Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 115, pp. 1–18.

ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, DOI: 10.58997/ejde.2025.115

DYNAMICS OF A MAY-LEONARD ASYMMETRIC SYSTEM OF ORDINARY

DIFFERENTIAL EQUATIONS

FABIO SCALCO DIAS, REGILENE OLIVEIRA, CLÁUDIA VALLS

Abstract. We study the May-Leonard asymmetric model in R3 which was introduced in [3, 8].
It is the celebrated classical May-Leonard model incorporating asymmetric competitive effects

instead of requiring equal intrinsic growth rates for each competing population. We study this

system when it has an invariant of Darboux type and for these values of the parameters we
shall describe its global dynamics in the compactification of the sphere, adding its infinity. In

particular, we study the dynamics of that system on the invariant planes and we complete the

study describing the dynamics at infinity. We also prove that the system is completely integrable
and describe the α and ω limits of all the orbits of the system.

1. Introduction and statement of the main results

We study the May-Leonard asymmetric model in R3 introduced in [3, 8] which is given by

x′ = x(1− x− α1y − β1z),

y′ = y(1− y − β2x− α2z),

z′ = z(1− z − α3x− β3y)

(1.1)

where αi, βi, i = 1, 2, 3 are real parameters. This model describes the competition between
three species and depends on six parameters. It is the celebrated classical May-Leonard model,
introduced by May and Leonard in 1975 in [6] incorporating asymmetric competitive effects instead
of requiring equal intrinsic growth rates for each competing population as it is required in the
classical May-Leonard model. In such a way the May-Leonard asymmetric model has a wider
range of applications. In contrast with the classical May-Leonard system whose global dynamics
is quite known and has been widely investigated, this is not the cases for the asymmetric model
for which there are few contributions. In particular system (1.1) was investigated in [2, 3, 8, 10]
where the authors study the conditions for the existence and stability of limit cycles, non-periodic
oscillations, existence of first integrals, Hopf bifurcations and the stability of steady states. It is
well-known that solutions of that system cannot be, in general, written in terms of elementary
functions, so the study of qualitative properties of solutions is a well tool to be performed. Since
the existence and uniqueness of solutions is obvious, other properties, such as the existence of first
integrals (which allow to reduce one dimension of the system) or the knowledge of the qualitative
behavior of the orbits (which allow to solve the problem qualitatively) are very desirable.

We note that the May-Leonard asymmetric model has a large number of parameters, so in
order to gain insight on the behavior of this model one had to begin by exploring subfamilies
depending on less parameters. That is one of the reasons why in this paper we will focus on the
May-Leonard asymmetric model when it has an invariant of Darboux type (see Section 2 for the
definition). For these subfamilies we shall describe their global dynamics in the compactification
of R3, adding the infinity. Such subfamilies have at most two parameters. The exact knowledge of
the dynamics for these values of the parameters provide many information about the dynamics of
the May-Leonard asymmetric models when the parameters are sufficient close to the selected ones

2020 Mathematics Subject Classification. 34A34, 34D23.

Key words and phrases. Poincaré compactification; phase portraits; dynamics at infinity; integrability.
©2025. This work is licensed under a CC BY 4.0 license.
Submitted August 26, 2025. Published December 16, 2025.

1



2 F. S. DIAS, R. OLIVEIRA, C. VALLS EJDE-2025/115

of our study. In particular, we study the dynamics of these models on the invariant planes and we
complete the study describing the dynamics at infinity. We also prove that two of these subfamilies
are completely integrable and because they have a Darboux invariant we can describe the α- and
ω-limits of all their orbits. The main tool will be the Poincaré compactification. Roughly speaking
the Poincaré ball is obtained identifying R3 with the interior of the 3-dimensional ball of radius
one centered at the origin of coordinates, and extending analytically the flow of the asymmetric
May–Leonard system to the boundary §2 of this ball, and consequently to the infinity. In this
way we can study the behavior of that model in a neighborhood of the infinity, and describe
completely the global dynamics of it. For a precise information on the Poincaré compactification
see [4, Chapter 5].

The main results of this paper are the following. For a definition of Darboux polynomial,
invariant algebraic surface, cofactor, invariant as well as the precise statement of Theorem 2.1, see
Section 2. For a precise definition of ω-limit, α-limit and Poincaré sphere, we refer the reader to
[4].

Theorem 1.1. The following holds for system (1.1).

(i) It has a Darboux invariant of the form I(x, y, z, t) = estf(x, y, z, t), where f(x, y, z) is given
by the product of the invariant planes x = 0, y = 0, z = 0 and one of the invariant planes
described in Theorem 2.1 if and only if conditions (1)–(6) of Table 1 holds, respectively.

(ii) For any orbit of system (1.1) restricted to the conditions of Table 1 we have
(ii.1) Its ω-limit is contained at infinity in the Poincaré compactification in R3.
(ii.2) For each j, from (1)–(6), its α-limit is contained, respectively, in Ω(j) union with its

boundary at infinity in the Poincaré compactification in R3, where Ω(j) is equal to

{(x, y, z) ∈ R3 : x = 0}∪{(x, y, z) ∈ R3 : y = 0}∪{(x, y, z) ∈ R3 : z = 0}∪{(x, y, z) ∈
R3 : Fj = 0}.

Theorem 1.1 is proved in Section 3.

Table 1. Conditions on the parameters of system (1.1) to the existence of a
Darboux invariant of the form I(x, y, z, t) = estf(x, y, z, t)

Invariant Planes Conditions on the parameters: s = −4 and
(1) x, y, z, F1 β1 = α2, α2 = −1/3, β2 = −(α3 + 2), β3 = −(α1 + 2), β2 ̸= 1
(2) x, y, z, F2 β3 = α1, α1 = −1/3, β1 = −(α2 + 2), β2 = −(α3 + 2), α3 ̸= 1
(3) x, y, z, F3 β2 = α3, α3 = −1/3, β1 = −(α2 + 2), β3 = −(α1 + 2), β3 ̸= 1
(4) x, y, z, F4 β1 = α3 = 1, α2 = −3, β2 = −3, β3 = −(α1 + 2), (α1 − 1)(β3 − 1) ̸= 0
(5) x, y, z, F5 β2 = α1 = 1, α3 = −3, β3 = −3, β1 = −(α2 + 2), (α2 − 1)(β1 − 1) ̸= 0

(6) x, y, z, F6 β3 = − (sα1−1)(α2−1)(α3−1)
(β1−1)(β2−1) + 1, α1 = − 7+2α3+2α2+α2α3

α2+α3+2 ,

β1 = −(α2 + 2), β2 = −(α3 + 2).

We now examine system (1.1) under each of the parameter conditions specified in Table 1. We
first note that when restricted to conditions (1), (2), and (3) (or alternatively conditions (4) and
(5)) of Table 1, system (1.1) becomes equivalent through the following coordinate transformations:

(1) → (2) (x, y, z, α1, α3) → (x, z, y,−(α2 + 2), α3 − 2),

(2) → (3) (x, y, z, α3, α2) → (y, x, z,−(α1 + 2),−(α2 + 2)),

(4) → (5) (x, y, z, α1) → (x, z, y, β1).

Hence, we shall study only system (1.1) restricted to the conditions (1), (4), and (6) of Table 1.
We start with restricting to conditions (1). System (1.1) restricted to conditions (1) of Table 1 is

x′ = x(1− x− α1y + z/3),

y′ = y(1− y + (α3 + 2)x+ z/3),

z′ = z(1− z − α3x+ (α1 + 2)y),

(1.2)



EJDE-2025/115 DYNAMICS OF A MAY-LEONARD ASYMMETRIC SYSTEM 3

where α3 ̸= −3, α1 are real parameters. Consider U = {(α1, α3) ∈ R2 : α3 ̸= −3} the parameter
space of system (1.2). The next result describes the global dynamics of system (1.2). The definition
of Jacobi multiplier is given in Section 2.

Theorem 1.2. The following statements hold for system (1.2) for the parameters in U .

(a) The phase portraits in the Poincaré sphere are topologically equivalent to one of the phase
portraits of Figure 1. Moreover, the ones in the curves L6,7, L8,9, L12,13 and for the points
P5 and P6 in Figure 1 occur only when α3 = −3.

(b) The phase portraits in the Poincaré disc of system (1.2) restricted to the invariant planes
are topologically equivalent to one of the phase portraits of Figure 2.

(c) It has a Jacobi multiplier.

Theorem 1.2 is proved in Section 4.

Figure 1. Phase portrait of system (1.2) in U on the Poincaré sphere. Phase
portraits for parameters in L6,7, L12,13, P5 and P6 occur only in U \ U .

In the next theorem we study system (1.1) restricted to the conditions (4) of Table 1. The
definition of completely integrability is given in Section 2. System (1.1) restricted to conditions
(4) of Table 1 is given by

x′ = x(1− x− z − α1y), y′ = y(1− y + 3x+ 3z), z′ = z(1− z − x+ (α1 − 2)y), (1.3)

where α1 ̸= {1,−3} is a real parameter. Consider U = {α1 ∈ R : α1 ̸= 1,−3} the parameter space
of system (1.3). The next result describes the global dynamics of system (1.3).



4 F. S. DIAS, R. OLIVEIRA, C. VALLS EJDE-2025/115

Figure 2. Phase portraits of system (4.5) with parameters in U on the Poincaré disc.

Theorem 1.3. The following statements hold for system (1.3) for the parameters in U .

(a) The phase portraits in the Poincaré sphere are topologically equivalent to one of the phase
portraits of Figure 3.

(b) The phase portraits in the Poincaré disc of system (1.3) restricted to the invariant planes
are topologically equivalent to one of the phase portraits of Figure 2.

(c) It is completely integrable (see the proof for a precise definition).

The proof of Theorem 1.3 is given in Section 5.

Figure 3. Phase portrait of system (1.3), with parameters in U , on the Poincaré sphere.

We study system (1.1) restricted to the conditions (6) of Table 1, that is, we consider system

x′ = x
(
1− x+

(α2α3 + 2α2 + α3 + 7)

(α2 + α3 + 2)
y + (α2 + 2)z

)
,

y′ = y(1 + (α3 + 2)x− y − α2z),

z′ = z
(
1− α3x− (α2α3 + 3)

(α2 + α3 + 2)
y − z

)
,

(1.4)

where α2 ̸= −3, α3 ̸= −3 and α2 + α3 + 2 ̸= 0. Consider U = {(α2, α3) ∈ R2 : α2 ̸= −3, α3 ̸=
−3, and α2 + α3 + 2 ̸= 0} the parameter space of system (1.4). The next result describes the
global dynamics of system (1.4).

Theorem 1.4. The following statements hold for system (1.4) for the parameters in U .

(a) The phase portraits in the Poincaré sphere are topologically equivalent to one of the phase
portraits of Figure 4.

(b) The phase portraits in the Poincaré disc of system (1.4) restricted to the invariant planes
are topologically equivalent to one of the phase portraits of Figure 2.



EJDE-2025/115 DYNAMICS OF A MAY-LEONARD ASYMMETRIC SYSTEM 5

(c) In region R1 (see Figure 4), the boundary of the infinity of the second octant is a hete-
roclinic cycle formed by three equilibrium points coming from the ones located at the end
of the x-negative half-axis, and the y and z positive half-axes, and three orbits connecting
these equilibria, each one coming from the orbit at the end of every plane of coordinates.
In the interior of the infinity of the second octant there is an attractor whose orbits fill
completely this interior. The same happens for the fourth octant in the region L34 and in
the third octant in the region L56. In region R3 (see again Figure 4) the heteroclinic cycle
is formed by the orbits connecting the origin with the point p1, the point p1 with the point
p2 and the point p2 with the origin instead of the three orbits connecting the equilibria at
the end of the z-negative half axis, and the x and y positive half-axes.

(d) It is completely integrable.

The proof of Theorem 1.4 is given in Section 6. We have also introduced a section, Section 2,
with the notions and results that will be used to prove Theorems 1.1–1.4.

Figure 4. Phase portraits of system (1.4) on the Poincaré sphere.

2. Preliminaries

We introduce some notion and results that will be used in the proof of the main theorems given
in the introduction.

Let R[x, y, z] be the ring of the real polynomials in the variables x, y and z. We say that
F = F (x, y, z) ∈ R[x, y, z] is a Darboux polynomial or a invariant algebraic surface of the system
(1.1) if there exist K ∈ C[x, y, z] such that ∇F · (P,Q,R) = KF . K is called cofactor of the
invariant surface F (x, y, z) = 0. When K ≡ 0 then F is a first integral of system (1.1).

We say that a nonconstant C1 function I(x, y, z, t) is an invariant of the differential system
(1.1) on an open subset U of R3 if dI/dt = 0 on the trajectories of the system contained in U , i.e.

x(1− x− α1y − β1z)
∂I

∂x
+ y(1− y − β2x− α2z)

∂I

∂y
+ z(1− z − α3x− β3y)

∂I

∂z
+

∂I

∂t
= 0. (2.1)

Then a first integral is an invariant of the system that is independent of the time t. When
a system has a Darboux polynomial F with a constant factor k = k0 ∈ R, then the function
I(x, y, z, t) = F (x, y, z)e−k0t is called a Darboux invariant of that system, see [4, Chapter 8].

The following result was proved in [8].



6 F. S. DIAS, R. OLIVEIRA, C. VALLS EJDE-2025/115

Theorem 2.1. System (1.1) has an invariant plane Fi = 0 different from the planes x = 0, y = 0
and z = 0 passing through the origin if one of the conditions (1)–(6) of Table 2 holds on the
parameter space.

Table 2. Invariant algebraic surfaces and cofactors according to conditions on
the parameters.

Invariant Plane Cofactor Parameters
F1 = (β2 − 1)x+ (1− α1)y K = 1− x− y − β1z, α2 = β1, β2 ̸= 1
F2 = (α3 − 1)x+ (1− β1)z K = 1− x− z − β3y, α1 = β3, α3 ̸= 1
F3 = (β3 − 1)y + (1− α2)z K = 1− y − z − α3x, α3 = β2, β3 ̸= 1
F4 = −(1− β2)(1− β3)x
+(1− α1)(1− β3)y
−(1− α1)(1− α2)z K = 1− x− y − z β1 = α3 = 1, (α1 − 1)(β3 − 1) ̸= 0,

F5 = −(1− α2)(1− α3)x
−(1− β1)(1− β3)y
+(1− α2)(1− β1)z, K = 1− x− y − z, β2 = α1 = 1, (α2 − 1)(β1 − 1) ̸= 0

F6 = −(1− α3)(1− β2)x
+(1− α1)(1− α3)y

+(1− β1)(1− β2)z K = 1− x− y − z, β3 = − (α1−1)(α2−1)(α3−1)
(β1−1)(β2−1) + 1

Let J = J(x, y, z) be a non-negative C1 function non-identically zero on any open subset whose
domain of definition is an open and dense subset of R3. Then J is a Jacobi multiplier of the
differential system (1.1) if∫

Ω

J(x, y, z) dx dy dz =

∫
ϕt(Ω)

J(x, y, z) dx dy dz,

where Ω is any open subset of R3 and ϕt is the flow defined by the differential system (1.1). In
general Jacobi multipliers are very difficult to detect but we have the following result from [9] that
provides a good way for obtaining them.

Proposition 2.2. Let J = J(x, y, z) be a non–negative C1 function non-identically zero on any
open subset of R3 whose domain of definition is dense in R3. Then J is a Jacobi multiplier of the
differential system (1.1) if and only if the divergence of the below differential system (2.2) is zero

x′ = J(x, y, z)x(1− x− α1y − β1z),

y′ = J(x, y, z)y(1− y − β2x− α2z),

z′ = J(x, y, z)z(1− z − α3x− β3y),

(2.2)

Let Hi : Ui → R for i = 1, 2 be two first integrals of the differential system (1.1). We say they
are independent in U1 ∩ U2 if their gradients are independent in all the points of U1 ∩ U2 except
perhaps in a zero Lebesgue measure set. System (1.1) is completely integrable in R3 if it has two
independent first integrals.

We have the following result, which goes back to Jacobi (for a proof see [5, Theorem 2.7], that
is a good tool to determine when a differential is completely integrable.

Theorem 2.3. Consider the differential system (1.1) in R3, and assume that it admits a Jacobi
multiplier J = J(x, y, z) and one first integral H. Then the system admits an additional first inte-
gral functionally independent with the previous one, i.e. the differential system (1.1) is completely
integrable in R3.

3. Proof of Theorem 1.1

We separate the proof of the two statements.



EJDE-2025/115 DYNAMICS OF A MAY-LEONARD ASYMMETRIC SYSTEM 7

Proof of Theorem 1.1(i). We present only the proof of the first case as the proof of the other cases
are similar and hence we have omitted them here. Taking into account that α2 = β1 and β2 ̸= 1,
it follows from Theorem 2.1 that system (1.1) has F1(x, y, z) = 0, x = 0, y = 0 and z = 0 as
invariant planes passing through the origin whose cofactors are given in Table 2.

System (1.1) admits a Darboux invariant of the form I(x, y, z, t) = estf(x, y, z), where f(x, y, z)
is given by the product of invariant planes F1(x, y, z) = 0, x = 0, y = 0 and z = 0, if and only if,
equation (2.1) is satisfied for a real s ̸= 0. Doing so we get

s = −4, α2 = −1/3, β2 = −α3 − 2, β3 = −α1 − 2.

This concludes the proof of statement (i) of the theorem.
The Proof of Theorem 1.1(ii) follows directly from [7, Proposition 6] in the case s < 0.

4. Proof of Theorem 1.2

We separate the proof of each of the statements of the theorem in different subsections.

Proof of Theorem 1.2 (a). We present the Poincaré compactification of system (1.2) in the
local charts Ui, Vi for i = 1, 2, 3 in order to understand the global behavior of the solutions near
infinity. See [1] for the definition of these charts and more details about them.

The expression of the Poincaré compactification of system (1.2) in the local chart U1 is given
by

z′1 = z1(3 + α3 + α1z1 − z1),

z′2 = z2(1− α3 + 2α1z1 + 2 z1 − 4 z2/3),

z′3 = z3(1− z3 + α1z1 − z2/3).

(4.1)

For z3 = 0 (which correspond to the points on the sphere S2 at infinity) system (4.1) becomes

z′1 = z1(3 + α3 + α1z1 − z1), z′2 = z2(1− α3 + 2α1z1 + 2 z1 − 4 z2/3). (4.2)

System (4.2) has the following possible equilibrium points.

p0 = (0, 0), p1 =
(
− α3 + 3

α1 − 1
, 0
)
, p2 =

(
0,−3

4
(α3 − 1)

)
,

p3 =
(
− α3 + 3

α1 − 1
,−3

4

(3α1α3 + 5α1 + α3 + 7)

α1 − 1

)
.

The eigenvalues of the Jacobian matrix evaluated in each of the equilibria are α3 + 3 and 1− α3

for p0, −(3 + α3) and −(3α1α3 + 5α1 + α3 + 7)/(α1 − 1) for p1, (α3 + 3) and α3 − 1 for p2 and
−(3 + α3) and (3α1α3 + 5α1 + α3 + 7)/(α1 − 1) for p3.

When α3 = 1, p2 coalesce with p0 and have a saddle-node. When α1 = −(7+α3)
(5+3α3)

, p3 coalesce

with p1 and have a saddle-node. When (α1 − 1)2 + (α3 + 3)2 = 0 system (4.2) has two lines filled
up of equilibria and it is topologically equivalent to the phase portrait P6 in Figure 1.

The flow in the local chart V1 is the same as the one in U1 because the compactified vector field
p(X) in V1 coincides with the compactified vector field in U1 multiplied by −1. Hence the phase
portrait on V1 is the same as the one in U1 reserving the sense of the flow.

The expression of the Poincaré compactification of system (1.2) in the local chart U2 restricted
to infinity (that is with z3 = 0) becomes

z′1 = −z1(−1 + α1 + α3z1 + 3z1), z′2 = −z2(−3− α1 + 2α3z1 + 2z1 + 4z2/3). (4.3)

In this case system (4.3) has the following possible equilibrium points.

q0 = (0, 0), q1 =
(
0,

3α1 + 9

4

)
, q2 =

(
− α1 − 1

α3 + 3
, 0
)
,

q3 =
(
− (α1 − 1)

(α3 + 3)
,
3

4

(3α1α3 + α3 + 5α1 + 7)

(α3 + 3)

)
.

The equilibria q2 and q3 were studied in the local chart U1. The eigenvalues of the equilibria q0
are 1− α1 and 3 + α1 and the eigenvalues of the equilibria q1 are 1− α1 and −(α1 + 3).

The flow in V2 is the same as the one in U2 reserving the sense of the flow.



8 F. S. DIAS, R. OLIVEIRA, C. VALLS EJDE-2025/115

Finally, the expression of the Poincaré compactification in the local chart U3 is

z′1 = z1(4/3 + α3z1 − z1 − 2z2 − 2α1z2),

z′2 = z2(4/3 + 2α3z1 + 2z1 − 3z2 − α1z2),

z′3 = z3(1 + α3z1 − 2z2 − α1z2 − z3).

(4.4)

System (4.4) restricted to infinity (that is with z3 = 0 ) becomes

z′1 = z1(4/3 + α3z1 − z1 − 2z2 − 2α1z2),

z′2 = z2(4/3 + 2α3z1 + 2z1 − 3z2 − α1z2).

Now the only point of the local chart U3 which is not covered by the local charts U1, V1, U2 and
V2 is the origin of coordinates of U3. The eigenvalues of the origin of U3 are 4/3 and 4/3.

The flow at infinity in the local chart V3 is the same as the flow in the local chart U3 reversing
the time. Accordingly, we define the following bifurcation curves and regions, as in Figure 5.

L1,2 =
{
(α1, α3) ∈ R2 : α1 = 1, α3 > 1

}
, L2,3 =

{
(α1, α3) ∈ R2 : α3 > 1,L = 0

}
,

L3,4 =
{
(α1, α3) ∈ R2 : α1 = −3, α3 > 1

}
, L4,5 =

{
(α1, α3) ∈ R2 : α3 = 1, α1 < −3

}
,

L5,6 =
{
(α1, α3) ∈ R2 : α1 < −3,L = 0

}
, L6,7 =

{
(α1, α3) ∈ R2 : α3 = −3, α1 < −3

}
,

L6,9 =
{
(α1, α3) ∈ R2 : α1 = −3,−3 < α3 < −1

}
, L7,8 =

{
(α1, α3) ∈ R2 : α1 = −3, α3 < −3

}
,

L8,9 =
{
(α1, α3) ∈ R2 : α3 = −3,−3 < α1 < 1

}
, L9,10 =

{
(α1, α3) ∈ R2 : −3 < α1 < −1,L = 0

}
,

L3,10 =
{
(α1, α3) ∈ R2 : α3 = 1,−3 < α1 < −1

}
,

L5,10 =
{
(α1, α3) ∈ R2 : α1 = −3,−3 < α3 < −1

}
,

L2,9 =
{
(α1, α3) ∈ R2 : α3 = 1,−1 < α1 < 1

}
, L8,11 =

{
(α1, α3) ∈ R2 : α3 < −1,L = 0

}
,

L11,12 =
{
(α1, α3) ∈ R2 : α1 = 1, α3 < −3

}
, L12,13 =

{
(α1, α3) ∈ R2 : α3 = −3, α1 > 1

}
,

L13,14 =
{
(α1, α3) ∈ R2 : α1 > 1,L = 0

}
, L9,14 =

{
(α1, α3) ∈ R2 : α1 = 1,−3 < α3 < 1

}
,

L1,14 =
{
(α1, α3) ∈ R2 : α3 = 1, α1 > 1

}
where L = 3α1α3 + α3 + 5α1 + 7.

In Table 3 we provide the description of the topological type of each equilibria (pi and qj , i =
0, 1, 2, 3 and j = 0, 2) according to the values of the parameters α1 and α3 in each the local charts
U1 U2 and U3, respectively. This table use the following notations: S (saddle), UN (unstable
node), SN (stable node), S-N (saddle-node) and PP (phase portrait). The corresponding phase
portraits are given in Figure 1.

Proof of Theorem 1.2 (b). In this section we study the dynamics of system (1.2) restricted to
the invariants planes.
Restriction of system (1.2) to the invariant plane z = 0. On the invariant plane z = 0 system 1.2
becomes

x′ = x(1− x− α1y), y′ = y(1− y + (α3 + 2)x). (4.5)

The equilibrium points of systems (4.5) are

r0 = (0, 0), r1 = (0, 1), r2 = (1, 0), r3 =
(
− α1 − 1

α1α3 + 2α1 + 1
,

α3 + 3

α1α3 + 2α1 + 1

)
.

The eigenvalues of the Jacobian matrix evaluated at each of the equilibria are 1, 1 for r0; 1−α1,−1

for r1; −1, α3 + 3 for r2; and −1, (α3+3)(α1−1)
α1α3+2α1+1 for r3. When α1 = 1, r3 coalesce with r1 and have

a saddle-node. Finally, when α1 = 1 and α3 = −3 system (4.5) becomes

x′ = x(1− x− y), y′ = y(1− x− y). (4.6)

The phase portrait of system (4.6) has a line of equilibria and it is given in Figure 2.
To study the infinite equilibrium points of system (4.5) we use the Poincaré compactification

for a polynomial vector field in R2 which is described in [4, Chapter 5].



EJDE-2025/115 DYNAMICS OF A MAY-LEONARD ASYMMETRIC SYSTEM 9

Figure 5. Bifurcation diagram of system (1.2) for the parameters in U . Note
that the transitions L6,7, L8,9, L12,13 and the points P5 and P6 belong to α3 = −3.

The Poincaré compactification of system (4.5) in the local chart U1 is given by

u′ = u((α1 − 1)u+ α3 + 3), v′ = v(1 + α1u− v).

So, there are two equilibria on v = 0, namely s0 = (0, 0) and s1 =
(
− (α3 + 3), (α1 − 1)

)
. The

eigenvalues of the Jacobian matrix evaluated at these two equilibria are 1, (α3 + 3) for s0 and
−(α3 + 1),−(α1α3 + 2α1 + 1)/(α1 − 1) for s1.

The Poincaré compactification of system (4.5) in the local chart U2 is given by

u′ = u((α3 + 3)u+ α1 − 1), v′ = −v(v − 1 + (α3 + 2)u).

In this case the origin is an equilibrium point whose eigenvalues of its Jacobian matrix are 1, α3+3.
We present the bifurcation diagram and the phase portraits of system (4.5) in Figures 6 and 2,

respectively. In Figure 6 we have the following curves:

L1,2 =
{
(α1, α3) ∈ R2 : α1 = 1, α3 > −3

}
, L2,3 =

{
(α1, α3) ∈ R2 : α1 < 0,L = 0

}
,

L2,4 =
{
(α1, α3) ∈ R2 : α3 = −3, α1 < 1

}
, L4,5 =

{
(α1, α3) ∈ R2 : 0 < α1 < 1,L = 0

}
,

L5,6 =
{
(α1, α3) ∈ R2 : α1 = 1, α3 < −3

}
, L6,7 =

{
(α1, α3) ∈ R2 : α3 = −3, α1 > 1

}
,

L1,7 =
{
(α1, α3) ∈ R2 : α1 > 1,L = 0

}
where L = α1α3 + 2α1 + 1 = 0. The topological type of each equilibrium point of system (4.5) is
given in Table 4.
Restriction of system (1.2) to the invariant planes x = 0, y = 0 and F1(x, y) = 0. System (1.2)
restricted to the planes x = 0 and y = 0 is equivalent to system 4.5 by the change of coordinates
(x, y, α1, α3) → (y, z,−1/3, α1) and (x, y, α1, α3) → (x, z,−1/3,−(α3 + 2)), respectively. On the
other hand, system (1.2) restricted to the plane F1(x, y) = 0 is equivalent to system 4.5 by the
change of coordinates

(x, y, α1, α3) →
( (α1α3 + 2α1 + 1)

(α3 + 3)
y, z,−1/3,

α3 + 2α1α3 + 3α1 + 6

1 + 2α1 + α1α3
− 2

)
.



10 F. S. DIAS, R. OLIVEIRA, C. VALLS EJDE-2025/115

Table 3. Topological type of each equilibria of system (1.2) on the Poincaré
sphere according to the values (α1, α3) ∈ U .

p0 p1 p2 p3 q1 q2 w0 PP
R1 S SN UN S S SN UN R1

R2 S S UN SN UN S UN R2

R3 S SN UN S UN S UN R3

R4 S SN UN S S UN UN R1

R5 UN SN S S S UN UN R3

R6 UN S S SN S UN UN R1

R7 S UN SN S S UN UN R1

R8 S UN SN S UN S UN R3

R9 UN S S SN UN S UN R3

R10 UN SN S S UN S UN R3

R11 S S SN UN UN S UN R2

R12 S UN SN S S SN UN R1

R13 UN S S SN S SN UN R1

R14 UN SN S S S SN UN R3

P1 S-N ∄ p0 ∄ S-N S-N UN P1

P2 S-N S-N p0 p1 UN S UN L8,9

P3 S-N SN p0 S S-N q1 UN P3

P4 UN S-N S p1 S-N q1 UN P3

P5 S-N p0 S-N p2 S-N q1 UN P5

P6 - - - - - - - P6

p0 p1 p2 p3 q1 q2 w0 PP
L1,2 S ∄ UN ∄ S-N S-N UN L1,2

L2,3 S S-N UN p1 UN S UN L2,3

L3,4 S SN UN S S-N q1 UN L3,4

L4,5 S-N SN p0 S S UN UN L4,5

L5,6 UN S-N S p1 S UN UN L4,5

L6,7 S-N p0 S-N p2 S UN UN L6,7

L7,8 UN S S SN S-N q1 UN L3,4

L6,9 UN SN S S S-N q1 UN L3,4

L8,9 S-N p0 S-N p2 UN S UN L8,9

L9,10 UN S-N S p1 UN S UN L3,10

L5,10 UN SN S S S-N q1 UN L5,10

L8,11 S S-N S p1 UN S UN L2,3

L11,12 S ∄ SN ∄ S-N S-N UN L1,2

L12,13 S-N p0 S-N p2 S SN UN L6,7

L13,14 UN S-N S p1 S SN UN L4,5

L9,14 UN ∄ S ∄ S-N S-N UN L9,14

L2,9 S-N S p0 SN UN S UN L2,3

L3,10 S-N SN p0 S UN S UN L3,10

L1,14 S-N SN p0 S S SN UN L4,5

Therefore, this new system has only the previously studied phase portraits except when the pa-
rameters α1 and α3 belong to the curve L1 given by 2α1 + α1α3 + 1 = 0. This system for the
parameters α1 and α3 on this curve is given by

y′ = y
(z
3
+ 1

)
, z′ = z(1− z +Ay), (4.7)



EJDE-2025/115 DYNAMICS OF A MAY-LEONARD ASYMMETRIC SYSTEM 11

Figure 6. Bifurcation diagram of system (4.5). Note that transitions L2,4, L6,7

and the point P1 belong to the line α3 = −3.

Table 4. Topological type of each equilibria of system (4.5) on the Poincaré disc
according to the values (α1, α3) ∈ U .

Regions r0 r1 r2 r3 s0 s1 w0 PP
R1 UN SN S S UN SN S R1

R2 UN S S SN UN S UN R1

R3 UN S S S UN SN UN R3

R4 UN S SN S S UN UN R1

R5 UN S SN SN S S UN R5

R6 UN SN SN S S UN S R5

R7 UN SN S SN UN S S R5

L1,2 UN S-N S r1 UN ∄ S-N L1,2

L2,3 UN S S ∄ UN S-N UN L2,3

L2,4 UN S S-N r2 S-N s0 UN L1,2

L4,5 UN S SN ∄ S S-N UN L4,5

L5,6 UN S-N SN r1 S ∄ S-N L5,6

L6,7 UN SN S-N r2 S-N s1 S L5,6

L1,7 UN SN S ∄ UN S-N S L4,5

P1 - - - - - - - P1

where A = (α3 + 3)/(α3 + 2). Note α3 ̸= −2, otherwise (α1,−2) /∈ L1. Since α3 ̸= −3, system
(4.7) is equivalent to

y′ = y(
z

3
+ 1), z′ = z(1− z + y). (4.8)

The phase portrait of system (4.8) is topologically equivalent to L2,3 of Figure 2.



12 F. S. DIAS, R. OLIVEIRA, C. VALLS EJDE-2025/115

Proof of Theorem 1.2 (c). System (1.1) becomes system (1.2) and so system (2.2) becomes

x′ = J(x, y, z)x(1− x− α1y + z/3),

y′ = J(x, y, z)y(1− y + (α3 + 2)x+ z/3),

z′ = J(x, y, z)z(1− z − α3x+ (α1 + 2)y).

(4.9)

We define

J = x
7α1+9

4(α1−1) y
7α3+5

4(α3+3) z7/4((1− α1)y − (3 + α3)x)
− 9α1α3+11α1+7α3+37

4(α1−1)(α3+3) .

It is easy to see that J is a Jacobi multiplier of system (1.2) because one can check directly that
the divergence of system (4.9) is zero. This completes the proof of Theorem 1.2(c) and so the
proof of Theorem 1.2.

5. Proof of Theorem 1.3

We separate the proof of each of the statements of the theorem in different subsections.

Proof of Theorem 1.3(a). The Poincaré compactification of system (1.3) in the local chart U1

is given by
z′1 = z1(4 + (α1 − 1)z1 + 4z2),

z′2 = 2(α1 + 1)z1z2,

z′3 = z3(−z3 + 1 + α1z1 + z2).

(5.1)

For z3 = 0 system (5.1) becomes

z′1 = z1(4 + (α1 − 1)z1 + 4z2), z′2 = 2(α1 + 1)z1z2. (5.2)

So z1 = 0 is a line of equilibrium points. After a convenient rescaling of the time we eliminate
the common factor z1 and we conclude that system (5.2) admits a unique equilibrium point (with
α1 ̸= 1) described by p0 = (−4/(α1 − 1), 0). The eigenvalues of the Jacobian matrix evaluated at
p0 are −4 and −8(α1 + 1)/(α1 − 1). If α1 = 1, system (5.2) has no equilibrium points. Moreover,
when α1 = −1 system (5.1) becomes

z′1 = 2z1(2− z1 + 2z2), z′2 = 0.

The Poincaré compactification of system (1.3) in U2 restricted to the infinity becomes

z′1 = −z1(α1 − 1 + 4(z1 + z2)), z′2 = −z2(−α1 − 3 + 4(z1 + z2)). (5.3)

In this case system (5.3) (with α1 ̸= −1) has the following equilibrium points

q0 = (0, 0), q1 =
(
0,

α1 + 3

4

)
, q2 =

(
− α1 − 1

4
, 0
)
.

The equilibria q2 has already been studied in the local chart U1. The eigenvalues of the equilibria
q0 are 1 − α1 and 3 + α1 and for the equilibria q1 we have the following eigenvalues −2(1 + α1)
and −(α1 + 3). When α1 = −1 system (5.3) becomes

z′1 = 2z1(1− 2(z1 + z2)), z′2 = 2z2(1− 2(z1 + z2)).

So 2(z1 + z2) = 1 is a line of equilibrium points. The Poincaré compactification in the local chart
U3 restricted to the infinity is

z′1 = −2(α1 + 1)z1z2, z′2 = z2(4 + 4z1 − α1z2 − 3z2). (5.4)

So z2 = 0 is a line of equilibrium points, that after a convenient rescaling of time, it can be
eliminated and the new resulting system does not has any distinguished equilibrium point in the
local chart U3. When α1 = −1 system (5.4) becomes

z′1 = 0, z′2 = 2z2(2z1 − z2 + 2).

In short we have Table 5. This table uses the following notations: S (saddle), UN (unstable node),
SN (stable node), S-N (saddle-node) and PP (phase portrait). The corresponding phase portraits
are given in Figure 3.



EJDE-2025/115 DYNAMICS OF A MAY-LEONARD ASYMMETRIC SYSTEM 13

Table 5. Topological type of each equilibria of system (1.3) on the Poincaré
sphere according to the values (α1, α3) ∈ U .

Regions p0 q0 q1 PP
α1 < −3 SN S UN R1

α1 = −3 SN S-N q0 L1,2

−3 < α1 < −1 SN UN S R1

α1 = −1 - - - L2,3

−1 < α1 < 1 S UN SN R2

α1 = 1 ∄ S-N SN L3,4

α1 > 1 SN S SN R1

Proof of Theorem 1.3(b). System (1.3) restricted to the planes x = 0, y = 0 and z = 0 is equiv-
alent to system (4.5) by the change of coordinates (x, y, α1, α3) → (y, z,−3, α1), (x, y, α1, α3) →
(x, z, 1,−3) and (x, y, α1, α3) → (x, y, α1, 1), respectively. On F4 = 0 system (1.3) becomes

x′ = x
(
1− 2

(α1 + 1)

(α1 − 1)
x− (5α1 + 3)

4
y
)
, y′ = y

(
1− 6

(α1 + 1)

(α1 − 1)
x+

(3α1 + 5)

4
y
)
. (5.5)

System (5.5) is equivalent to system (4.5) by the change of coordinates

(x, y, α1, α3) →
(
2
(α1 + 1

α1 − 1

)
x,− (3α1 + 5)

4
y,−5α1 + 3

3α1 + 5
,−5

)
.

This new system has only the previously studied phase portraits except when the parameter
α1 ∈ {−1, 1,−5/3,−3/5}. When α1 = −1,−5/3,−3/5, the phase portraits are topologically
equivalent to P1, L2,3, R1 of Figure 2, respectively and system (5.5) is not defined for α1 = 1.

Proof of Theorem 1.3(c). System (1.1) becomes (1.3). It is easy to see that

H = xα1+3zα1−1(−4(α1 + 3)x− (α1 − 1)((α1 + 3)y − 4z))−2(α1+1)

is a first integral of system (1.3). Since now we are working with system (1.3), system (2.2)
becomes

x′ = J(x, y, z)x(1− x− z − α1y),

y′ = J(x, y, z)y(1− y + 3x+ 3z),

z′ = J(x, y, z)z(1− z − x+ (α1 − 2)y).

(5.6)

We define
J = y

3
4 z

3
α1+3 (−4(α1 + 3)x− (α1 − 1)((α1 + 3)y − 4z))

9
4−

3
α1+3 .

It is easy to see that J is a Jacobi multiplier of system (1.3) because one can check directly that the
divergence of system (5.6) is zero. Now, since system (1.3) has the first integral H and a Jacobi
multiplier, by Theorem 2.3, it is completely integrable. This completes the proof of Theorem
1.3(c) and so the proof of Theorem 1.3.

6. Proof of Theorem 1.4

We separate the proof of each of the statements in different subsections.

Proof of Theorem 1.4 (a). The Poincaré compactification of system (1.4) in the local chart U1

restricted to infinity becomes

z′1 = z1

(
− (α3 + 3)(α2 + 3)

α2 + α3 + 2
z1 − 2(α2 + 1)z2 + α3 + 3

)
,

z′2 = z2

(
− 2

(α3 + α2α3 + α2 + 5)

α2 + α3 + 2
z1 − (α2 + 3)z2 − α3 + 1

)
.

(6.1)

The equilibrium points on the local chart U1 are

p0 = (0, 0), p1 =
(α2 + α3 + 2

α2 + 3
, 0
)
, p2 =

(
0,−α3 − 1

α2 + 3

)
,



14 F. S. DIAS, R. OLIVEIRA, C. VALLS EJDE-2025/115

p3 =
(
− α2 + α3 + 2

α2 − 1
,
α3 + 3

α2 − 1

)
.

The eigenvalues of the Jacobian matrix evaluated in each of the equilibria are α3 + 3 and 1− α3

for p0, −(3 + α3) and −(3α2α3 + 5α3 + α2 + 7)/(α2 + 3) for p1, (3α2α3 + 5α3 + α2 + 7)/(α2 + 3)

and (α3 − 1) for p2 and ±
√

−(α2 − 1)(α3 + 3)(3α2α3 + 5α3 + α2 + 7)/(α2 − 1) for p3.
To study the local behavior of p3, we must follow the following steps: shift p3 to the origin;

diagonalize the linear part (with a linear change of coordinates) and apply [4, Theorem 8.15 (iii)].
From this, we conclude that the point p3 is a center. The stability for each of these equilibria is
presented in Table 6.

We observe that when 3α2α3 + 5α3 + α2 + 7 = 0 system (6.1) becomes

z′1 = z1(α3 + 3)
( −4

(3α3 + 5)
z1 +

4

(3α3 + 1)
z2 + 1

)
,

z′2 = −z2(α3 − 1)
( −4

(3α3 + 5)
z1 +

4

(3α3 + 1)
z2 + 1

)
.

(6.2)

For system (6.2), we have the origin and the line

z1 =
(3α3 + 5)(4 z2 + 3α3 + 1)

4(3α3 + 1)

as equilibrium points. Finally, when α3 = −3 system (6.2) has a line z2 = 0 filled up of equilibria.
The Poincaré compactification of system (1.4) in the local chart U2 restricted to infinity becomes

z′1 = z1

(
− (α3 + 3)z1 + 2(α2 + 1)z2 +

(α3 + 3)(α2 + 3)

α2 + α3 + 2

)
,

z′2 = z2

(
− 2(α3 + 1)z1 + (α2 − 1)z2 −

(α3 − 1)(α2 − 1)

α2 + α3 + 2

)
.

(6.3)

So, in the local chart U2, there are two equilibrium points not yet studied on the chart U1

q0 = (0, 0), q1 =
(
0,

α3 − 1

α2 + α3 + 2

)
.

The eigenvalues of the equilibria q0 are (α3 +3)(α2 +3)/(α2 +α3 +2) and (α3 − 1)(α2 − 1)/(α2 +
α3 + 2) and the eigenvalues of the equilibria q1 are (α2 + 3α2α3 + 5α3 + 7)/(α2 + α3 + 2) and
(α3−1)(α2−1)/(α2+α3+2). Again, when 3α2α3+5α3+α2+7 = 0 and α2+α3+2 ̸= 0 system
(6.3) becomes

z′1 = −z1(α3 + 3)
(
z1 +

4

(3α3 + 1)
z2 −

4

3α3 + 5

)
,

z′2 = −2z2(α3 + 1)
(
z1 +

4

(3α3 + 1)
z2 −

4

3α3 + 5

)
.

(6.4)

In this case, system (6.4) has as equilibrium points the origin and the line z1 = − 4
(3α3+1)z2+

4
3α3+5 .

In the chart U3 the origin is an equilibrium point whose eigenvalues of the Jacobian matrix are
α2 + 3 and 1− α2.

In summary, we present the bifurcation diagram and the topological type of each equilibrium
point of system (1.4) in Figure 7 and Table 4, respectively. Consider the following curve L =
3α2α3 + 5α3 + α2 + 7 = 0 in Figure 7.

L1,2 =
{
(α2, α3) ∈ R2 : α2 = 1, α3 > 1

}
, L2,3 =

{
(α2, α3) ∈ R2 : α3 > 1,L = 0

}
,

L3,4 =
{
(α2, α3) ∈ R2 : α2 = −3, α3 > 1

}
, L5,6 =

{
(α2, α3) ∈ R2 : α3 = 1, α2 < −3

}
,

L6,7 =
{
(α2, α3) ∈ R2 : α2 < −3,L = 0

}
, L7,8 =

{
(α2, α3) ∈ R2 : α3 = −3, α2 < −3

}
,

L8,9 =
{
(α2, α3) ∈ R2 : α2 = −3, α3 < −3

}
, L9,10 =

{
(α2, α3) ∈ R2 : α3 = −3,−3 < α2 < −1

}
,

L9,14 =
{
(α2, α3) ∈ R2 : α3 < −3,L = 0

}
, L13,14 =

{
(α2, α3) ∈ R2 : α3 = −3,−1 < α2 < 1

}
,

L10,13 =
{
(α2, α3) ∈ R2 : −3 < α3 < −5/3,L = 0

}
,

L11,12 =
{
(α2, α3) ∈ R2 : −5/3 < α3 < −1,L = 0

}
,



EJDE-2025/115 DYNAMICS OF A MAY-LEONARD ASYMMETRIC SYSTEM 15

L2,11 =
{
(α2, α3) ∈ R2 : α3 = 1,−3 < α2 < 1

}
, L11,18 =

{
(α2, α3) ∈ R2 : α2 = 1,−1 < α3 < 1

}
,

L12,17 =
{
(α2, α3) ∈ R2 : α2 = 1,−3 < α3 < −1

}
, L14,15 =

{
(α2, α3) ∈ R2 : α2 = 1, α3 < −3

}
,

L16,17 =
{
(α2, α3) ∈ R2 : α3 = −3, α1 > 1

}
, L17,18 =

{
(α2, α3) ∈ R2 : α2 > 1,L = 0

}
,

L1,18 =
{
(α2, α3) ∈ R2 : α3 = 1, α2 > 1

}
.

The phase portraits of system (1.4) are presented in Figure 4.

Figure 7. Bifurcation diagram of system (1.4). Note that transitions L3,4, L7,10

and L8,9 belong to the line α2 = −3, while transitions L7,8, L9,10, L13,14 and L16,17

belong to the line α3 = −3.

Proof of Theorem 1.4 (b). System (1.4) restricted to the planes x = 0, y = 0 and z = 0 is equiv-
alent to system (4.5) by the change of coordinates (x, y, α1, α3) → (y, z, α2, γ), (x, y, α1, α3) →
(x, y, γ, α3) and (x, y, α1, α3) → (x, z,−(α2+2),−(α3+2)), respectively, where γ = − α2α3+3

α2+α3+2 −2

On the invariant plane F6 = 0 system (1.4) becomes

x′ = x
( (2α2α3 + 4α3 + α2 + 5)

α2 + α3 + 2
y − (1 + α2α3 + 2α3)

α2 + 3
x+ 1

)
,

y′ = y
(
− (α2α3 + α3 + 2)

α2 + α3 + 2
y +

(3α3 + 2α2α3 + α2 + 6)

α2 + 3
x+ 1

)
.

(6.5)

System (6.5) is equivalent to (4.5) by the change of coordinates (x, y, α1, α3) → (Ax,By,C,D),
where

A =
α2α3 + 2α3 + 1

α2 + 3
, B =

α3 + α2α3 + 2

α2 + α3 + 2
,

C = −2α2α3 + 4α3 + α2 + 5

α3 + α2α3 + 2
, D =

α2 + 6 + 2α2α3 + 3α3

1 + α2α3 + 2α3
.

Therefore, system (6.5) has only the previously studied phase portraits except when the parameters
α2 and α3 belong the curve L1 = α2α3 + 2α3 + 1 = 0 or L2 = α3 + α2α3 + 2 = 0.



16 F. S. DIAS, R. OLIVEIRA, C. VALLS EJDE-2025/115

Table 6. Topological type of each equilibria of system (1.4) on the Poincaré
sphere according to the values of α2 and α3.

Region p0 p1 p2 p3 q0 q1 w0 PP
R1 S SN UN C S UN S R1

R2 S SN UN S UN S UN R2

R3 S S S C UN SN UN R3

R4 S SN UN C S SN S R1

R5 S SN UN C S UN S R1

R6 UN SN S C UN S S R1

R7 UN S SN S UN SN S R2

R8 S UN SN C S SN S R1

R9 S S S C UN SN UN R3

R10 UN SN S S S SN UN R2

R11 UN SN S S S UN UN R2

R12 UN S SN C S S UN R1

R13 UN S SN C S S UN R1

R14 S UN SN S UN S UN R2

R15 S UN SN C S UN S R1

R16 S UN SN C S SN S R1

R17 UN S SN S UN SN S R2

R18 UN SN S C UN S S R1

P1 S-N SN p0 ∄ - - - P1

P2 - - - - ∄ SN S-N P2

P3 - - - - ∄ ∄ UN P3

P4 UN - - - - - - P4

Region p0 p1 p2 p3 q0 q1 w0 PP
L1,2 S SN UN ∄ - - - L1,2

L2,3 S - - - UN ∄ UN L2,3

L3,4 S ∄ ∄ C S-N SN S-N L3,4

L5,6 S-N SN p0 C S-N q0 S L5,6

L2,11 S-N SN p0 S S-N q0 UN L2,11

L1,18 S-N SN p0 C S-N q0 S L5,6

L6,7 UN - - - UN ∄ S L2,3

L9,14 S - - - UN ∄ UN L2,3

L10,13 UN - - - S ∄ UN L2,3

L11,12 UN - - - S ∄ UN L2,3

L17,18 UN - - - UN ∄ S L2,3

L7,10 UN ∄ ∄ S S-N SN S-N L7,10

L8,9 S ∄ ∄ C S-N SN S-N L3,4

L7,8 - - SN - ∄ SN S L7,8

L9,10 - - S - ∄ SN UN L2,3

L13,14 - - SN - ∄ S UN L7,8

L16,17 - - SN - ∄ SN S L7,8

L11,18 UN SN S ∄ - - - L1,2

L12,17 UN S SN ∄ - - - L1,2

L14,15 S UN SN ∄ - - - L1,2

System (6.5) for the parameters α2 and α3 on the curve L1 = 0, after a trivial change of
coordinates, is given by

x′ = x(1 + y), y′ = y
(
1 + x− 1

(α2 + 2)
y
)
. (6.6)



EJDE-2025/115 DYNAMICS OF A MAY-LEONARD ASYMMETRIC SYSTEM 17

When α2 < −3 or −3 < α2 < −2, the phase portrait is topologically equivalent to L4,5 of Figure
2 and when α2 > −2, it is topologically equivalent to L2,3 of Figure 2.

Finally, system (6.5) for the parameters α2 and α3 on the curve L2 = 0, after a trivial change
of coordinates (considering α2 ̸= 1), is given by

x′ = x(1 + y), y′ = y
(
1 + x+

1

α2
y
)
. (6.7)

System (6.7) is equivalent to system (6.6). If α2 = 1 and α3 = −1, system (6.5) is given by

x′ = x
(
1 +

x

2

)
, y′ = y

(
1 +

x

2

)
. (6.8)

The phase portrait of system (6.8) is topologically equivalent to P1 of Figure 2.

Proof of Theorem 1.4 (c). We only prove the case when the parameters belong to the region
R1 since the other cases can be done in the same manner. In that case, the open arc of the infinity
of the second octant in the local chart U1 corresponding to the end of the plane z = 0 is formed
by an orbit having as ω-limit the equilibrium (0, 0, 0). The open arc of the infinity of the second
octant corresponding to the end of the plane y = 0 is formed by an orbit having as α-limit the
equilibrium (0, 0, 0). The orbit on the x-axis near of (0, 0, 0) has this equilibrium as its α-limit.
Similar studies can be done for the equilibria located at the origin of the local charts U2 and U3

that is, at the end of the y- and z-axes, respectively. Hence the boundary of the infinity of the
second octant is formed by a heteroclinic cycle formed by three equilibria coming from the ones
located at the end of the negative x half-axis and the positive y and z half-axes, and the three
orbits living on the three open arcs connecting these three points and contained in the boundary
of the infinity of the second octant. From the study of the global dynamics on the Poincaré sphere
we see that there is an additional equilibria in the interior of the heteroclinic cycle that is an stable
attractor and it is the α-limit set of each point on the interior of the heteroclinic cycle.

Proof of Theorem 1.4 (d). System (1.1) becomes system (1.4). It is easy to see that

H = x(α2−1)(α3−1)y(α2+3)(α3−1)z(α2+3)(α3+3)T (x, y, z)−α2(3α3+1)−5α3−7,

where T (x, y, z) = (α3 − 1)(α2 +α3 +2)x+ (α2 +3)
(
(1−α3)y+ (α2 +α3 +2)z

)
is a first integral

of system (1.4). Since now we are working with system (1.4), system (2.2) becomes

x′ = J(x, y, z)x
(
1− x+

(α2α3 + 2α2 + α3 + 7)

(α2 + α3 + 2)
y + (α2 + 2)z

)
,

y′ = J(x, y, z)y(1 + (α3 + 2)x− y − α2z),

z′ = J(x, y, z)z
(
1− α3x− (α2α3 + 3)

(α2 + α3 + 2)
y − z

)
.

(6.9)

We define

J = y−
3

α2−1 z
− 3(α2+α3+2)

(α2−1)(α3−1)R(x, y, z)
3(α2α3+α3+2)

(α2−1)(α3−1) ,

where R(x, y, z) = (α3−1)(α2+α3+2)x− (α2+3)(α3−1)y+(α2+3)(α2+α3+2)z. It is easy to
see that J is a Jacobi multiplier of system (1.4) because one can check directly that the divergence
of system (6.9) is zero. Now, since system (1.4) has the first integral H and a Jacobi multiplier,
by Theorem 2.3, it is completely integrable. This completes the proof of Theorem 1.4(d) and so
the proof of Theorem 1.4.

Acknowledgements. F. S. Dias was supported by FAPEMIG grant APQ-00628-24. R. Oliveira
was supported by CNPq grants 310857/2023-6 and 407454/2023-3, DSYREKI- HORIZON-MSCA-
2023-SE-01 101183111, and FAPESP grant 2019/21181-0. C. Valls was supported by FCT/Portugal
through CAMGSD, IST-ID by UIDB/04459/2020 and UIDP/04459/2020.



18 F. S. DIAS, R. OLIVEIRA, C. VALLS EJDE-2025/115

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Fabio Scalco Dias

Instituto de Matemática e Computação, Universidade Federal de Itajubá, Avenida BPS 1303, Pinheir-
inho, CEP 37.500-903, Itajubá, MG, Brazil

Email address: scalco@unifei.edu.br

Regilene Oliveira
Departamento de Matemática, ICMC-Universidade de São Paulo, Avenida Trabalhador São-carlense,

400 - 13566-590, São Carlos, SP, Brazil
Email address: regilene@icmc.usp.br

Cláudia Valls

Departamento de Matemática, Instituto SuperiorTécnico, Universidade Técnica de Lisboa, Av. Rovisco
Pais 1049-001, Lisboa, Portugal

Email address: claudia.valls@tecnico.pt


	1. Introduction and statement of the main results
	2. Preliminaries
	3. Proof of Theorem ??
	4. Proof of Theorem ??
	Proof of Theorem ?? (a)
	Proof of Theorem ?? (b)
	Proof of Theorem ?? (c).

	5. Proof of Theorem ??
	Proof of Theorem ??(a)
	Proof of Theorem ??(b)
	Proof of Theorem ??(c)

	6. Proof of Theorem ??
	Proof of Theorem ?? (a)
	Proof of Theorem ?? (b)
	Proof of Theorem ?? (c)
	Proof of Theorem ?? (d)
	Acknowledgements

	References