact guarantees the existence of the orbit that plays the role of γ2 on S
2, because there

is exactly one orbit departing from the repellor and arriving to the attractor passing through the
point r .

In the sequel we establish these conditions. First, we choose the cross section Σ of the orbit
{(x,0,0)} as the plane x = 0 in a neighborhood of (0,0,0). We want that for ε = 0 there exist
invariant cylinders surrounding the x-axis and that the restriction of system (8) on Σ has a
nonisochronous center at the origin. In order to obtain this we need to take b2 = c2 = 0.

The restriction of system (8) on Σ becomes

ẏ = b1z + b3yz,

ż = c1y + c3y
2 + c4z

2. (14)

In similar way of the proof of Lemma 7 we obtain that if b1c1 < 0, then the origin is a
nonisochronous center of system (14) and for system (8) we have invariant cylinders surrounding
the x-axis.

System (8) satisfying the condition b1c1 < 0, according to (3), on the chart U1 has the expres-
sion

ż1 = (b2 − a2)z1 + (b3 − a3)z1z2 + b1z2z3 − a4z
3
1 − a5z1z

2
2 − a1z

2
1z3 − a0z1z

2
3,

ż2 = (c2 − a2)z2 + c3z
2
1 + (c4 − a3)z

2
2 + c1z1z3 − a4z

2
1z2 − a5z

3
2 − a1z1z2z3 − a0z2z

2
3,

ż3 = −a2z3 − a3z2z3 − a4z
2
1z3 − a5z

2
2z3 − a1z1z

2
3 − a0z

3
3.

So, the conditions to have (0,0,0) as an attractor node on S2 and repellor on R3 are b2 − a2 < 0,
c2 − a2 < 0 and a2 < 0.

Lemma 8. If a5(a2 − b2) − (a3 − b3)
2/4 > 0 and a4 = 0 or a4(a5 + a2 − b2) > 0, then there are

no singular points on U2 for z3 = 0.

Proof. The system (8) on the chart U2 for z3 = 0 is

ż1 = a4 + (a2 − b2)z
2
1 + (a3 − b3)z1z2 + a5z

2
2,

ż2 = c3 + (c4 − b3)z
2
2 + (c2 − b2)z1z2,

ż3 = 0.

We recall that for a conic f (z1, z2) = ∑2
i+j�0 aij z

i
1z

j

2 = 0, we have that

D3 = det

⎛
⎝a20

a11
2

a10
2

a11
2 a02

a01
2

a10 a01

⎞
⎠ , D2 = det

(
a20

a11
2

a11
2 a02

)
.

2 2 a00



1346 C.A. Buzzi et al. / J. Math. Anal. Appl. 335 (2007) 1335–1346
It is well known that if D2 > 0 and D3 = 0, then f (z1, z2) = 0 is a pair of imaginary parallels
straight lines, and if D2 > 0 and D3(a20 + a02) > 0, then f (z1, z2) = 0 is an imaginary ellipse.

The equation ż1 = 0 is a conic in the plane z1z2 and we get that D2 = a5(a2 − b2) −
(a3 − b3)

2/4 > 0 and D3 = a4D2. In short ż1 = 0 has no real solutions. �
Lemma 9. If (a3 − c4)

2 − 4a5(a2 − c2) < 0, then there are no singular points on U3 for z2 =
z3 = 0.

Proof. System (8) on the chart U2 for z3 = 0 is

ż1 = a5 + (a3 − c4)z1 + (a2 − c2)z
2
1,

ż2 = 0,

ż3 = 0.

So, by hypothesis the discriminant of the ż1 = 0 is negative and we conclude the proof. �
With Lemmas 8 and 9 we conclude that on the Poincaré sphere S

2 there are only the two
singular points at (0,0,0) of U1 and V1. So, the proof of Theorem 3 is completed.

We observe here that the conditions presented in Lemmas 8 and 9 are sufficient, but not
necessary, conditions.

Acknowledgments

The first and the third authors wish to thank to the dynamical system research group of the Universitat Autònoma de
Barcelona, where the paper was written, for the invitation and hospitality.

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