A simple test for asymptotic stability in some dynamical systems

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

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

Jhoana Patricia Romero Leiton
Instituto de Matemáticas, Universidad de Antioquia, Medellín - Colombia.
E-mail: patirom3@udea.edu.co

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

Abstract

In this paper we analyze asymptotic stability of the dynamical system =f(x) defined by a C¹ function is and open set. We ontain a criterion of satabilty for the equilibrium solution when the vector field f satisfies

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

1. Introduction

In 1982, A.M. Lyapunov developed his stability theory for nonlinear ordinary differential equiations which chaeacterices the behavior of the dinamical systems trajetories in the sense that nearby solutions remain that way from naow on (Hirsch and Smale, (Hirsch & Smale, 1974). .He esablished very useful stabilty criterua for dynamical systems of the form:

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

where G(y) = O(||y||)². Using the IML, it is possible prove that y = 0 is asymptotically satble if and only if ℌ (λ) < 0 for any eigenvalue λ of the matrix Df(), and so unstable, if there exist an eigenvalue λ of the matrix Df() whit ℌ (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() has a realk part zero, ℌ (λ) = 0 (Khalil, [12]).

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 requires the fiow associated with the dynamical system (1) being decreased on sorne scalar function V for which is an isolated minimum. This function is known as the Lyapunov function.

For the Lyapunov function V : where containing the origin, and its orbital derivative defined by

the DML establishes:

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

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

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

Theorem l. Let be an equilibrium of (1). Let be a continuous function defined on a neigborhood differentiable on such that

then 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 propase a function and check if this candidate satisfies the stability conditions (Perko, 1991) .

While the intention of A. M. Lyapunov was to study movement stability (Taylor & Francis, 1992), the DML found a wide range of applications. For example, in problems related with automatic regulation and control of dynamical processes (Rouche et al., 1977; Vasilév, 1981; Yoshizawa, 1966; Artstein, 1978; Barbastin,1970); in competition models (Goh, 1979, 1980; Takeuchi, 1996); in SIR models (Mena-Lorca & Hethcote, 1992; Safi & Garba, 2012); in SIRS models (O'Regan et al., 2010); in models with two compartments (A. Yu, 2003) , and in the proof of the Hopf bifurcation theorem (cited by O'Regan et al., [2010]).

Recently, Lyapunov functions are being applied within the fractional calculus to analyze the stability of dynamical systems. In this field, the method is called Fractional Lyapunov Direct Method (Yan Li et al., 2010; Momani & Hadid, 2004; Zhang et al., 2005; Tarasov, 2007). 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 & Gonzáles, 2011, Giesl & Hafstein, 2010, 2012). In 2012 the same authors designed an algorithm to explain the construction of these functions.

There are sorne 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

where α_i 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 α_i that will satisfy them.

B. S. Goh (Goh, 1979) used the function defined in (2) to prove global stability in mutualism models of the form

In this paper we establish global stability properties for the dynamical system (1) following the same ideas of S. B. Goh in (Goh, 1979). That is, we use the Lyapunov function (2) with specific values of the constants α_i to determine the stability conditions.

2. Calculus and linear algebra

Theorem 2. Let E be an open subset of ℝⁿ containing x_o, 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 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 is an open subset of

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

Proposition 1. Let be an open subset of containing Suppose that the function defined in (1) satisfies and be the determinants defined by

where a_j is a positive constant.

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

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

Proof. Let a₁,...,a_n be positive constants, for = (x₁,...,x_n) ∈ , then the function defined in (2) satisfies the condition V() = 0. On the other hand, the i-th term of (2) is:

Observe that which implies that η' (x_i) > if and only if < x_i and η' (x_i) < 0 if and only if > x_i Thus is a global minimun of η defined in (4). Since η'() = 0, then η'() > 0 for all x_i ≠ _i therefore V(x) > 0 for al x ≠ . From DML we conclude that if its orbital derivative is negative ( (x) < 0) for all x ∈ / {}, then is asymptotically stable on , while is unstable when (x) > 0 (see Theorem 1).

Observe that (x) = -g(x) where

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

for k = 1,...,n. From (5) we have that ∂g()/∂x_k = 0 for k = 1,...,n, which implies ∇g() = 0. Therefore is a critical point of g.

Since Hg() is a symmetric matrix, and assuming that all its principal minors are positive, then from Theorem 3 we have that is a local minimum of g on . Now, suppose that y ∈ is another equilibrium solution of (1) then g(y) = g() = 0. Similarly it is verified that if its principal minors have alternating signs for k = 1,...,n, starting with a negative value, then 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 is positive definite, then is globally asymptotically stable on and unstable when Hg() 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 a_i defined in (2) for conditions easy to verify.

Theorem 5 (Stability Test). Let be an equilibrium solution of nonlinear system ( 1). If

then is globally asymptotically stable.

In consequence, from Corollary 4 we conclude that is globally asymnt.nt.irallv stable in .

4. Application of main result

4.1 Numerical solutions

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

The following lemma ensures that all solutions of (14) starting in The following lemma ensures that all solutions of (14) starting in t ≤ 0

Lemma 6. The set ₁ defined in (15) is positively invariant for the solutions of the system (14).

Proof. Let x (x⁰₁, x⁰₂,...,x⁰_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 x_j(t) ≡ 0 for all t ≥ 0, and so for k ≠ j such x⁰_k ≠ 0, we have that x_k(t) satisfies the logic differential equation:

for which we know that 0 ≤ x_k(t) ≤ 1. In other words, if there is 1 ≤ j ≤ n such that x⁰_j = 0, we have that 0 ≤ x_k(t) ≤ 1 for 1 ≤ k ≤ n. Now, we assume that x (x⁰₁, x⁰₂,...,x⁰_n) ∈ ₁ is such that x⁰_j ≠ 0 for any 1 ≤ j ≤ n In this case, we know that x_j(t) for any ≤ 0 and 0 ≤ j ≤ n. Then, from (14) we obtain:

or equivalently

Let z = x^-1j, then dz/dt = -x^-2_jdx_j/dj. Substituting z and dz/dt in (16) we have

Multiplying the above inequality by e^αjt we obtain:

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

Substituting z = x^-1_j in (18) we obtain:

Therefore, we conclude that:

O ≤ x_j(t) ≤ 1 for all t ≤ O.

maening x = (x⁰₁, x⁰₂,...,x⁰_n) ∈ ₁ 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 ₁

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

Observe that in the following cases, a)x_j = 0, b)x_j = 1 and x_k = 0 for j ≠ k, the equations (19) are satisfied, which implies the existence of 2ⁿ - 1 equilibrium of the form x₀ = (p₁,...,p_n) where p_j = 0 or p_j = 1.

One of the possible applications for system (14) when n = 3 could be the On the other hand, from (19) we obtain:

The above implies that x_j > O if and only if O < k < α_n/4σ_n. Therefore, there are at least two equilibriums in int (₁). This completes the proof.

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

Proposition 3. Suppose that the system (14) has an interior steady state ∈ ₂ ⊂ ₁ where

₂ = {x ∈ ℝⁿ : 0 ≤ x_i ≤ 1,0 ≤ x_i + x_j, ≤ i j = 1,2,...,n}.

then this steady state is globally asymptotically stable on the interior set of ¹.

Proof. From (14) we conclude that:

From hypothesis ∈ ², results that 0 < 1 + _j < 1 which implies (_i + _j)² < _i + _j, or equivalently (1-_i) + (1-)_j > 2_ij. The above implies that the second hypothesis of Theorem 5 is satisfied. That is l_ij > 2. Therefore is globally asymptotically stable on interior set of ₁.

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.

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., 2011). 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 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).

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