21Volumen 18 No. 1, agosto 2014

1

A SIMPLE TEST FOR ASYMPTOTIC STABILITY IN SOME
DYNAMICAL SYSTEMS

Eduardo Ibargüen-Mondragón Miller Cerón Gomez
Universidad de Nariño Universidad de Nariño

Jhoana Patricia Romero Leitón
Universidad de Antioquia

Received: September 20, 2013 Accepted: November 23, 2013

Abstract

In this paper we analyze asymptotic stability of the dynamical system ẋ = f(x)

defined by a C1 function f : D → Rn where D ∈ Rn
+ is an open set. We obtain a

criterion of stability for the equilibrium solution x̄ ∈ D when the vector field f satisfies

a) ∂ifi(x̄) < 0 and b) (∂ifi(x̄))
−1∂ifj(x̄) + (∂jfj(x̄))

−1∂jfi(x̄) > 2 for i, j = 1, . . . , n.

Keywords: ordinary differential equations, asymptotic stability, equilibrium solution.

1 Introduction

In 1892, A. M. Lyapunov developed his stability theory for nonlinear ordinary
differential equations which characterizes the behavior of the dynamical systems
trajectories in the sense that nearby solutions remain that way from now on (Hirsch
and Smale, [9]). He established very useful stability criteria for dynamical systems
of the form:

ẋ = f(x), (1)

where f : D → Rn is a C1 map and D ⊂ Rn is an open set.

The first Lyapunov method, also known as Indirect Method of Lyapunov (IML)
allows to studying the stability of the equilibrium points for a dynamical system
of the type (1) by analyzing the stability of the trivial solution for the linearized
system:

dy(t)

dt
= Df(x̄)y +G(y),

where G(y) = O(‖y‖2). Using the IML, it is possible prove that y = 0 is
asymptotically stable if and only if R(λ) < 0 for any eigenvalue λ of the matrix
Df(x̄), and so unstable, if there exists an eigenvalue λ of the matrix Df(x̄) with
R(λ) > 0. We note that the IML does not allow us to obtain a conclusion if one of
the eigenvalues λ of the matrix Df(x̄) has real part zero, R(λ) = 0 (Khalil, [12]).

Facultad de Ciencias Naturales y Exactas
Universidad del Valle

Pag. 21-32

1   Introduction

 
 Eduardo Ibargüen Mondragón       Miller Cerón Gómez

Universidad del Nariño

 Received: September 20, 2013          Accepted: November 23, 2013

Jhoana Patricia Romero Leiton
Universidad del Antioquia

A simple test for asymptotic stability in some dynamical systems

1

A SIMPLE TEST FOR ASYMPTOTIC STABILITY IN SOME
DYNAMICAL SYSTEMS

Eduardo Ibargüen-Mondragón Miller Cerón Gomez
Universidad de Nariño Universidad de Nariño

Jhoana Patricia Romero Leitón
Universidad de Antioquia

Received: September 20, 2013 Accepted: November 23, 2013

Abstract

In this paper we analyze asymptotic stability of the dynamical system ẋ = f(x)

defined by a C1 function f : D → Rn where D ∈ Rn
+ is an open set. We obtain a

criterion of stability for the equilibrium solution x̄ ∈ D when the vector field f satisfies

a) ∂ifi(x̄) < 0 and b) (∂ifi(x̄))
−1∂ifj(x̄) + (∂jfj(x̄))

−1∂jfi(x̄) > 2 for i, j = 1, . . . , n.

Keywords: ordinary differential equations, asymptotic stability, equilibrium solution.

1 Introduction

In 1892, A. M. Lyapunov developed his stability theory for nonlinear ordinary
differential equations which characterizes the behavior of the dynamical systems
trajectories in the sense that nearby solutions remain that way from now on (Hirsch
and Smale, [9]). He established very useful stability criteria for dynamical systems
of the form:

ẋ = f(x), (1)

where f : D → Rn is a C1 map and D ⊂ Rn is an open set.

The first Lyapunov method, also known as Indirect Method of Lyapunov (IML)
allows to studying the stability of the equilibrium points for a dynamical system
of the type (1) by analyzing the stability of the trivial solution for the linearized
system:

dy(t)

dt
= Df(x̄)y +G(y),

where G(y) = O(‖y‖2). Using the IML, it is possible prove that y = 0 is
asymptotically stable if and only if R(λ) < 0 for any eigenvalue λ of the matrix
Df(x̄), and so unstable, if there exists an eigenvalue λ of the matrix Df(x̄) with
R(λ) > 0. We note that the IML does not allow us to obtain a conclusion if one of
the eigenvalues λ of the matrix Df(x̄) has real part zero, R(λ) = 0 (Khalil, [12]).

1

A SIMPLE TEST FOR ASYMPTOTIC STABILITY IN SOME
DYNAMICAL SYSTEMS

Eduardo Ibargüen-Mondragón Miller Cerón Gomez
Universidad de Nariño Universidad de Nariño

Jhoana Patricia Romero Leitón
Universidad de Antioquia

Received: September 20, 2013 Accepted: November 23, 2013

Abstract

In this paper we analyze asymptotic stability of the dynamical system ẋ = f(x)

defined by a C1 function f : D → Rn where D ∈ Rn
+ is an open set. We obtain a

criterion of stability for the equilibrium solution x̄ ∈ D when the vector field f satisfies

a) ∂ifi(x̄) < 0 and b) (∂ifi(x̄))
−1∂ifj(x̄) + (∂jfj(x̄))

−1∂jfi(x̄) > 2 for i, j = 1, . . . , n.

Keywords: ordinary differential equations, asymptotic stability, equilibrium solution.

1 Introduction

In 1892, A. M. Lyapunov developed his stability theory for nonlinear ordinary
differential equations which characterizes the behavior of the dynamical systems
trajectories in the sense that nearby solutions remain that way from now on (Hirsch
and Smale, [9]). He established very useful stability criteria for dynamical systems
of the form:

ẋ = f(x), (1)

where f : D → Rn is a C1 map and D ⊂ Rn is an open set.

The first Lyapunov method, also known as Indirect Method of Lyapunov (IML)
allows to studying the stability of the equilibrium points for a dynamical system
of the type (1) by analyzing the stability of the trivial solution for the linearized
system:

dy(t)

dt
= Df(x̄)y +G(y),

where G(y) = O(‖y‖2). Using the IML, it is possible prove that y = 0 is
asymptotically stable if and only if R(λ) < 0 for any eigenvalue λ of the matrix
Df(x̄), and so unstable, if there exists an eigenvalue λ of the matrix Df(x̄) with
R(λ) > 0. We note that the IML does not allow us to obtain a conclusion if one of
the eigenvalues λ of the matrix Df(x̄) has real part zero, R(λ) = 0 (Khalil, [12]).

1

A SIMPLE TEST FOR ASYMPTOTIC STABILITY IN SOME
DYNAMICAL SYSTEMS

Eduardo Ibargüen-Mondragón Miller Cerón Gomez
Universidad de Nariño Universidad de Nariño

Jhoana Patricia Romero Leitón
Universidad de Antioquia

Received: September 20, 2013 Accepted: November 23, 2013

Abstract

In this paper we analyze asymptotic stability of the dynamical system ẋ = f(x)

defined by a C1 function f : D → Rn where D ∈ Rn
+ is an open set. We obtain a

criterion of stability for the equilibrium solution x̄ ∈ D when the vector field f satisfies

a) ∂ifi(x̄) < 0 and b) (∂ifi(x̄))
−1∂ifj(x̄) + (∂jfj(x̄))

−1∂jfi(x̄) > 2 for i, j = 1, . . . , n.

Keywords: ordinary differential equations, asymptotic stability, equilibrium solution.

1 Introduction

In 1892, A. M. Lyapunov developed his stability theory for nonlinear ordinary
differential equations which characterizes the behavior of the dynamical systems
trajectories in the sense that nearby solutions remain that way from now on (Hirsch
and Smale, [9]). He established very useful stability criteria for dynamical systems
of the form:

ẋ = f(x), (1)

where f : D → Rn is a C1 map and D ⊂ Rn is an open set.

The first Lyapunov method, also known as Indirect Method of Lyapunov (IML)
allows to studying the stability of the equilibrium points for a dynamical system
of the type (1) by analyzing the stability of the trivial solution for the linearized
system:

dy(t)

dt
= Df(x̄)y +G(y),

where G(y) = O(‖y‖2). Using the IML, it is possible prove that y = 0 is
asymptotically stable if and only if R(λ) < 0 for any eigenvalue λ of the matrix
Df(x̄), and so unstable, if there exists an eigenvalue λ of the matrix Df(x̄) with
R(λ) > 0. We note that the IML does not allow us to obtain a conclusion if one of
the eigenvalues λ of the matrix Df(x̄) has real part zero, R(λ) = 0 (Khalil, [12]).

1

A SIMPLE TEST FOR ASYMPTOTIC STABILITY IN SOME
DYNAMICAL SYSTEMS

Eduardo Ibargüen-Mondragón Miller Cerón Gomez
Universidad de Nariño Universidad de Nariño

Jhoana Patricia Romero Leitón
Universidad de Antioquia

Received: September 20, 2013 Accepted: November 23, 2013

Abstract

In this paper we analyze asymptotic stability of the dynamical system ẋ = f(x)

defined by a C1 function f : D → Rn where D ∈ Rn
+ is an open set. We obtain a

criterion of stability for the equilibrium solution x̄ ∈ D when the vector field f satisfies

a) ∂ifi(x̄) < 0 and b) (∂ifi(x̄))
−1∂ifj(x̄) + (∂jfj(x̄))

−1∂jfi(x̄) > 2 for i, j = 1, . . . , n.

Keywords: ordinary differential equations, asymptotic stability, equilibrium solution.

1 Introduction

In 1892, A. M. Lyapunov developed his stability theory for nonlinear ordinary
differential equations which characterizes the behavior of the dynamical systems
trajectories in the sense that nearby solutions remain that way from now on (Hirsch
and Smale, [9]). He established very useful stability criteria for dynamical systems
of the form:

ẋ = f(x), (1)

where f : D → Rn is a C1 map and D ⊂ Rn is an open set.

The first Lyapunov method, also known as Indirect Method of Lyapunov (IML)
allows to studying the stability of the equilibrium points for a dynamical system
of the type (1) by analyzing the stability of the trivial solution for the linearized
system:

dy(t)

dt
= Df(x̄)y +G(y),

where G(y) = O(‖y‖2). Using the IML, it is possible prove that y = 0 is
asymptotically stable if and only if R(λ) < 0 for any eigenvalue λ of the matrix
Df(x̄), and so unstable, if there exists an eigenvalue λ of the matrix Df(x̄) with
R(λ) > 0. We note that the IML does not allow us to obtain a conclusion if one of
the eigenvalues λ of the matrix Df(x̄) has real part zero, R(λ) = 0 (Khalil, [12]).

1

A SIMPLE TEST FOR ASYMPTOTIC STABILITY IN SOME
DYNAMICAL SYSTEMS

Eduardo Ibargüen-Mondragón Miller Cerón Gomez
Universidad de Nariño Universidad de Nariño

Jhoana Patricia Romero Leitón
Universidad de Antioquia

Received: September 20, 2013 Accepted: November 23, 2013

Abstract

In this paper we analyze asymptotic stability of the dynamical system ẋ = f(x)

defined by a C1 function f : D → Rn where D ∈ Rn
+ is an open set. We obtain a

criterion of stability for the equilibrium solution x̄ ∈ D when the vector field f satisfies

a) ∂ifi(x̄) < 0 and b) (∂ifi(x̄))
−1∂ifj(x̄) + (∂jfj(x̄))

−1∂jfi(x̄) > 2 for i, j = 1, . . . , n.

Keywords: ordinary differential equations, asymptotic stability, equilibrium solution.

1 Introduction

In 1892, A. M. Lyapunov developed his stability theory for nonlinear ordinary
differential equations which characterizes the behavior of the dynamical systems
trajectories in the sense that nearby solutions remain that way from now on (Hirsch
and Smale, [9]). He established very useful stability criteria for dynamical systems
of the form:

ẋ = f(x), (1)

where f : D → Rn is a C1 map and D ⊂ Rn is an open set.

The first Lyapunov method, also known as Indirect Method of Lyapunov (IML)
allows to studying the stability of the equilibrium points for a dynamical system
of the type (1) by analyzing the stability of the trivial solution for the linearized
system:

dy(t)

dt
= Df(x̄)y +G(y),

where G(y) = O(‖y‖2). Using the IML, it is possible prove that y = 0 is
asymptotically stable if and only if R(λ) < 0 for any eigenvalue λ of the matrix
Df(x̄), and so unstable, if there exists an eigenvalue λ of the matrix Df(x̄) with
R(λ) > 0. We note that the IML does not allow us to obtain a conclusion if one of
the eigenvalues λ of the matrix Df(x̄) has real part zero, R(λ) = 0 (Khalil, [12]).

1

A SIMPLE TEST FOR ASYMPTOTIC STABILITY IN SOME
DYNAMICAL SYSTEMS

Eduardo Ibargüen-Mondragón Miller Cerón Gomez
Universidad de Nariño Universidad de Nariño

Jhoana Patricia Romero Leitón
Universidad de Antioquia

Received: September 20, 2013 Accepted: November 23, 2013

Abstract

In this paper we analyze asymptotic stability of the dynamical system ẋ = f(x)

defined by a C1 function f : D → Rn where D ∈ Rn
+ is an open set. We obtain a

criterion of stability for the equilibrium solution x̄ ∈ D when the vector field f satisfies

a) ∂ifi(x̄) < 0 and b) (∂ifi(x̄))
−1∂ifj(x̄) + (∂jfj(x̄))

−1∂jfi(x̄) > 2 for i, j = 1, . . . , n.

Keywords: ordinary differential equations, asymptotic stability, equilibrium solution.

1 Introduction

In 1892, A. M. Lyapunov developed his stability theory for nonlinear ordinary
differential equations which characterizes the behavior of the dynamical systems
trajectories in the sense that nearby solutions remain that way from now on (Hirsch
and Smale, [9]). He established very useful stability criteria for dynamical systems
of the form:

ẋ = f(x), (1)

where f : D → Rn is a C1 map and D ⊂ Rn is an open set.

The first Lyapunov method, also known as Indirect Method of Lyapunov (IML)
allows to studying the stability of the equilibrium points for a dynamical system
of the type (1) by analyzing the stability of the trivial solution for the linearized
system:

dy(t)

dt
= Df(x̄)y +G(y),

where G(y) = O(‖y‖2). Using the IML, it is possible prove that y = 0 is
asymptotically stable if and only if R(λ) < 0 for any eigenvalue λ of the matrix
Df(x̄), and so unstable, if there exists an eigenvalue λ of the matrix Df(x̄) with
R(λ) > 0. We note that the IML does not allow us to obtain a conclusion if one of
the eigenvalues λ of the matrix Df(x̄) has real part zero, R(λ) = 0 (Khalil, [12]).



22

Revista de Ciencias 	 E. Ibargüen Mondragón, M. Cerón Gómez y J. P. Romero
2

In this paper, we will consider the second Lyapunov method, also known as
Direct Method of Lyapunov (DML), in which the stability of an equilibrium point
x̄ requires the flow associated with the dynamical system (1) being decreased on
some scalar function V for which x̄ is an isolated minimum. This function is known
as the Lyapunov function.

For the Lyapunov function V : D ⊆ Rn → R where D containing the origin,
and its orbital derivative V̇ : D ⊆ Rn → R defined by

V̇ (x) = DV (x)(f(x)),

the DML establishes:

1. If V (x) is positive definite and V̇ (x) is negative semi-definite, then the origin
is stable.

2. If V (x) is positive definite and V̇ (x) is negative definite, then the origin is
asymptotically stable.

In general, for any equilibrium solution of (1) the DML states that:

Theorem 1. Let x̄ ∈ D be an equilibrium of (1). Let V : B → R be a continuous
function defined on a neigborhood B ⊂ D of x̄, differentiable on B − x̄, such that

a) V (x̄) = 0 and V (x) > 0 if x �= x̄;

b) V̇ (x) ≤ 0 in B − x̄,

then x̄ is stable. Furthemore, if

c) V̇ (x) < 0 in B − x̄,

then x̄ is asymptotically stable.

In the twentieth century, the DML became in the principal tool to analyze
global stability of dynamical systems applied to basic sciences and engineering.
The main setback of this method is precisely to find a Lyapunov function, because
there is not a systematic method for finding. The suggestion is to propose a
function and check if this candidate satisfies the stability conditions (Perko, [20]).

While the intention of A. M. Lyapunov was to study movement stability
(Taylor and Francis, [14]), the DML found a wide range of applications. For
example, in problems related with automatic regulation and control of dynamical
processes (Rouche et al., [21]; Vasilév, [25]; Yoshizawa, [26]; Artstein, [2];
Barbastin, [3]); in competition models (Goh, [7, 8]; Takeuchi, [23]); in SIR models
(Mena-Lorca and Hethcote, [16]; Safi and Garba, [22]), in SIRS models (O’Regan
et al., [19]); in models with two compartments (A. Yu, [1]), and in the proof of the
Hopf bifurcation theorem (cited by O?Regan et al., [19]).

Recently, Lyapunov functions are being applied within the fractional calculus
to analyze the stability of dynamical systems. In this field, the method is called



23Volumen 18 No. 1, agosto 2014

3

Fractional Lyapunov Direct Method (Yan Li et al., [13]; Momani and Hadid, [17];
Zhang et al., [27]; Tarasov, [24]). In 2011, it was used Lyapunov functions to
analyze the dynamics of the Hopf bifurcation in a class of models that exhibit Zip
bifurcation (Escobar and Gonzáles, [4], Giesl and Hafstein, [5, 6]). In 2012 the
same authors designed an algorithm to explain the construction of these functions.

There are some systems where the Lyapunov function is defined in a natural
way, like in the case of electrical or mechanical systems where energy is often
a Lyapunov function. In mathematical biology, more precisely in population
dynamic modeled through the mass action law, the functions of Goh type

V (x) =

n∑

i=1

ai

[
xi − x̄i − x̄i ln

(
xi

x̄i

)]
, (2)

where ai for i = 1, . . . , n are positive constants that satisfy the first item of
Theorem 1 while the other items are reduced to find the constants ai that will
satisfy them.

B. S. Goh (Goh, [7]) used the function defined in (2) to prove global stability
in mutualism models of the form

ẋi = xifi(x1, x2, . . . , xn) i = 1, 2, . . . , n.

In this paper we establish global stability properties for the dynamical system (1)
following the same ideas of S. B. Goh in (Goh, [7]). That is, we use the Lyapunov
function (2) with specific values of the constants ai to determine the stability
conditions.

2 Calculus and linear algebra

Theorem 2. Let E be an open subset of Rn containing x0, if f : E ⊂ Rn → R
such that f ∈ C3(E), f(x0) = 0 and Hessian matrix Hf(x0) is positive definite,
then x0 is a relative minimum of f . Similarly, if Hf(x0) is negative definite, then
x0 is a relative maximum of f .

See [15] for proof of Theorem 2.

Theorem 3. (Sylvester’s Criterion). A real symmetric matrix is positive definite
if and only if all its principal minors are positive.

See [10] for proof of Theorem 3.

3 Test of stability

In this section we establish a test for the asymptotic stability of the system
(1) equilibrium when D is an open subset of

Rn
+ = {(x1, . . . , xn) ∈ Rn : xi ≥ 0 for i = 1, . . . , n}.

The following proposition relates the equilibrium stability with the sign of certain
determinants.

A simple test for asymptotic stability in some dynamical systems

2    Calculus and linear algebra

3    Test of stability



24

Revista de Ciencias 	 E. Ibargüen Mondragón, M. Cerón Gómez y J. P. Romero4

Proposition 1. Let D be an open subset of Rn
+ containing x̄ = (x̄1, . . . , x̄n).

Suppose that the function f : D → Rn defined in (1) satisfies f ∈ C1(D) and
f(x̄) = 0. Let ∆j(x̄) be the determinants defined by

∆j(x̄) = (−1)j
∣∣∣∣
aj
x̄j

∂fj(x̄)

∂xi
+

ai
x̄i

∂fi(x̄)

∂xj

∣∣∣∣
i=1,...,j

, j = 1, . . . , n (3)

where aj is a positive constant.

1. If ∆j(x̄) for j = 1, . . . , n are positive, then x̄ is globally asymptotically stable.

2. If ∆j(x̄) for j = 1, . . . , n has alternate signs starting with a negative value,
then x̄ is unstable.

Proof. Let a1, . . . , an be positive constants, for x̄ = (x̄1, . . . , x̄n) ∈ D, then the
function defined in (2) satisfies the condition V (x̄) = 0. On the other hand, the
i-th term of (2) is:

η(xi) = ai

[
xi − x̄i − x̄i ln

(
xi

x̄i

)]
. (4)

Observe that

η′(xi) = ai

(
1− x̄i

xi

)
,

Observe that which implies that η′(xi) > 0 if and only if x̄i < xi and η′(xi) < 0
if and only if x̄i > xi. Thus x̄i is a global minimum of η defined in (4). Since
η(x̄i) = 0, then η(xi) > 0 for all xi �= x̄i therefore V (x) > 0 for all x �= x̄.
From DML we conclude that if its orbital derivative is negative (V̇ (x) < 0) for
all x ∈ D\{x̄}, then x̄ is asymptotically stable on D, while x̄ is unstable when
V̇ (x) > 0 (see Theorem 1).

Observe that V̇ (x) = −g(x) where

g(x) =
n∑

i=1

ai

(
x̄i

xi
− 1

)
fi(x).

Since g(x̄) = 0, then to prove the stability of x̄ it is enough to verify that x̄ is a
minimum of g on D, and any other equilibrium solution y ∈ D of (1) satisfies that
g(y) ≥ g(x̄). The derivative of g is given by the gradient vector

�g(x) =

(
∂g(x)

∂x1
, . . . ,

∂g(x)

∂xn

)
,

where
∂g(x)

∂xk
= −ak

x̄k

x2
k

fk(x) +
n∑

i=1

ai

(
x̄i

xi
− 1

)
∂fi(x)

∂xk
, (5)

for k = 1, . . . , n. From (5) we have that ∂g(x̄)/∂xk = 0 for k = 1, . . . , n, which

implies �g(x̄) = 0. Therefore x̄ is a critical point of g.



25Volumen 18 No. 1, agosto 2014

A simple test for asymptotic stability in some dynamical systems5

The Hessian matrix of g(x) is

Hg(x) =

(
∂2g(x)

∂xi∂xj

)

i,j=1,...,n

, (6)

where

∂2g(x)

∂xj∂xk
= −ak

x̄k

x2
k

∂fk(x)

∂xj
+

∂

∂xj

[
n∑

i=1

ai

(
x̄i

xi
− 1

)
∂fi(x)

∂xk

]
(7)

= −
(
ak

x̄k

x2
k

∂fk(x)

∂xj
+ aj

x̄j

x2
j

∂fj(x)

∂xk

)
+

n∑

i=1

ai

(
x̄i

xi
− 1

)
∂2fi(x)

∂xj∂xk
,

for j, k = 1, . . . , n and j �= k. As a result

∂2g(x̄)

∂xj∂xk
= −

(
ak
x̄k

∂fk(x̄)

∂xj
+

aj
x̄j

∂fj(x̄)

∂xk

)
. (8)

Therefore, the determinant of Hessian matrix of g(x) evaluated at x̄ is:

|Hg(x̄)| =

∣∣∣∣
∂2g(x)

∂xi∂xj

∣∣∣∣
i,j=1,...,n

=

∣∣∣∣−
(
aj
x̄j

∂fj(x̄)

∂xi
+

ai
x̄i

∂fi(x̄)

∂xj

)∣∣∣∣
i,j=1,...,n

= (−1)j
∣∣∣∣
aj
x̄j

∂fj(x̄)

∂xi
+

ai
x̄i

∂fi(x̄)

∂xj

∣∣∣∣
i,j=1,...,n

= ∆n(x̄).

Since Hg(x̄) is a symmetric matrix, and assuming that all its principal minors
are positive, then from Theorem 3 we have that x̄ is a local minimum of g
on D. Now, suppose that y ∈ D is another equilibrium solution of (1) then
g(y) = g(x̄) = 0. Similarly it is verified that if its principal minors have alternating
signs for k = 1, . . . , n, starting with a negative value, then x̄ is unstable, which
completes the proof.

From the above proposition, the following corollary is derived:

Corollary 4. If the Hessian matrix Hg(x) defined in (6) evaluated at x̄ is positive
definite, then x̄ is globally asymptotically stable on D and unstable when Hg(x̄) is
negative definite.

The following theorem summarizes the main result of this work. The novelty
of next test consists in replacing the expertise of the authors to find the constants
ai defined in (2) for conditions easy to verify.

Theorem 5 (Stability Test). Let x̄ ∈ D ⊂ Rn
+ be an equilibrium solution of

nonlinear system (1). If



26

Revista de Ciencias 	 E. Ibargüen Mondragón, M. Cerón Gómez y J. P. Romero6

1.
∂fi(x̄)

∂xi
< 0 for i = 1, 2, . . . , n. and

2. lij(x̄) > 2 for i, j = 1, . . . , n with i �= j, where

lij(x̄) =

(
∂fi(x̄)

∂xi

)−1
∂fi(x̄)

∂xj
+

(
∂fj(x̄)

∂xj

)−1
∂fj(x̄)

∂xi
,

then x̄ is globally asymptotically stable.

Proof. Let x = (x1, . . . , xn) ∈ D, we will prove that the quadratic form

G(x̄, x) = xTHg(x̄)x

=
n∑

j=1

(
n∑

k=1

∂2g(x̄)

∂xj∂xk
xjxk

)
, (9)

is positive. Since
∂2g(x̄)

∂xj∂xk
=

∂2g(x̄)

∂xk∂xj
,

for j, k = 1, . . . , n, then (9) is rewritten as:

G(x̄, x) =
n∑

k=1

∂2g(x̄)

∂x2
k

x2
k + 2

n∑

k=2

∂2g(x̄)

∂x1∂xk
x1yk + 2

n∑

k=3

∂2g(x̄)

∂x2∂xk
x2yk

+ · · ·+ 2
∂2g(x̄)

∂xn−1∂xn
xn−1xn. (10)

Substituting (8) in (10) we have:

G(x̄, x) = −
n∑

k=1

2
ak
x̄k

∂fk(x̄)

∂xk
x2
k −

n∑

k=2

2

(
ak
x̄k

∂fk(x̄)

∂x1
+

a1
x̄1

∂f1(x̄)

∂xk

)
y1xk

−
n∑

k=3

2

(
ak
x̄k

∂fk(x̄)

∂x2
+

a2
x̄2

∂f2(x̄)

∂xk

)
x2xk + · · ·+

−2

(
an
x̄n

∂fn(x̄)

∂xn−1
+

an−1

x̄n−1

∂fn−1(x̄)

∂xn

)
xn−1xn. (11)

Let

ak = −x̄k

[
2
∂fk(x̄)

∂xk

]−1

, k = 1, . . . , n. (12)



27Volumen 18 No. 1, agosto 2014

7

Since ∂fk(x̄)/∂xk < 0, then ak > 0. Substituting (12) in (11) we have:

G(x̄, x) =
n∑

k=1

x2
k +

n∑

k=2

[(
∂fk(x̄)

∂xk

)−1
∂fk(x̄)

∂x1
+

(
∂f1(x̄)

∂x1

)−1
∂f1(x̄)

∂xk

]
x1xk

+
n∑

k=3

[(
∂fk(x̄)

∂xk

)−1
∂fk(x̄)

∂x2
+

(
∂f2(x̄)

∂x2

)−1
∂f2(x̄)

∂xk

]
x2xk

+ · · ·+
[(

∂fn(x̄)

∂xn

)−1
∂fn(x̄)

∂xn−1
+

(
∂fn−1(x̄)

∂xn−1

)−1
∂fn−1(x̄)

∂xn

]
xn−1xn

>
n∑

k=1

x2
k +

n∑

k=2

2x1xk +
n∑

k=3

2x2xk + · · ·+ 2xn−1xn

= (x1 + x2 + · · ·+ xn)
2

> 0. (13)

In consequence, from Corollary 4 we conclude that x̄ is globally asymptotically
stable in D.

4 Application of main result

In this section we will apply the Theorem 5 to prove the asymptotic stability
of nontrivial equilibrium of the nonlinear system

dxj

dt
= αjxj(1− xj)− σj

n∏

i=1

xi, j = 1, 2, . . . , n, (14)

where 0 < αj < 1 and 0 < σj < 1 for j = 1, 2, . . . , n. Our set of interest is:

D1 = {x ∈ Rn : 0 ≤ xi ≤ 1, i, j = 1, 2, . . . , n}. (15)

The following lemma ensures that all solutions of (14) starting in D1 remain there
for all t ≥ 0.

Lemma 6. The set D1 defined in (15) is positively invariant for the solutions of
the system (14).

Proof. Let x = (x0
1, x

0
2, . . . , x

0
n) be given. If there is 1 ≤ j ≤ n such that x0

j = 0,
then we see directly from the unique and existent result that xj(t) ≡ 0 for all
t ≥ 0, and so for k �= j such that x0

k �= 0, we have that xk(t) satisfies the logic
differential equation:

dxk

dt
= αkxk(1− xk),

for which we know that 0 ≤ xk(t) ≤ 1. In other words, if there is 1 ≤ j ≤ n such
that x0

j = 0, we have that 0 ≤ xk(t) ≤ 1 for 1 ≤ k ≤ n. Now, we assume that

x = (x0
1, x

0
2, . . . , x

0
n) ∈ D1 is such that x0

j �= 0 for any 1 ≤ j ≤ n. In this case, we
know that xj(t) for any t ≥ 0 and 0 ≤ j ≤ n. Then, from (14) we obtain:

dxj

dt
≤ αjxj(1− xj), j = 1, 2, . . . , n

A simple test for asymptotic stability in some dynamical systems

4     Application of main result



28

Revista de Ciencias 	 E. Ibargüen Mondragón, M. Cerón Gómez y J. P. Romero8

or equivalently

−x−2
j

dxj

dt
+ αjx

−1
j ≥ αj , j = 1, 2, . . . , n. (16)

Let z = x−1
j , then dz/dt = −x−2

j dxj/dt. Substituting z and dz/dt in (16) we have

dz

dt
+ αjz ≥ αj .

Multiplying the above inequality by eαjt we obtain:

d (eαjtz)

dt
≥ αje

αjt. (17)

Integrating the inequality (17) between 0 and t we have:

z(t) ≥ 1 + (z(0)− 1)e−αjt. (18)

Substituting z = x−1
j in (18) we obtain:

xj(t) ≤
1

1 +
[
x−1
j (0)− 1

]
e−αjt

.

Therefore, we conclude that:

0 ≤ xj(t) ≤ 1 for all t ≥ 0.

meaning x = (x0
1, x

0
2, . . . , x

0
n) ∈ D1 as desired.

The next proposition summarizes existent results of the equilibrium solutions of
(14).

Proposition 2. The system (14) has at least 2n+1−1 equilibrium solution in D1.

Proof. The equilibrium solutions of (14) are given by the solutions of the algebraic
system

αjxj(1− xj)− σj

n∏

i=1

xi = 0, j = 1, 2, . . . , n. (19)

Observe that in the following cases, a)xj = 0, b)xj = 1 and xk = 0 for j �= k, the
equations (19) are satisfied, which implies the existence of 2n − 1 equilibrium of
the form x0 = (p1, . . . , pn) where pj = 0 or pj = 1. On the other hand, from (19)
we obtain:

αj

σj
xj(1− xj) = k, j = 1, 2, . . . , n, (20)

where k =
∏n

i=1 xi. The solutions of (20) are:

xj =
1±

√
1− 4kσj/αj

2
, j = 1, 2, . . . , n.

The above implies that xj > 0 if and only if 0 < k < αn/4σn. Therefore, there
are at least two equilibriums in int (D1). This completes the proof.



29Volumen 18 No. 1, agosto 2014

A simple test for asymptotic stability in some dynamical systems9

The following proposition summarizes stability results of the equilibrium of
(14).

Proposition 3. Suppose that the system (14) has an interior steady state x̄ ∈
D2 ⊂ D1 where

D2 = {x ∈ Rn : 0 ≤ xi ≤ 1, 0 ≤ xi + xj ,≤ 1 i, j = 1, 2, . . . , n}.
then this steady state is globally asymptotically stable on the interior set of D1.

Proof. From (14) we conclude that:

fj(x) = αjxj(1− xj)− σj

n∏

k=1

xk, j = 1, 2, . . . , n,

which implies that

∂fj(x̄)

∂x̄j
= αj(1− x̄j)− αj x̄j − σj

n∏

k=1,k �=j

x̄k, j = 1, 2, . . . , n. (21)

From equilibrium equations we have:

αj(1− x̄j)− σj

n∏

k=1,k �=j

x̄k = 0, j = 1, 2, . . . , n. (22)

Therefore, substituting (22) in (21) we verify the first hypothesis of Theorem 5,
that is

∂fj(x̄)

∂x̄j
= −αj x̄j < 0, j = 1, 2, . . . , n.

On the other hand,

lij(x̄) =

(
∂fi(x̄)

∂xi

)−1
∂fi(x̄)

∂xj
+

(
∂fj(x̄)

∂xj

)−1
∂fj(x̄)

∂xi

= (−αix̄i)
−1


−σi

n∏

k=1,k �=j

x̄k


+ (−αj x̄j)

−1


−σj

n∏

k=1,k �=i

x̄k




=

(
σi

αi
+

σj

αj

) n∏

k=1,k �=i,k �=j

x̄k. (23)

From (22) we have:

σj

αj
=

1− x̄j∏n
k=1,k �=j x̄k

, j = 1, 2, . . . , n. (24)

Substituting (24) in (23) we obtain:

lij(x̄) =
1− x̄i

x̄j
+

1− x̄j

x̄i
, for i �= j.

From hypothesis x̄ ∈ D2, results that 0 < x̄i + x̄j < 1 which implies (x̄i + x̄j)
2 <

x̄i + x̄j , or equivalently (1 − x̄i)x̄i + (1 − x̄j)x̄j > 2x̄ix̄j . The above implies that
the second hypothesis of Theorem 5 is satisfied. That is lij > 2. Therefore x̄ is
globally asymptotically stable on interior set of D1.



30

10

4.1 Numerical solutions

One of the possible applications for system (14) when n = 3 could be the
competition among three species with logistic growth. The simulation of Figure
1 was made with the following data: α1 = 0.1, α2= 0.2, α3 = 0.15, σ1 = 0.08,
σ2 = 0.15 and σ3 = 0.14. In this case the solutions of (14) tend to the coexistent
equilibrium P1= (0.68,0.72,0.53) which agrees with the theoretical results.

0 20 40 60 80 100 120 140 160 180 200
0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Time

Po
pu

lat
io

n 
de

ns
ity

 

 

x
1

(t)

x
2

(t)

x
3

(t)

Figure 1: Graphs of the component solutions x1, x2 and x3 of (14) for n = 3. In
this case, α1 =0.1, α2=0.2, α3 =0.15, σ1 =0.08, σ2 =0.15, σ3 =0.14.

5 Conclusion

In certain areas of applied mathematics such as Biomathematics, the
qualitative analysis of the solutions of dynamical systems defined by ordinary
differential equations is fundamental to understand problems in biology (Ibargüen
et al. [11]). In this sense, the DML is very practical and widely used to analyze the
stability of dynamical systems. In this article we use the DML to establish easier
conditions to verify the assurance of global asymptotic stability of the equilibrium
solutions of some dynamical systems. The fact that these conditions are defined
in terms of ∂fi(x̄)/∂xi for i = 1, . . . , n, suggest the possibility that the stability
test (Theorem 5) can be used to numerically verify asymptotic stability.

Acknowledgements

We want to thank to anonymous referees and Dr. L. Esteva for their valuable
comments and suggestions that helped us to improve the paper. E. Ibarguen
acknowledges support from project No 082-16/08/2013 (VIPRI-UDENAR).

Revista de Ciencias 	 E. Ibargüen Mondragón, M. Cerón Gómez y J. P. Romero

4.1  Numerical solutions

5     Conclusión

10

4.1 Numerical solutions

One of the possible applications for system (14) when n = 3 could be the
competition among three species with logistic growth. The simulation of Figure
1 was made with the following data: α1 = 0.1, α2= 0.2, α3 = 0.15, σ1 = 0.08,
σ2 = 0.15 and σ3 = 0.14. In this case the solutions of (14) tend to the coexistent
equilibrium P1= (0.68,0.72,0.53) which agrees with the theoretical results.

0 20 40 60 80 100 120 140 160 180 200
0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Time

Po
pu

lat
io

n d
en

sit
y

 

 

x
1

(t)

x
2

(t)

x
3

(t)

Figure 1: Graphs of the component solutions x1, x2 and x3 of (14) for n = 3. In
this case, α1 =0.1, α2=0.2, α3 =0.15, σ1 =0.08, σ2 =0.15, σ3 =0.14.

5 Conclusion

In certain areas of applied mathematics such as Biomathematics, the
qualitative analysis of the solutions of dynamical systems defined by ordinary
differential equations is fundamental to understand problems in biology (Ibargüen
et al. [11]). In this sense, the DML is very practical and widely used to analyze the
stability of dynamical systems. In this article we use the DML to establish easier
conditions to verify the assurance of global asymptotic stability of the equilibrium
solutions of some dynamical systems. The fact that these conditions are defined
in terms of ∂fi(x̄)/∂xi for i = 1, . . . , n, suggest the possibility that the stability
test (Theorem 5) can be used to numerically verify asymptotic stability.

Acknowledgements

We want to thank to anonymous referees and Dr. L. Esteva for their valuable
comments and suggestions that helped us to improve the paper. E. Ibarguen
acknowledges support from project No 082-16/08/2013 (VIPRI-UDENAR).

Figure 1:



31Volumen 18 No. 1, agosto 2014

A simple test for asymptotic stability in some dynamical systems

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11

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Author’s address

Eduardo Ibargüen-Mondragón
Departamento de Matemática y Estad́ıstica, Universidad de Nariño,
San Juan de Pasto - Colombia
edbargun@udenar.edu.co

Miller Cerón Gómez
Departamento de Matemática y Estad́ıstica, Universidad de Nariño,
San Juan de Pasto - Colombia
millercg@udenar.edu.co

Jhoana Patricia Romero Leitón
Instituto de Matemáticas, Universidad de Antioquia, Medelĺın - Colombia
patirom3@udea.edu.co