Research Article | Open Access
An Algebraic Criterion of Zero Solutions of Some Dynamic Systems
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.
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 , Boese , Kuang , 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
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 .
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
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 , 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.
This research was supported by the National Natural Science Foundation of China (no. 10871056).
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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.