Abstract and Applied Analysis

Volume 2012 (2012), Article ID 956359, 13 pages

doi:10.1155/2012/956359

Research Article

An Algebraic Criterion of Zero Solutions of Some Dynamic Systems

Ying Wang,¹ Baodong Zheng,¹ and Chunrui Zhang²

¹Department of Mathematics, Harbin Institute of Technology, Harbin, Heilongjiang 150001, China

²Department of Mathematics, Northeast Forestry University, Harbin, Heilongjiang 150040, China

Received 19 October 2012; Revised 7 December 2012; Accepted 7 December 2012

Academic Editor: Allan Peterson

Copyright © 2012 Ying Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We establish some algebraic results on the zeros of some exponential polynomials and a real coefficient polynomial. Based on the basic theorem, we develop a decomposition technique to investigate the stability of two coupled systems and their discrete versions, that is, to find conditions under which all zeros of the exponential polynomials have negative real parts and the moduli of all roots of a real coefficient polynomial are less than 1.

1. Introduction

For an ordinary (delay) differential equation, the trivial solution is asymptotically stable if and only if all roots of the corresponding characteristic equation of the linearized system have negative real parts while the moduli of all roots of a real coefficient polynomial less than 1 mean the trivial solution is asymptotically stable for the difference equation. However, it is difficult to obtain the expression of the characteristic equation corresponding to the linearized systems. Special cases of the characteristic equation have been discussed by many authors. For example, Bellman and Cooke [1], Boese [2], Kuang [3], and Ruan and Wei [4–8] studied some exponential polynomials and used the results to investigate the stability and bifurcations for some systems. The well-known Jury criterion can be used to determine the moduli of the roots of a real coefficient polynomial less than one [9, 10], but the calculation is prolixly.

The purpose of this paper is to provide a new algebraic criterion of zero for some exponential polynomials and a real coefficient polynomial.

2. Some Algebraic Results

Let be a linear space over a number field and a subspace of . A complement space of in is a linear space of such that . A vector is said to be the projection of a vector along at if , where and .

Let be a linear transformation on and a subspace of . A subspace is said to be a invariant subspace if for any . Let be a -complement space of for , if

(1)is a invariant subspace;(2) is a complement space of in , that is, ;(3)for any , the projection of along at is always 0.

Let , be two linear transformations on . We consider that and are concordant if there exist a nontrivial invariant subspaces of and , and a complement space of in , such that

(1)at least one of constraints of on is a scalar transformation ;(2) is a -complement space of or for .

Let . We say that and are concordant if linear transformations and of

  ,  for all ,  ,  for all

are concordant.

For example, and a reducible matrix are always in concordance.

Theorem 2.1. Let be concordant. Then there exist an invertible matrix , such that

or

or

or

where .

Proof. Let be the base of and of . If the constraints of on are and is a -complement space of for , then and there exists making , that is, . So

Because a subspace is a invariant subspace, we can write , , . So

If the constraint of on is and is a -complement space of for , then . . So

Because a subspace is a invariant subspace, we can write , . Because (), , then there exists making . So

Other situations are similar.

3. Algebraic Criterion of Zero for Some Exponential Polynomials

From Section 2, we have the following theorem on the roots of exponential polynomial.

Theorem 3.1. Let be concordant, a constant, , a polynomial about . If is a root of

then

is an eigenvalue of , or

is an eigenvalue of , where or .

Proof. Without loss of generality, assume that

Because is a root of

then

  If , then is an eigenvalue of .  If , then is an eigenvalue of .

As a application of Theorem 3.1, consider a BAM neural network model with delays:

where . Assume that , and .

Under the hypothesis, the origin is an equilibrium of (3.7), and the linearization of system (3.7) at the origin is

where are Jacobi matrices.

The associated characteristic equation of (3.8) is

Since have no influence on the stability of system (3.7), then let . We have

Let be eigenvalues of . From Theorem 3.1, we have

where , that is,

Consider

Lemma 3.2. If , then (3.12) has a root , and

are determined by

Proof. Let be a root of (3.13), then

Separating the real and imaging parts, the roots can be obtained.

Lemma 3.3.  .

Proof. Differentiating both sides of (3.13) with respect to gives

that is,

where .

Theorem 3.4. Let .

(1)If , then the zero solution of (3.7) is absolutely stable.(2)If , then the zero solution of (3.7) is asymptotically stable when and unstable when , and (3.7) undergoes a Hopf bifurcation at the origin 0 when , where is defined in Lemma 3.2.(3)If , then the zero solution of (3.7) is unstable for all .

Proof. For , (3.13) becomes

Let be a root of (3.19). Separating the real and imaginary parts, we can obtain

Hence, . If , then . Using Lemmas 3.2 and 3.3 the conclusions follow.

4. Algebraic Criterion of Zero for Some Real Polynomial

Theorem 4.1. Let be concordant. If be a polynomial about , is a root of

then

is an eigenvalue of , or

is an eigenvalue of , where or .

Proof. Let us assume that

Because is a root of

then

  If , then is an eigenvalue of .  If , then is an eigenvalue of .

For the application of Theorem 4.1, consider the discretization of BAM neural network (3.7). Let , then (3.7) can be rewritten as

Let , then (4.7) can be rewritten as

Let , using an Euler method to (4.8), we obtain

Let . Using the notation, (4.9) can be expressed as

where and

It is clear that 0 is also an equilibrium of (4.10). The linearization of (4.10) at the origin 0 is

The characteristic equation of (4.12) is

Since , hence there is no influence on the stability of (4.10). Let , then (4.14) can be rewritten as

Let be an eigenvalue of . Using Theorem 4.1, we have

It is clear that the stability of system (4.10) is determined by the distribution of the roots of (4.16). Next, we consider a special case of system (3.7)

where , which is a Hopfiled neural network with delay. Its discrete system is

The linear part is

It is clear that we only need to discuss the distribution of the roots

where is an eigenvalue of .

Lemma 4.2. Let . There exists a such that for , all roots of (4.20) have moduli less than one.

Proof. When , (4.20) has an m-fold root and a simple root at . Consider the root such that . This root depends continuously on and is a differential function of :

Hence, for all sufficiently small . Thus, all roots of (4.20) lie in for sufficiently small positive and existence of the maximal follows.

Lemma 4.3. Assume that the seep size h is sufficiently small. Let , then there are no roots of (4.20) with moduli one for all .

Proof. Let be a root of (4.20) when , then

where , hence

Let , for sufficiently small , , which yields a contradiction. The proof is complete.

Lemma 4.4. If the seep size is sufficiently small, then

where , satisfy (4.22).

Proof. Consider the following equation:

Using Lemmas 4.1–4.4 and Theorem 1 of [11], we have the following results.

Theorem 4.5. (1) If , then the zero solution of (4.10) is unstable for .

(2) If , , then the zero solution of (4.18) is asymptotically stable when , where , and (4.10) undergoes a Naimark-Sacker bifurcation.

(3) If , then the zero solution of (4.10) is unstable for all .

5. Computer Simulation

To illustrate the analytical results found, let us consider the following particular case of (3.7):

where are parameters.

Using the conclusions of Theorem 3.4, the phase-locked periodic solutions appear with See Figure 1.

(a)

(b)

Figure 1: Two phase-locked oscillations: , .

For the special case of (4.8), we have the following equations:

where . The phase-locked periodic solutions appear. See Figure 2.

(a)

(b)

Figure 2: Two phase-locked oscillations: , .

Acknowledgment

This research was supported by the National Natural Science Foundation of China (no. 10871056).

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