Abstract
Many processes are mathematically simulated by systems of discrete equations with quadratic right-hand sides. Their stability is thought of as a very important characterization of the process. In this paper, the method of Lyapunov functions is used to derive classes of stable quadratic discrete autonomous systems in a critical case in the presence of a simple eigenvalue of the matrix of linear terms. In addition to the stability investigation, we also estimate stability domains.
1. Introduction
The main results on the stability theory of difference equations are presented, for example, by Agarwal [1], Agarwal et al. [2], Chetaev [3], Elaydi [4], Halanay and Răsvan [5], Lakshmikantham and Trigiante [6], and Martynjuk [7]. Instability problems are considered, for example, in [8–10] by Slyusarchuk. Note that stability and instability results often have a local character and are usually obtained without any estimation of the stability domain, or without investigating the character of instability. Moreover, it should be emphasized that global instability questions have only been discussed for linear systems.
Many processes and phenomena are described by differential or difference systems with quadratic nonlinearities. Among others, let us mention epidemic and populations models, models of chemical reactions, and models for describing convection currents in the atmosphere.
The stability of a zero solution of difference systems where , and with differentiable , is very often investigated by linearly approximating system (1.1) in question by using the matrix of linear terms where is the Jacobian matrix of at , and . This approach becomes unsuitable in what is called a critical case, that is, when the spectral radius of the matrix because, among all systems (1.2), there are classes of stable systems as well as classes of unstable systems. Concerning this, we formulate the following known results (see, e.g., Corollary [4, page 222] and Theorem [4, page 226]).
Theorem 1.1.
() If , then the zero solution of (1.2) is exponentially stable.
() If , then the zero solution of (1.2) may be stable or unstable.
() If and is as , then the zero solution of (1.2) is unstable.
In this paper, we consider a particular critical case when there exists a simple eigenvalue of the matrix of linear terms and the remaining eigenvalues lie inside a unit circle centered at origin. The purpose of this paper is to obtain (using the method of Lyapunov functions) conditions for the stability of a zero solution of difference systems with quadratic nonlinearities in the above case and derive classes of stable systems. In addition to the stability investigation, we estimate the stability domains as well. The domains of stability obtained are also called guaranteed domains of stability. Preliminary results in this direction were published in [11].
1.1. Quadratic System and Preliminary Consideration
In the sequel, the norms used for vectors and matrices are defined as for a vector and for any matrix . Here and in the sequel, (or ) is the maximal (or minimal) eigenvalue of the corresponding symmetric and positive- (semi-) definite matrix (see, e.g., [12]).
Consider a nonlinear autonomous discrete system with a quadratic right-hand side where and the coefficients and (we assume that ) are constant. As emphasized, for example, in [3, 7, 12], system (1.5) can be written in a general vector-matrix form where (a), , is an constant square matrix, (b)matrix is rectangular and all the elements of the matrices , , are equal to zero except the th row with entries , that is, (c)matrix is rectangular and the constant matrices , , are symmetric.
The stability of the zero solution of (1.6) depends on the stability of the matrix . If , then the zero solution of (1.6) is exponentially stable for an arbitrary matrix by Theorem 1.1. In this case, matrix only impacts on the shape of the stability domain of the equilibrium state. If the zero solution of (1.6) is investigated on stability by the second Lyapunov method and an appropriate Lyapunov function is taken as the quadratic form with a suitable constant real symmetric positive-definite matrix , which is defined below, then the first difference along the trajectories of (1.6) equals since .
Since , for arbitrary positive-definite symmetric matrix , the matrix Lyapunov equation has a unique solution —a positive-definite symmetric matrix (e.g., [4, Theorem , page 216]). We use such matrix to estimate the stability domain. Then, as follows from (1.8), Analysing (1.10), we deduce that the first difference will be negative definite if that is, it will be negative definite in a neighborhood of the steady-state , if is sufficiently small. In the case considered, the domain of stability can be described by means of two inequalities. The first inequality (1.11) defines a part of the space , where the first difference is negative definite. The second inequality describes points inside a level surface. The guaranteed domain of stability is given by inequality (1.12) if is taken so small that the domain described by (1.12) is embedded in the domain described by inequality (1.11).
Considering the investigated critical case, we will deal with a different structure of the right-hand side of the inequality from that seen in (1.10). Namely, we will show that, unlike the right-hand side of the inequality for that is multiplied by with in (1.10), in the critical case considered, the right-hand side of the inequality (or equality) for will be multiplied only by a term with (see (2.21) in the case and (2.69) in the general case below).
2. Main Results
In this section we derive the classes of the stable systems (1.6) in a critical case when the matrix has one simple eigenvalue .
2.1. Instability in One-Dimensional Case
We start by discussing a simple scalar equation with the eigenvalue of matrix equaling one, that is, . Then (1.6) takes the form and it is easy to see that the trivial solution is unstable for an arbitrary (to show this, we can apply, e.g., Theorem [4, page 29]).
This elementary example shows that stability in the case of system (1.6) has an extraordinary significance and the results on stability (for ) lose their meaning for when we deal with instability. We show that, if and satisfies certain assumptions, the zero solution is stable. Moreover, the shape of the guaranteed domain of stability will be given.
We divide our forthcoming analysis into two parts. In the first one we give an explicit coefficient criterion in the subcase of . Then we consider the general -dimensional case.
2.2. Stability in the General Two-Dimensional Case
Let . Then system (1.6) with the matrix having a simple eigenvalue reduces (after linearly transforming the dependent variables if necessary) to We will assume that . Define auxiliary numbers as follows: where is a positive number.
Theorem 2.1. Let and be positive numbers. Assume that and . Then the zero solution of system (2.2) is stable in the Lyapunov sense and a guaranteed domain of stability is given by the inequality if is taken so small that the domain described by (2.4) is embedded in the domain If, moreover, , then a guaranteed domain of stability can be described using inequality with where runs over all real solutions of the nonlinear system with unknowns and :
Proof. Define
We rewrite system (2.2) as
To investigate the stability of the zero solution, we use, in accordance with the direct Lyapunov method, an appropriate Lyapunov function . Let a matrix , defined as
where instead of the entry we put the number , be positive definite. We set
The first difference of the function along the trajectories of system (2.10) equals
It is easy to see that does not preserve the sign if . Therefore, we put and reduces to
In the polynomial , with respect to and , we will put together the third-degree terms (the expression below) and the fourth-degree terms (the expression below). In the computations we use the formulas
We get
where
Analysing the increment of , we see that, if , will be nonpositive in a small neighborhood of the zero solution if the multipliers of the terms , and are equal to zero and the multiplier of the term is nonpositive, that is, if
As long as the Lyapunov function is positive definite, and . Therefore, conditions (2.18) hold if and only if
Then, system (2.2) turns into
and (without loss of generality, we put , i.e., ) into
The first difference of the Lyapunov function is nonpositive in a sufficiently small neighborhood of the origin (this is because , , and ). In other words, the zero solution is stable in the Lyapunov sense.
Now we will discuss the shape of the guaranteed domain of stability. It can be defined by the inequalities
where . This means that inequalities (2.4) and (2.5) are correct. Both inequalities geometrically express closed ellipses if . For the second inequality, this is obvious. For the first one, this follows from the following inequalities: , and
Moreover, for , the ellipse (2.4)
is contained (because it shrinks to the origin) in the ellipse (2.5), that is, there exists such that, for , the ellipse (2.4) lies inside the ellipse (2.5) without any intersection points and, for , there exists at least one common boundary point of both ellipses. Let us find the value . It is characterized by the requirement that the slope coefficients and of both ellipses are the same at the point of contact. Therefore
where we assume (without loss of generality) that the denominators are nonzero. Thus, we get a quadratic system of two equations to find the contact points :
For the corresponding values of , we have
In accordance with the geometrical meaning of the above quadratic system, we take such a solution as a defintion of the minimal positive value of and set .
Example 2.2. Consider a system In our case, , , and . Therefore, by Theorem 2.1, the zero solution of system (2.28) is stable in the Lyapunov sense. We will find the guaranteed domain of stability. We have and . Set . Then That is, the guaranteed domain of stability is given by the inequalities if is so small that the domain described by inequality (2.32) is embedded in the domain described by inequality (2.31). We consider the case when the ellipse (2.32) is embedded in the ellipse (2.31) and the boundaries of both ellipses have only one intersection point. Solving the system (2.8), that is, the system with Mathematica software, we get the solutions (see Figure 1 where the -axis is identified with the horizontal line and the -axis is identified with the vertical line, the blue ellipse graphically depicts equation (2.33), and the red hyperbola graphically depicts equation (2.34)): Then, in accordance with (2.7), and the guaranteed domain of stability obtained from (2.31), (2.32) is depicted in Figure 2 (as an ellipsoidal domain shaded in red and bounded by the thick red ellipse, with the identification of -axis and -axis being the same as before). Here, the domain (2.31) is bounded by the blue ellipse (2.33).
2.3. Stability in the General -Dimensional Case
Consider system (1.6) in . Assume that the matrix has a simple eigenvalue that is equal to unity with the others lying inside the unit circle. After linearly transforming the dependent variables if necessary, we can assume, without loss of generality, that the matrix of the linear terms in a block form, that is, where , is the -dimensional zero vector and all the eigenvalues of the matrix lie inside the unit circle. In order to formulate the next result and its proof, we have to introduce some new definitions (they copy the ones used in Section 1.1, but we use dimension or size instead of and note this change as a subscript if necessary): Matrices , , are symmetric since , (see Section 1.1). Moreover, we assume that there exists a symmetric positive definite matrix such that the symmetric matrix is positive definite. Let be a positive number and
Theorem 2.3. Let and be positive numbers. Assume that Then the zero solution of system (1.6) is stable by Lyapunov and the guaranteed domain of stability is described by the inequalities if is so small that the domain described by inequality (2.44) is embedded into the domain described by inequality (2.43).
Proof. We will perform auxiliary matrix computations. With this in mind, we have defined an matrix as where all the elements of the matrices , are equal to zero except the row , which equals , that is, Moreover, we define (a)vectors , , as a row -dimensional vector with coordinates equal to zero except the th element, which equals , that is, (b) zero matrix , (c)vectors ,, (d)vector . It is easy to see that Now we are able to rewrite system (1.6) in an equivalent form where Before the following computations, for the reader's convenience, we recall that for the matrices , , vectors , , vectors , and “matrices" , , we have To investigate the stability of system (1.6), we use the Lyapunov function where is an constant real symmetric and positive definite matrix. Let us find the first difference of the Lyapunov function (2.52) along the solutions of (2.49). We get or, using formula (2.51), Using formulas (2.50), we have where Then After further computation, we get Now we can represent as where contains only second-order polynomial terms, third-order polynomial terms and fourth-order polynomial terms with respect to the dependent variables. For , we have For , we get where Finally, can be represented as where Assume the parameters of the system and the coefficients of matrix to be such that Identities (2.65) hold if and only if As the matrix is positive definite, (2.66) hold if (these relations coincide with (2.42)). Hence The first difference (2.59) of can be estimated as and is nonpositive if The stability domain is defined by the inequality supposing that is so small that the domain (2.71) is embedded in the domain (2.70).
3. Concluding Remarks
Since 1892, when the general problem of stability of differential equations by the linear approximation was considered by A.M. Lyapunov, investigation of stability by linear approximation has been attracting permanent interest. For example, Malkin [13, Chapter 3] considered a general case of stability by linear approximation and derived stability criteria. In our paper we deal with systems of difference equations of a special form (with quadratic right-hand sides). Comparing our results with possible extensions of Malkin's results to difference equations, we point out that the primary purpose of our results, unlike those of Malkin, is to select such terms of the quadratic right-hand sides as contributing as much as possible to the loss of stability. The domains of stability are described in terms of the coefficients.
Since the matrix has one simple eigenvalue , a question on asymptotic stability is not asked. In fact, we are looking for a domain , (embedded in the domain described by inequality ), of admissible initial perturbations, that is, a set of invariance. All solutions generated by the initial data within this set remain in an neighborhood of the zero steady state provided that the initial perturbations are in a neighborhood of the zero steady state in the phase plane.
The method presented can be extended to further classes of systems. It is, for example, possible to consider the case of the multiplicity of eigenvalue being more than one or the case of and get stability criteria. But the computations needed are too cumbersome. Other questions are related to the results of Theorem 2.1 and Theorem 2.3. Is the zero solution of system (2.2) or of system (1.6) unstable when the coefficients indicated are not necessarily zero coefficients? We formulate them as open problems—prove or disprove the following conjecture.
Conjecture 3.1.
(a) Let and . Then the zero solution of system (2.2) is unstable.
(b) Let all the eigenvalues of the matrix lie inside the unit circle and
Then the zero solution of system (1.6) is unstable.
Finally, we formulate an open problem related to the shape of the guaranteed domain of stability.
Problem 1. The guaranteed domain of stability of the zero solution of system (2.2) is described by inequality (2.6) with defined by (2.7) where runs over all real solutions of the nonlinear system (2.8). Is it possible to derive an analogous shape of the guaranteed domain of stability for the zero solution of system (1.6) using inequalities (2.43), (2.44) and an analogy of method applied in the proof of Theorem 2.1?
Acknowledgments
This research was supported by the Grant no. P201/10/1032 of Czech Grant Agency, by the grant FEKT-S-10-3 of Faculty of Electrical Engineering and Communication, Brno University of Technology and by the Council of Czech Government MSM 0021630503, MSM 0021630519, and MSM 0021630529.