Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 36, pp. 1–12.

ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu

DOI: 10.58997/ejde.2025.36

GLOBAL ASYMPTOTIC STABILITY IN QUADRATIC SYSTEMS

JAUME LLIBRE, CLAUDIA VALLS

Abstract. A classical problem in the qualitative theory of differential systems that is relevant
for its applications, is to characterize the differential systems which are globally asymptotically

stable, that is differential systems having a unique equilibrium point for which all their orbits,

with the exception of the equilibrium point, tend in forward time to this equilibrium point.
Here we provide three conditions that characterize the global asymptotic stability for planar

quadratic polynomial differential systems. Using these three conditions we characterize all
planar quadratic polynomial differential systems that are globally asymptotically stable.

1. Introduction and statement of main results

Let f = (f1, f2) : R2 → R2 be a polynomial map. In this article we deal with the polynomial
differential systems

ẋ = f1(x, y), ẏ = f2(x, y), (1.1)

using a dot for the derivative with respect to t.
These systems are globally asymptotically stable if they have an equilibrium point p such that

any orbit (x(t), y(t)) with maximal interval (α, ω) different from p satisfies that (x(t), y(t)) → p
as t → ω.

To find conditions which guarantee global asymptotic stability of an equilibrium point in a
planar polynomial differential system is, in general, a difficult problem. Lyapunov’s function
method is probably the most common method used, but in general to find a Lyapunov function
is not easy.

There is a result proven in 1993 and known as the Markus-Yamabe conjecture which provides
sufficient but not necessary conditions for the global asymptotic stability in the plane. These
conditions are that the differential system has a unique equilibrium point, the trace of the Jacobian
matrixDf is negative and its determinant is positive for all (x, y) ∈ R2. While the Markus-Yamabe
conditions for C1 differential systems in R2 guarantee asymptotic stability in R2 (see [9, 11, 12])
this is not the case in Rn with n > 2, see [2, 4].

In this article we provide necessary and sufficient conditions that characterize the global as-
ymptotic stability of planar quadratic polynomial differential systems. Using this characterization
we are able to characterize all quadratic systems that are globally asymptotically stable. We recall
that the polynomial differential system (1.1) has degree n if the maximum of the degrees of the
polynomials f1(x, y) and f2(x, y) is n. When n = 2 the planar polynomial differential system (1.1)
is called simply a quadratic system.

We recall that a polynomial differential system in the plane is bounded if there are no orbits
(x(t), y(t)) with maximal interval of definition (α, ω) such that x(t)2 + y(t)2 → ∞ when t → ω,
i.e. the ω-limit of any orbit never is the infinity. We recall that a limit cycle is a periodic orbit
isolated in the set of periodic orbits of the differential system. The first result of the paper is the
following proposition.

2020 Mathematics Subject Classification. 34C05.

Key words and phrases. Global asymptotic stability; planar polynomial vector fields; quadratic systems.
©2025. This work is licensed under a CC BY 4.0 license.
Submitted June 2, 2024. Published April 7, 2025.

1



2 J. LLIBRE, C. VALLS EJDE-2025/36

Proposition 1.1. The quadratic system (1.1) is globally asymptotically stable if and only if it
satisfies the following three conditions:

(C1) The differential system (1.1) has a unique equilibrium point p ∈ R2.
(C2) The differential system (1.1) has no periodic orbits.
(C3) The differential system (1.1) is bounded.

As we shall see the proof of Proposition 1.1 mainly follows from previous results in [6], summa-
rized in [7, 14].

One can say that the Poincaré disc is the closed unit disc centered at 0 ∈ R2 whose interior
is identified with R2 and its boundary S1 is identified with the infinity of R2. A polynomial
differential system in R2 can always be extended analytically to the Poincaré disc, and in this way
we can study the orbits of the polynomial differential system in a neighbourhood of the infinity.
For more details see Section 2 and [8, Chapter 5].

Figure 1. Orbits drawn as continuous curves are the separatrices of the differ-
ential system, while broken curves are not separatrices

The next result follows from [14, Theorem 2 of Chapter 4], also stated in [7, Theorem 2.14],
both without proof. For a “proof” of these mentioned two theorems modulo limit cycles see [6].
More precisely, the proof of [6, Theorem 1.2] assumes (without proof) that the bounded quadratic
systems with a unique finite equilibrium point either has no limit cycles, or it has at most one
limit cycle, see [6, page 267].

Theorem 1.2. The phase portraits in the Poincaré disc of the bounded quadratic systems with
a unique equilibrium point and without limit cycles are topologically equivalent to one the phase
portraits of Figure 1.

Using Proposition 1.1 we characterize all quadratic systems that are globally asymptotically
stable. Without loss of generality we can always assume that p is the origin of coordinates. We
recall that a quadratic system with the origin as an equilibrium point can be written as

(ẋ, ẏ) = f(x, y) = (f1(x, y), f2(x, y)) (1.2)

with

f1(x, y) = a10x+ a01y + a20x
2 + a11xy + a02y

2,

f2(x, y) = b10x+ b01y + b20x
2 + b11xy + b02y

2,

where aij , bij ∈ R for i = 0, 1, 2 and j = 0, 1, 2.
Our main result is the following one. The statement of the next theorem coincides with the

statements of [14, Theorem 2 of Chapter 4], or [7, Theorem 2.14], but in the present article this
theorem is proved without the assumption that the bounded quadratic systems with a unique
finite equilibrium point either has no limit cycles, or it has at most one limit cycle.

Theorem 1.3. The unique quadratic systems that are globally asymptotically stable are the fol-
lowing ones.



EJDE-2025/36 GLOBAL ASYMPTOTIC STABILITY 3

(i) ẋ = a10x, ẏ = b10x+ b01y + xy, with a10 < 0 and b01 < 0.
(ii) ẋ = a10x+ a01y + y2, ẏ = b01y, with a10 < 0 and b01 < 0.
(iii) ẋ = a01y + y2, ẏ = −a01x+ b01y − xy + cy2,

with a01 ̸= 0, b01 < 0, c ∈ (−2, 2) and ca01 − b01 > 0.
(iv) ẋ = a01y + y2, ẏ = −a01x− xy + cy2, with ca01 > 0 and c ∈ (−2, 2).
(v) ẋ = a10x+ a01y + y2, ẏ = b10x+ b01y − xy + cy2, with a10 < 0, c ∈ (−2, 2), a10 + b01 < 0

and (a01 − b10 + a10c)
2 + 4(a01b10 − a10b01) < 0.

(vi) ẋ = a10x+ a01y + y2, ẏ = b10x+ b01y − xy + cy2, with a10 < 0, c ∈ (−2, 2), a10 + b01 = 0
and (a01 − b10 + a10c)

2 + 4(a01b10 − a10b01) < 0.
(vii) ẋ = a10x + a01xy + y2, ẏ = b10x + b01y − xy + cy2, with a10 < 0, a01 = b10 − a10c,

b01 = b10(b10 − a10c)/a10 and c ∈ (−2, 2).

In the next section we state several preliminary results that we need for proving Theorem 1.3.
In section 3 we prove Proposition 1.1 and Theorem 1.3.

2. Preliminaries

In this section we introduce some preliminary results that will be used during the proof of
Theorem 1.3. The following theorem characterizes the quadratic systems that are bounded, for a
proof see [6, Lemmas 1–4].

Theorem 2.1. A quadratic system is bounded if and only if it is one of the following systems.

(I) ẋ = a10x, ẏ = b10x+ b01y + xy, with a10 < 0 and b01 ≤ 0.
(II) ẋ = a10x+ a01y + y2, ẏ = b01y, with a10 ≤ 0, b01 ≤ 0 and a10 + b01 < 0.
(III) ẋ = a01y + y2, ẏ = b01y − xy + cy2, with c ∈ (−2, 2).
(IV) ẋ = a01y+ y2, ẏ = −a01x+ b01y− xy+ cy2, with a01 ̸= 0, c ∈ (−2, 2) and ca01 − b01 ≥ 0.
(V) ẋ = a10x+ a01y + y2, ẏ = b10x+ b01y − xy + cy2, with a10 < 0 and c ∈ (−2, 2).

The following theorem computes the Lyapunov constants for quadratic systems having a center
at the origin of coordinates, for a proof see [13].

Theorem 2.2. Consider a quadratic system

ẋ = −y + a20x
2 + a11x+ a02y

2, ẏ = x+ b20x
2 + b11x+ b02y

2.

Let w1 = Aα−Bβ, w2 = γ(β(5A− β) + α(5B − α)) and w3 = γδ(Aβ +Bα), where

A = a20 + a02, B = b20 + b02, α = a11 + 2b02, β = b11 + 2a20,

γ = b20A
3 − (a20 − b11)A

3B + (b02 − a11)AB2 − a02B
3,

δ = a202 + b220 + a02A+ b20B.

Then the following statements hold.

(i) The origin is a center if and only if w1 = w2 = w3 = 0.
(ii) The origin is a weak focus of order 1 if w1 ̸= 0 (stable if w1 < 0 and unstable if w1 > 0).

The next result, proved in [5, Theorem 6], shows that for a quadratic system inside the region
bounded by a limit cycle there is a focus.

Theorem 2.3. An equilibrium point in the interior of a periodic orbit of a quadratic system must
be either a focus or a center.

The following result characterizes the quadratic systems that can have limit cycles (see the
beginning of [18, Chapter 12]).

Theorem 2.4. Any quadratic system that can have a limit cycle after an affine transformation
can be written in one of the following forms

(a) ẋ = yP1(x, y), ẏ = Q2(x, y), being P1(x, y) a linear polynomial and Q2(x, y) a quadratic
one;

(b) ẋ = P2(x, y), ẏ = xQ1(x, y), being Q1(x, y) a linear polynomial and P2(x, y) a quadratic
one.



4 J. LLIBRE, C. VALLS EJDE-2025/36

The following theorem provides the local phase portraits of semi–hyperbolic equilibrium points
for planar polynomial differential systems, for a proof of it see [1] or [8, Theorem 2.19].

Theorem 2.5. Let (0, 0) be an isolated equilibrium point of the planar polynomial differential
system

ẋ = F (x, y), ẏ = y +G(x, y)

with F and G being polynomials with at least second degree terms in x and y. Let y = g(x) be
the solution of y′ = y +G(x, y) = 0 and assume that F (x, g(x)) = amxm + · · · , where m ≥ 2 and
am ̸= 0. Then

(i) If m is odd and am > 0, then (0, 0) is an unstable node.
(ii) If m is odd and am < 0, then (0, 0) is a saddle.
(iii) If m is even, then (0, 0) is a saddle-node.

The following theorem was proved in [10, Theorem 4(x)].

Theorem 2.6. Consider the Abel differential system

dρ

dθ
= A(θ)ρ3 +B(θ)ρ2. (2.1)

If there exists two real numbers a and b such that aA(θ) + bB(θ) ̸≡ 0 and aA(θ) + bB(θ) is either
≥ 0, or ≤ 0 for all θ ∈ [0, 2π], then the differential system (2.1) has at most one limit cycle in the
region ρ > 0, and if it exists it is hyperbolic.

The Poincaré compactified vector field p(X) of the polynomial vector fieldX = (f1(x, y), f2(x, y))
is an analytic vector field on the 2-dimensional sphere

S2 =
{
x = (x1, x2, x3) ∈ R3 : x2

1 + x2
2 + x2

3 = 1
}

as follows (for more details, see [8, Chapter 5]). Let TxS2 be the tangent plane to S2 at the point x.
We identify R2 with T(0,0,1)S2 and we consider the central projection P : T(0,0,1)S2 → S2. The map

P defines two copies of X on S2, one on the southern hemisphere and the other on the northern
hemisphere. Denote by X̄ the vector field D(P ◦X) defined on S2 \ S1, where the equator

S1 =
{
(x1, x2, x3) ∈ S2 : x3 = 0

}
of the sphere S2 is identified with the infinity of R2. Since the degree of X is 2, p(X) is the
unique analytic extension of x2

3X̄ to S2. On S2 \ S1 there are two symmetric copies of X, and
once we know the behavior of p(X) near S1, we know the behavior of X near the infinity. The
projection of the closed northern hemisphere of S2 on x3 = 0 under (x1, x2, x3) 7→ (x1, x2) is called
the Poincaré disc D and the boundary is S1 (note that the Poincaré compactification leads S1
invariant by p(X)).

Let X be a polynomial vector field and let φ(t, p) be the orbit of X passing through the point
p at time t = 0, defined on its maximal interval (α, ω). If ω = ∞ we define the set

ω(p) =
{
q ∈ R2 : there exist {tn} with tn → ∞ and φ(tn) → q when n → ∞

}
.

The set ω(p) is called the ω-limit set of the orbit φ(t, p). If α = −∞ we define the set

α(p) =
{
q ∈ R2 : there exist {tn} with tn → −∞ and φ(tn) → q when n → ∞

}
.

The set α(p) is called the α-limit set of the orbit φ(t, p). If p is a point of the orbit γ, then we
define α(γ) = α(p) and ω(γ) = ω(p), note that these definitions do not depend on the point p of
the orbit γ

For a proof of the next result see for instance [8, Chapter 1].

Theorem 2.7 (Poincaré-Bendixson Theorem). Let φ(t, p) be an orbit of a polynomial vector field
X defined for all t ≥ 0, such that γ+

p = {φ(t, p) : t ≥ 0} is contained in a compact set. Assume
that the vector field X has a finite number of equilibrium points in ω(p). Then one of the following
statements holds.

(i) If ω(p) does not contain equilibrium points, then ω(p) is a periodic orbit.



EJDE-2025/36 GLOBAL ASYMPTOTIC STABILITY 5

(ii) If ω(p) contains equilibrium points and points which are non-equilibrium, then ω(p) is
formed by a finite number of orbits γ1, . . . , γn and a finite number of equilibrium points
p1, . . . , pn such that α(γi) = pi, ω(γi) = pi+1 for i = 1, . . . , n − 1, α(γn) = pn and
ω(γn) = p1. Possibly, some of the equilibrium points pi are the same.

(iii) If ω(p) does not contain non-equilibrium points, then ω(p) is an equilibrium point.

3. Proof of Proposition 1.1 and Theorem 1.3

Proof of Proposition 1.1. Assume first that a quadratic polynomial differential system (1.1) is
globally asymptotically stable. Then conditions (C1), (C2) and (C3) hold because such a system
only has one equilibrium point, it does not have periodic orbits and has no orbits going to infinity,
respectively.

Now we shall prove that a quadratic system satisfying conditions (C1), (C2) and (C3) is globally
asymptotically stable. Indeed, a quadratic system satisfying conditions (C1), (C2) and (C3) verifies
the assumptions of Theorem 1.2, so its phase portrait in the Poincaré disc is one of the three phase
portraits of Figure 1, and consequently such quadratic system is globally asymptotically stable. □

In what follows we say that a quadratic system with a unique equilibrium point satisfies condi-
tion (C4) if such equilibrium is locally asymptotically stable. We shall need the following result.

Proposition 3.1. The unique equilibrium point of a globally asymptotically stable quadratic system
is locally asymptotically stable, i.e. it satisfies condition (C4).

Proof. Since the phase portraits of the globally asymptotically stable quadratic systems are (a), (b)
and (c) of Figure 1, it follows that their equilibrium points are locally asymptotically stable. □

Now we start the proof of Theorem 1.3, i.e. we want to characterize the quadratic systems which
are globally asymptotically stable. So, by Proposition 1.1 we shall assume that the quadratic
system satisfies conditions (C1), (C2) and (C3), and by Proposition 3.1 such a quadratic system
also satisfies condition (C4). Using these four conditions we shall prove Theorem 1.3.

In view of Theorem 2.1 we can restrict to study the quadratic systems given in the cases (I)–(V)
of that theorem, because are the unique ones that satisfy that the differential systems (1.2) are
bounded. We study each case in a separate subsection.

3.1. Case (I) in Theorem 2.1. We consider the system

ẋ = a10x, ẏ = b10x+ b01y + xy, (3.1)

with a10 < 0 and b01 ≤ 0.
This system satisfies condition (C3). In order that it satisfies condition (C1) we shall see that

the origin is the unique equilibrium point. Indeed the equilibrium points of system (3.1) satisfy
x = 0 and b01y = 0. So in order that the origin is the unique finite equilibrium point we must have
b01 ̸= 0. So b01 < 0 and condition (C1) holds. Since the straight line x = 0 is invariant, system
(3.1) has no periodic orbits, and condition (C2) holds. Hence from Proposition 1.1 we obtain that
system (3.1) under the condition (i) of Theorem 1.3 is globally asymptotically stable.

3.2. Case (II) in Theorem 2.1. We consider the system

ẋ = a10x+ a01y + y2, ẏ = b01y, (3.2)

with a10 ≤ 0, b01 ≤ 0 and a10 + b01 < 0.
This system satisfies condition (C3). In order that it satisfies condition (C1) we shall see

that the origin is the unique equilibrium point. First note that b01 ̸= 0 otherwise we have a
continuum of equilibrium points. When b01 ̸= 0, the equilibrium points of system (3.2) satisfy
y = 0 and a10x = 0. So, in order that the origin is the unique finite equilibrium point we must
have a10 < 0. Therefore condition (C1) holds. Since the straight line y = 0 is invariant, system
(3.2) has no periodic orbits, and condition (C2) holds. Again from Proposition 1.1 system (3.2)
under conditions (ii) of Theorem 1.3 is globally asymptotically stable.



6 J. LLIBRE, C. VALLS EJDE-2025/36

3.3. Case (III) in Theorem 2.1. We consider the system

ẋ = a01y + y2, ẏ = b01y − xy + cy2,

with c ∈ (−2, 2). Note that this system has a continuum of equilibrium points, so it is not globally
asymptotically stable.

3.4. Case (IV) in Theorem 2.1. We consider the system

ẋ = a01y + y2, ẏ = −a01x+ b01y − xy + cy2, (3.3)

with a01 ̸= 0, c ∈ (−2, 2) and ca01 − b01 ≥ 0. This system satisfies condition (C3).
In order that it satisfies condition (C1) we shall see that the origin is the unique equilibrium

point. The equilibrium points satisfy y(y + a01) = 0. In order that the origin is the unique
equilibrium point we must have ca01 − b01 > 0 (if ca01 − b01 = 0 then there is a continuum of
equilibria). Hence condition (C1) holds.

Now we study when the origin is locally asymptotically stable. We compute the eigenvalues of
the Jacobian matrix at that point and we see that they are λ± = (b01±

√
b201 − 4a201)/2. If b01 ̸= 0

the origin is locally asymptotically stable if and only if b01 < 0 in which case it is a hyperbolic
stable node (if b201−4a201 ≥ 0), or a hyperbolic stable focus (if b201−4a201 < 0). On the other hand,
if b01 = 0 then it is either a weak focus or a center. To see when the origin is a stable focus we use
Theorem 2.2. For this we rescale system (3.3) by setting the independent variable dt1 = −a01 dt
and we obtain

x′ = −y − 1

a01
y2, y′ = x− b01

a01
y +

1

a01
xy − c

a01
y2, (3.4)

where the prime means derivative in the new time t1. Applying Theorem 2.2 to system (3.4) we
obtain that the Lyapunov constants are

w1 =
3c

a201
, w2 = −6c(c2 − 1)

a601
, w3 = −2c(2c2 − 1)

a801
.

Note that c ̸= 0, otherwise the origin is a center. So the origin of system (3.4) is a stable focus
if either c < 0 and a01 < 0, or c > 0 and a01 > 0. Hence, the origin of system (3.3) is locally
asymptotically stable if and only if either b01 < 0, or b01 = 0, and ca01 > 0. Hence condition (C4)
holds.

Finally, we see when there are no limit cycles. We claim that

System (3.3) has no limit cycles under the assumptions that condition (C4) holds. (3.5)

Now we prove the claim. Note that if we prove the claim we will prove Theorem 1.3 in this
case (see statements (iii) and (iv) of Theorem 1.3) since the unique global asymptotically stable
systems in case (IV) are the ones with either a01 ̸= 0, c ∈ (−2, 2), ca01 − b01 > 0 and b01 < 0, or
ca01 > 0, b01 = 0 with c ∈ (−2, 2).

Note that system (3.3) is invariant under the change

(x, y, a01, b01, c) 7→ (x,−y,−a01, b01,−c)

and so without loss of generality we can assume that a01 < 0. Now we introduce the change of
variables and the rescaling of the independent variable

X = − 1

a01
x, Y = − 1

a01
y, T = −a01t.

With this change system (3.3) becomes

X ′ = −Y + Y 2, Y ′ = X + bY −XY + cY 2, (3.6)

where b = −b01/a01 ≤ 0, and the prime denotes the derivative in the new time T .
We note that the straight line Y = 1 is either transversal (that is all the orbits of system (3.6)

cross it in the same direction), or invariant. Indeed

Y ′|Y=1 = b+ c.



EJDE-2025/36 GLOBAL ASYMPTOTIC STABILITY 7

Therefore, the possible periodic orbits of system (3.6) must be contained in the half-plane Π of
R2 defined by

Π = {(X,Y ) ∈ R2 : Y ≤ 1}.
Now we consider two cases.

Case b ̸= 0. In this case since we are assuming that condition (C4) holds we have that b < 0. Note
that

det

(
Y − Y 2 X + b1Y −XY + cY 2

Y − Y 2 X + b2Y −XY + cY 2

)
= (b2 − b1)Y

2(1− Y ),

and so it follows from [8, (7.19)] and the references [15, 16, 17] that the vector field

(X ′, Y ′) = (Y − Y 2, X + bY −XY + cY 2) (3.7)

is a generalized rotated vector field in the half-plane Π with respect to the parameter b < 0, and
in the half-planes of R2 separated by the straight line Y = 1 (note that c ∈ (−2, 2)). So, the
vector field (3.7) is a generalized rotated vector field in the half-plane Π. We shall prove that in
the half-plane Π system (3.6) has no limit cycles.

Assume that the differential system (3.6) has limit cycles and we shall arrive to a contradiction.
Then such limit cycle must surround the equilibrium point (0, 0) (see [8, Theorem 1.31]). Moreover,
since b < 0 these limit cycles are travelled in counterclockwise sense. Let γ be the closest limit cycle
to the stable equilibrium point (0, 0). This limit cycle is internally unstable. By the properties of
the rotated vector fields (see [8, Section 7.4]) we see that when b increases the limit cycle contracts
tending to the origin of coordinates but since the origin is a hyperbolic stable equilibrium point
for any value of b < 0 such a limit cycle cannot exist. This proves the claim (3.5) when b < 0.

Case b = 0. In this case system (3.6) becomes

X ′ = −Y + Y 2, Y ′ = X −XY + cY 2. (3.8)

We introduce polar coordinates

X = r cos θ, Y = r sin θ

and system (3.8) becomes

r′ = cr2 sin θ3, θ′ = 1 + r sin θ(−1 + c cos θ sin θ) (3.9)

that we write as
dr

dθ
=

cr2 sin θ3

1 + r sin θ(−1 + c cos θ sin θ)
=

α

1 + βr
,

with α = cr2 sin θ3 and β = sin θ(−1 + c cos θ sin θ). It is proved in [3] that the limit cycles of a
quadratic polynomial differential system surrounding a focus localized at the origin of coordinates
are contained in the region θ′ > 0, or θ′ < 0. So the change of variables r 7→ ρ given by

ρ =
r

1 + r sin θ(−1 + c cos θ sin θ)

is well-defined. In these new variable system (3.9) becomes the Abel differential system

dρ

dθ
= A(θ)ρ3 +B(θ)ρ2 (3.10)

with

A(θ) = c sin4 θ
(
1− c

2
sin θ

)
and B(θ) = cos θ + c(sin θ − sin(3θ)).

Note that A(θ) ≤ 0 for any θ ∈ [0, 2π] because c ∈ (−2, 0) and so it follows from Theorem 2.6 (with
a = 1 and b = 0) that the differential system (3.10) has at most one limit cycle and if it exists it is
hyperbolic. But if such a limit cycle is hyperbolic, it persists for values of b < 0 sufficiently small
but this is not possible due to the fact that it was proved in Case b < 0 that system (3.6) has no
limit cycles. This proves the claim (3.5) when b = 0 and concludes the proof of the theorem in
Case (IV).



8 J. LLIBRE, C. VALLS EJDE-2025/36

3.5. Case (V) in Theorem 2.1. We consider the system

ẋ = a10x+ a01y + y2, ẏ = b10x+ b01y − xy + cy2, (3.11)

where a10 < 0 and c ∈ (−2, 2). This system satisfies condition (C3).
In order that it satisfies condition (C1) we shall see that the origin is the unique equilibrium

point. The equilibrium points of system (3.11) satisfy

x = −y(a01 + y)

a10

and

y(a10b01 − a01b10 + (a01 − b10 + a10c)y + y2) = 0. (3.12)

So in order that the origin is the unique equilibrium point the unique solution of (3.12) must be
y = 0. We have two cases:

1. (a01 − b10 + a10c)
2 + 4(a01b10 − a10b01) < 0,

2. a01 = b10 − a10c and

a10b01 − b10(b10 − a10c) = 0,

i.e. b01 = (b10(b10 − a10c))/a10.

Now we study when conditions (C4) and (C2) are satisfied. We consider the cases 1 and 2
separately.

Case 1: Condition 1 holds.
First we prove that system (3.11) under conditions 1 do not have limit cycles. We will apply

Theorem 2.4 by showing that system (3.11) cannot be written with any affine change of variables
in the form (a) or in the form (b). Since (0, 0) is the unique equilibrium point it is sufficient to
do a linear change of variables instead of an affine change of variables. Hence we apply to system
(3.11) the general linear change of variables

X =
by − dx

bc1 − ad
, Y =

c1x− ay

bc1 − ad
, bc1 − ad ̸= 0,

which maintains the origin as an equilibrium point. With this linear change of variables system
(3.11) becomes

Ẋ = (abb10 + bb01c1 − aa10d− a01c1d)X + (b2b10 − d(a10b− bb01 + a01d))Y

− c1(ab− bcc1 + c1d)X
2 − (b2c1 + b(a− 2cc1)d+ 2c1d

2)XY

− d(b2 − bcd+ d2)Y 2,

Ẏ = −(a2b10 − a(a10 − b01)c1 − a01c
2
1)X + (c1(a10b+ a01d)

− a(bb10 + b01d))Y + c1(a
2 − acc1 + c21)X

2

+ (a2d+ 2c21d+ ac1(b− 2cd))XY + d(c1d+ a(b− cd))Y 2.

(3.13)

We first impose that system (3.13) can be written in the form (a) of Theorem 2.4, that is, that
satisfies

abb10 + bb01c1 − aa10d− a01c1d = c1(ab− bcc1 + c1d) = 0, ad− bc ̸= 0.

Doing so we obtain two solutions (a+, d+) and (a−, d−) where

a± = − c1
2a10

(a01 + b10 − a10c∓
√
(a01 − b10 + a10c)2 + 4(a01b10 − a10b01)),

d± =
b

2a10
(a01 + b10 + a10c±

√
(a01 − b10 + a10c)2 + 4(a01b10 − a10b01)).

Note that these solutions are never real because condition 1 implies that

(a01 − b10 + a10c)
2 + 4(a01b10 − a10b01) < 0.

Now we impose that system (3.13) can be written in the form (b) of Theorem 2.4, that is, that
satisfies

c1(a10b+ a01d)− a(bb10 + b01d) = d(c1d+ a(b− cd)) = 0, ad− bc ̸= 0.



EJDE-2025/36 GLOBAL ASYMPTOTIC STABILITY 9

Doing so we obtain two solutions (b+, c
+) and (b−, c

−) where

b± = − d

2a10
(a01 + b10 − a10c±

√
(a01 − b10 + a10c)2 + 4(a01b10 − a10b01)),

c± =
a

2a10
(a01 + b10 + a10c±

√
(a01 − b10 + a10c)2 + 4(a01b10 − a10b01))

which are also not real because condition 1 implies that

(a01 − b10 + a10c)
2 + 4(a01b10 − a10, b01) < 0.

Therefore, in view of Theorem 2.4 system (3.11) under the assumptions of Case 1 does not have
limit cycles, and condition (C2) is verified.

Now we compute when condition (C4) holds, that is, when the origin is locally asymptotically
stable. In this case the determinant of the Jacobian matrix at the origin is different from zero so
the origin is a hyperbolic equilibrium point. The eigenvalues of the Jacobian matrix at the origin
are

λ± =
1

2

(
a10 + b01 ±

√
(a10 − b01)2 + 4(a01b10 − a10b01)

)
.

By condition 1 we have a01b10 − a10b01 < 0, so in order that it is locally asymptotically stable we
must have that either a10 + b01 < 0 (in which case is a hyperbolic stable node if (a10 − b01)

2 +
4(a01b10 − a10b01) ≥ 0 and a hyperbolic stable focus if (a10 − b01)

2 + 4(a01b10 − a10b01) < 0), or
a10 + b01 = 0 in which case it is either a weak focus or a center when a210 + a01b10 < 0.

In the first case when a10+ b01 < 0 condition (C4) is satisfied and since we have already proven
that conditions (C1), (C2) and (C3) are satisfied we obtain from Proposition 1.1 that the origin
is globally asymptotically stable proving Theorem 1.3(v).

Assume now that a10 + b01 = 0 and a210 + a01b10 < 0. In particular, a01 ̸= 0. To see when the
origin under these conditions is a stable focus we use Theorem 2.2. We need to write system (3.3)
as in Theorem 2.2. We introduce the change of variables and a rescaling of the time of the form

x = X, y = − 1

a01

(
a10X +

√
−a210 − a01b10Y

)
, t = T

√
−a210 − a01b10.

With these new variables and time system (3.3) becomes

X ′ = −Y +
a210

a201
√
−a210 − a01b10

X2 +
2a10
a201

XY +

√
−a210 − a01b10

a201
Y 2,

Y ′ = X +
a10(a

2
01 + a210 + a01a10c)

a201(a
2
10 + a01b10)

X2 − a201 + 2a210 + 2a01a10c

a201
√
−a210 − a01b10

XY

− (a10 + a01c)

a201
Y 2,

(3.14)

where the prime means derivative with respect to the new time T . Applying Theorem 2.2 to
system (3.14) we obtain that the Lyapunov constants are

w1 =
a10b10 − 2(a210 + b210)c+ 2a10b10c

2 + a01(−a10 + b10c)

a01(−a210 − a01b10)3/2
,

w2 = − (a10 − b10c)(a
2
10 + b210 − a10b10c)(a01 − 6a10c+ b10(6c

2 − 5))

a301(−a210 − a01b10)7/2
,

w3 =
(a10 − b10c)(a

2
10 + b210 − a10b10c)(b10 + 2a10c− 2b10c

2)

a601(−a210 − a01b10)11/2
(a410(−5b10 + 2a10c)

+ a301a10(2a10 − b10c) + a201(−2b310 + 3a310c− a210b10(1 + c2))

+ a01a
2
10(−5b210 − 2a10b10c+ a210(3 + c2))).

If a10 = b10c condition 1 becomes (a01+ b10(1+ c2))2 which is never negative and so we must have
a10 ̸= b10c. In this case we can solve w1 = 0 by setting

a01 =
a10b10 − 2a210c− 2b210c+ 2a10b10c

2

a10 − b10c
.



10 J. LLIBRE, C. VALLS EJDE-2025/36

Imposing the value of a01 in condition 1 together with b01 = −a10, condition 1 becomes

(a210 + b210 − a10b10c)(4a
2
10 − 12a10b10c+ a210c

2 + 9b210c
2 − a10b10c

3)

(a10 − b10c)2
< 0.

But this condition is never satisfied because when c ∈ (−2, 2) we have a210 + b210 − a10b10c > 0 and

4a210 − 12a10b10c+ a210c
2 + 9b210c

2 − a10b10c
3 > 0.

Indeed

a210 + b210 − a10b10c = 0 yields a10 = b10c± b10
√
−4 + c2,

which is not real for c ∈ (−2, 2) and additionally

a210 + b210 − a10b10c|c=0 = a210 + b210 > 0.

This proves that a210 + b210 − a10b10c > 0 on c ∈ (−2, 2).
Furthermore,

4a210 − 12a10b10c+ a210c
2 + 9b210c

2 − a10b10c
3 = 0

yields

a10 =
12b10c+ b10c

3 ± b10c
2
√
−12 + c2

2(4 + c2)
,

which is not real on c ∈ (−2, 2) and additionally

4a210 − 12a10b10c+ a210c
2 + 9b210c

2 − a10b10c
3|c=0 = 4a210 > 0.

This proves that

4a210 − 12a10b10c+ a210c
2 + 9b210c

2 − a10b10c
3 > 0

on c ∈ (−2, 2).
In short in this case w1 never vanishes. Imposing condition 1 with b01 = −a10, i.e.,

4a210 + 4a01b10 + (a01 − b10 + a10c)
2 < 0,

together with the condition a210 + a01b10 < 0 and w1 < 0 we obtain the following conditions that
the parameters must satisfy

(3) a01 < 0, a01 < a10/2 < 0,

−a01 − 2a10 − 2
√
2a01a10 − a210 < b10 < −a01 − 2a10 + 2

√
2a01a10 − a210,

and

−2 < c < (−a01 + b10)/a10 + 2
√
−(a210 + a01b10)/a210,

(4) a01 > 0, −2a01 < a10 < 0,

−a01 + 2a10 − 2
√
−2a01a10 − a210 < b10 < −a01 + 2a10 + 2

√
−2a01a10 − a210,

and

(b10 − a01)/a10 − 2
√

−(a210 + a01b10)/a210 < c < 2.

We note that we have verified conditions (3) and (4) using the instruction Reduce of the algebraic
manipulator Mathematica. These conditions are equivalent to condition (vi) in Theorem 1.3. This
proves Theorem 1.3 (vi).

Case 2: Condition 2 holds. In this case the Jacobian matrix at the origin is

J =

(
a10 b10 − a10c

b10
b10(b10−a10c)

a10

)
.

Since a10 ̸= 0, the origin is never a linearly zero equilibrium point. The eigenvalues of J are λ1 = 0
and λ2 = (a210 + b210 − a10b10c)/a10. Note that λ2 = 0 if and only if

a10 =
1

2
(b10c± b10

√
c2 − 4)



EJDE-2025/36 GLOBAL ASYMPTOTIC STABILITY 11

which is not possible because a10 < 0 and c ∈ (−2, 2). So the origin is a semihyperbolic equilibrium
point. In order to apply Theorem 2.5 we need to write system (3.11) in Jordan canonical normal
form.

If b10 = 0, system (3.11) has y = 0 as an invariant straight line and so it cannot have periodic
orbits implying that condition (C2) holds. Hence in this case conditions (C1)–(C3) of Proposi-
tion 1.1 are satisfied and so the origin is globally asymptotically stable. If b10 ̸= 0, we introduce
the change of variables

x =
1

a210 + b210 − a10b10c
((b210 − a10b10c)X + a210Y ),

y =
a10b10

a210 + b210 − a10b10c
(Y −X),

and a rescaling of time

dt1 =
a10

a210 + b210 − a10b10c
dt. (3.15)

With these new variables and time system (3.11) can be written as

X ′ = − a310
(a210 + b210 − a10b10c)2

(XY − Y 2),

Y ′ = Y +
1

(a210 + b210 − a10b10c)2
(a10b

2
10X

2 − a10b10(b10 + a10c)XY + a210b10cY
2),

where the prime means derivative in the new time t1. Note that

X ′∣∣
Y=− a10b210

(a2
10+b210−a10b10c)2

X2+···
=

a410b
2
10

(a210 + b210 − a10b10c)4
X3 + · · ·

and since a10 < 0, b10 ̸= 0 and a210+ b210−a10b10c > 0 for c ∈ (−2, 2), by Theorem 2.5 we conclude
that the origin is an unstable node. Going back to system (3.11) its origin is a stable node due
to the change in the time given in (3.15). So condition (C4) holds and by Theorem 2.3 condition
(C2) is also satisfied.

Since under condition 2 the three conditions (C1)–(C3) of Proposition 1.1 are satisfied, the
origin is globally asymptotically stable, proving Theorem 1.3(vii).

Acknowledgements. J. Llibre was partially supported by the Agencia Estatal de Investigación
of Spain grant PID2022-136613NB-100, AGAUR (Generalitat de Catalunya) grant 2021SGR00113,
and by the Reial Acadèmia de Ciències i Arts de Barcelona. C. Valls was partially supported by
FCT/Portugal through CAMGSD, IST-ID, projects UIDB/04459/2020 and UIDP/04459/2020.

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Jaume Llibre

Departament de Matemàtiques, Universitat Autònoma de Barcelona, 8193 Bellaterra, Barcelona, Cat-
alonia, Spain

Email address: jaume.llibre@uab.cat

Claudia Valls

Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais

1049-001, Lisboa, Portugal
Email address: claudia.valls@tecnico.pt


	1. Introduction and statement of main results
	2. Preliminaries
	3. Proof of Proposition ?? and Theorem ??
	3.1. Case (I) in Theorem ??
	3.2. Case (II) in Theorem ??
	3.3. Case (III) in Theorem ??
	3.4. Case (IV) in Theorem ??
	3.5. Case (V) in Theorem ??
	Acknowledgements

	References