An Algebraic Criterion of Zero Solutions of Some Dynamic Systems

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 𝑉 be a linear space over a number field 𝐹 and 𝑊 a subspace of 𝑉. A complement space of 𝑊 in 𝑉 is a linear space 𝑈 of 𝑉 such that 𝑉=𝑊⊕𝑈. A vector 𝛾 is said to be the projection of a vector 𝛼 along 𝑊 at 𝑈 if 𝛼=𝛽+𝛾, where 𝛽∈𝑊 and 𝛾∈𝑈.

Let 𝒜 be a linear transformation on 𝑉 and 𝑊 a subspace of 𝑉. A subspace 𝑊 is said to be a 𝒜 invariant subspace if 𝒜𝛼∈𝑊 for any 𝛼∈𝑊. Let 𝑈 be a 𝑘-complement space of 𝒜 for 𝑊, if(1)𝑊is a 𝒜 invariant subspace;(2)𝑈 is a complement space of 𝑊 in 𝑉, that is, 𝑉=𝑊⊕𝑈;(3)for any 𝛼∈𝑈, the projection of (𝒜−𝑘𝐼_𝑉)𝛼 along 𝑊 at 𝑈 is always 0.

Let 𝒜, ℬ be two linear transformations on 𝑉. We consider that 𝒜 and ℬ are (𝑘,𝑙) concordant if there exist a nontrivial invariant subspaces 𝑊 of 𝒜 and ℬ, and a complement space 𝑈 of 𝑊 in 𝑉, such that(1)at least one of constraints of 𝒜,ℬ on 𝑊 is a scalar transformation 𝑙𝐼_𝑊;(2)𝑈 is a 𝑘-complement space of 𝒜 or ℬ for 𝑊.

Let 𝐴,𝐵∈𝐑^𝑛×𝑛. We say that 𝐴 and 𝐵 are (𝑘,𝑙) concordant if linear transformations 𝒜 and ℬ of 𝐑^𝑛  𝒜:𝐑^𝑛→𝐑^𝑛,𝑋↦𝐴𝑋,  for all 𝑋∈𝐑^𝑛,  ℬ:𝐑^𝑛→𝐑^𝑛,𝑋↦𝐵𝑋,  for all 𝑋∈𝐑^𝑛are concordant.

For example, 𝐼_𝑛 and a reducible matrix 𝐴_𝑛×𝑛 are always in concordance.

Theorem 2.1. Let 𝐴,𝐵∈𝐑^𝑛×𝑛 be (𝑘,𝑙) concordant. Then there exist an invertible matrix 𝑃, such that 𝑃𝐴𝑃⁻¹=𝐴₁₁𝐴₁₂𝑂𝑘𝐼_𝑛2,𝑃𝐵𝑃⁻¹=𝑙𝐼_𝑛1𝐵₁₂𝑂𝐵₂₂, (2.1) or 𝑃𝐴𝑃⁻¹=𝑙𝐼_𝑛1𝐴₁₂𝑂𝑘𝐼_𝑛2,𝑃𝐵𝑃⁻¹=𝐵₁₁𝐵₁₂𝑂𝐵₂₂, (2.2) or 𝑃𝐴𝑃⁻¹=𝐴₁₁𝐴₁₂𝑂𝐴₂₂,𝑃𝐵𝑃⁻¹=𝑙𝐼_𝑛1𝐵₁₂𝑂𝑘𝐼_𝑛2, (2.3) or 𝑃𝐴𝑃⁻¹=𝑙𝐼_𝑛1𝐴₁₂𝑂𝐴₂₂,𝑃𝐵𝑃⁻¹=𝐵₁₁𝐵₁₂𝑂𝑘𝐼_𝑛2, (2.4) where 𝑛₁>0,𝑛₂>0,𝑛₁+𝑛₂=𝑛.

Proof. Let 𝛼₁,𝛼₂,…,𝛼_𝑟 be the base of 𝑊 and 𝛼_𝑟+1,…,𝛼_𝑛 of 𝑈. If the constraints of 𝒜 on 𝑊 are 𝑙𝐼_𝑊 and 𝑈 is a 𝑘-complement space of 𝒜 for 𝑊, then 𝐴𝛼_𝑖=𝑙𝛼_𝑖,𝑖=1,…,𝑟 and there exists 𝑎_1𝑗,…,𝑎_𝑟𝑗 making 𝐴𝛼_𝑗−𝑘𝛼_𝑗=𝑎_1𝑗𝛼₁+⋯+𝑎_𝑟𝑗𝛼_𝑟, that is, 𝐴𝛼_𝑗=𝑎_1𝑗𝛼₁+⋯+𝑎_𝑟𝑗𝛼_𝑟+𝑘𝛼_𝑗,𝑗=𝑟+1,…,𝑛. So 𝐴𝛼₁,…,𝛼_𝑟,𝛼_𝑟+1,…,𝛼_𝑛=𝛼₁,…,𝛼_𝑟,𝛼_𝑟+1,…,𝛼_𝑛𝑙⋯0𝑎_1,𝑟+1⋯𝑎_1𝑛⋮⋱⋮⋮⋮0⋯𝑙𝑎_𝑟,𝑟+1⋯𝑎_𝑟𝑛𝑘0𝑘⋯0⋱⋮. (2.5) Because a subspace 𝑊 is a ℬ invariant subspace, we can write 𝐵𝛼_𝑖=𝑏_1𝑖𝛼₁+⋯+𝑏_𝑟𝑖𝛼_𝑟, 𝑖=1,…,𝑟, 𝐵𝛼_𝑖=𝑏_1𝑗𝛼₁+⋯+𝑏_𝑟𝑗𝛼_𝑟+⋯+𝑏_𝑛𝑗𝛼_𝑛,𝑗=𝑟+1,…,𝑛. So 𝐵𝛼₁,…,𝛼_𝑟,𝛼_𝑟+1,…,𝛼_𝑛=𝛼₁,…,𝛼_𝑟,𝛼_𝑟+1,…,𝛼_𝑛𝑏₁₁⋯𝑏_1𝑟𝑏_1,𝑟+1⋯𝑏_1𝑛𝑏⋮⋱⋮⋮⋮_𝑟1⋯𝑏_𝑟𝑟𝑏_𝑟,𝑟+1⋯𝑏_𝑟𝑛0𝑏_{𝑟+1,𝑟+1}⋯𝑏_𝑟+1,𝑛𝑏⋮⋱⋮_𝑛,𝑟+1⋯𝑏_𝑛𝑛. (2.6) If the constraint of 𝒜 on 𝑊 is 𝑙𝐼_𝑊 and 𝑈 is a 𝑘-complement space of ℬ for 𝑊, then 𝐵𝛼_𝑖=𝑙𝛼_𝑖,𝑖=1,…,𝑟. 𝐴𝛼_𝑗=𝑎_1𝑗𝛼₁+⋯+𝑎_𝑟𝑗𝛼_𝑟+𝑎_𝑟+1,𝑗𝛼_𝑟+1+⋯+𝑎_𝑛𝑗𝛼_𝑛,𝑗=𝑟+1,…,𝑛. So 𝐴𝛼₁,…,𝛼_𝑟,𝛼_𝑟+1,…,𝛼_𝑛=𝛼₁,…,𝛼_𝑟,𝛼_𝑟+1,…,𝛼_𝑛𝑙⋯0𝑎_1,𝑟+1⋯𝑎_1𝑛⋮⋱⋮⋮⋮0⋯𝑙𝑎_𝑟,𝑟+1⋯𝑎_𝑟𝑛0𝑎_{𝑟+1,𝑟+1}⋯𝑎_𝑟+1,𝑛𝑎⋮⋱⋮_𝑛,𝑟+1⋯𝑎_𝑛𝑛. (2.7) Because a subspace 𝑊 is a ℬ invariant subspace, we can write 𝐵𝛼_𝑖=𝑏_1𝑖𝛼₁+⋯+𝑏_𝑟𝑖𝛼_𝑟, 𝑖=1,…,𝑟. Because 𝛼_𝑗∈𝑈 (𝑗=𝑟+1,…,𝑛), 𝐵𝛼_𝑗−𝑘𝛼_𝑗∈𝑊, then there exists 𝑏_1𝑗,…,𝑏_𝑟𝑗 making 𝐵𝛼_𝑗=𝑏_1𝑗𝛼₁+⋯+𝑏_𝑟𝑗𝛼_𝑟+𝑘𝛼_𝑗,𝑗=𝑟+1,…,𝑛. So 𝐵𝛼₁,…,𝛼_𝑟,𝛼_𝑟+1,…,𝛼_𝑛=𝛼₁,…,𝛼_𝑟𝛼_𝑟+1,…,𝛼_𝑛𝑏₁₁⋯𝑏_1𝑟𝑏_1,𝑟+1⋯𝑏_1,𝑛𝑏⋮⋱⋮⋮⋮_𝑟1⋯𝑏_𝑟𝑟𝑏_𝑟,𝑟+1⋯𝑏_𝑟,𝑛𝑘0𝑘⋯0⋱⋮. (2.8)
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 𝐴,𝐵∈𝐑^𝑛 be (𝑘,𝑙) concordant, 𝜏 a constant, 𝜏>0, 𝑓(𝜆) a polynomial about 𝜆. If 𝜆₀ is a root of 𝑓(𝜆)𝐼_𝑛−𝐴−𝐵𝑒^−𝜆𝜏=0, (3.1) then 𝑢=𝑓𝜆₀−𝑎𝑒^−𝜆0𝜏 (3.2) is an eigenvalue of 𝐴, or 𝑣=𝑓𝜆₀−𝑎𝑒^𝜆0𝜏 (3.3) is an eigenvalue of 𝐵, where 𝑎=𝑘 or 𝑙.

Proof. Without loss of generality, assume that 𝑃𝐴𝑃⁻¹=𝐴₁₁𝐴₁₂𝑂𝑘𝐼_𝑛2,𝑃𝐵𝑃⁻¹=𝑙𝐼_𝑛1𝐵₁₂𝑂𝐵₂₂. (3.4) Because 𝜆₀ is a root of 𝑓(𝜆)𝐼_𝑛−𝐴−𝐵𝑒^−𝜆𝜏𝑓𝜆=0,₀𝐼_𝑛1−𝐴₁₁−𝑙𝐼_𝑛1𝑒^−𝜆0𝜏∗𝑂𝑓𝜆₀𝐼_𝑛2−𝑘𝐼_𝑛2−𝐵₂₂𝑒^−𝜆0𝜏=0, (3.5) then 𝑓𝜆₀𝐼_𝑛1−𝐴₁₁−𝑙𝐼_𝑛1𝑒^−𝜆0𝜏=0or𝑓𝜆₀𝐼_𝑛2−𝑘𝐼_𝑛2−𝐵₂₂𝑒^−𝜆0𝜏=0. (3.6)  If |𝑓(𝜆₀)𝐼_𝑛1−𝐴₁₁−𝑙𝐼_𝑛1𝑒^−𝜆0𝜏|=0, then 𝑢=𝑓(𝜆₀)−𝑙𝑒^−𝜆0𝜏 is an eigenvalue of 𝐴.  If |𝑓(𝜆₀)𝐼_𝑛2−𝑘𝐼_𝑛2−𝐵₂₂𝑒^−𝜆0𝜏|=0, then 𝑣=(𝑓(𝜆₀)−𝑘)𝑒^𝜆0𝜏 is an eigenvalue of 𝐵.
As a application of Theorem 3.1, consider a BAM neural network model with delays: 𝑋̇=−𝑎𝑋+𝑓𝑌𝑡−𝜏₁,𝑌̇=−𝑏𝑌+𝑔𝑋𝑡−𝜏₂, (3.7) where 𝑋∈𝐑^𝑛,𝑌∈𝐑^𝑛,𝑎>0,𝑏>0. Assume that 𝑓,𝑔∈𝐶¹, and 𝑓(0)=0,𝑔(0)=0.
Under the hypothesis, the origin 𝑂 is an equilibrium of (3.7), and the linearization of system (3.7) at the origin 𝑂 is 𝑋̇=−𝑎𝑋+𝐵₁𝑌𝑡−𝜏₁,𝑌̇=−𝑏𝑌+𝐵₂𝑋𝑡−𝜏₂, (3.8) where 𝐵₁,𝐵₂ are Jacobi matrices.
The associated characteristic equation of (3.8) is 𝜆𝐼_𝑛+𝑎𝐼_𝑛−𝐵₁𝑒^−𝜆𝜏1−𝐵₂𝑒^−𝜆𝜏2𝜆𝐼_𝑛+𝑎𝐼_𝑛=0. (3.9) Since 𝜆=−𝑎,𝜆=−𝑏 have no influence on the stability of system (3.7), then let 𝜆≠−𝑎,𝜆≠−𝑏. We have (𝜆+𝑎)(𝜆+𝑏)𝐼_𝑛−𝑒^{−𝜆(𝜏1+𝜏2)}𝐵₁𝐵₂=0. (3.10)
Let 𝑢₁,𝑢₂,…,𝑢_𝑟 be eigenvalues of 𝐵₁𝐵₂. From Theorem 3.1, we have (𝜆+𝑎)(𝜆+𝑏)−𝑢_𝑘𝑒^−𝜆𝜏=0,(𝑘=1,2,…,𝑛), (3.11) where 𝜏=𝜏₁+𝜏₂, that is, 𝜆²+(𝑎+𝑏)𝜆+𝑎𝑏−𝑢_𝑘𝑒^−𝜆𝜏=0. (3.12) Consider 𝜆²+(𝑎+𝑏)𝜆+𝑎𝑏−𝑢_𝑘𝑒^−𝜆𝜏=0,𝑢_𝑘=𝑐_𝑘+𝑑_𝑘𝑖. (3.13)

Lemma 3.2. If max_{1≤𝑘≤𝑛}|𝑢_𝑘|>𝑎𝑏, then (3.12) has a root 𝑖𝑤₀^(𝑘)(𝑤₀>0), and 𝑤₀^(𝑘)=1√2−𝑎²+𝑏²+𝑎²−𝑏²²+4𝑢_𝑘^1/2, (3.14)𝜏₀^(𝑘)are determined by 𝑤^𝑘₀²=𝑐_𝑘cos𝑤₀^(𝑘)𝜏−𝑑_𝑘sin𝑤₀^(𝑘)𝜏,𝑎+𝑏=𝑐_𝑘sin𝑤₀^(𝑘)𝜏−𝑑_𝑘cos𝑤₀^(𝑘)𝜏. (3.15)

Proof. Let 𝑖𝑤(𝑤>0) be a root of (3.13), then −𝑤²+𝑖(𝑎+𝑏)𝑤+𝑎𝑏−𝑐_𝑘+𝑑_𝑘𝑖(cos𝑤𝜏+𝑖sin𝑤𝜏)=0. (3.16) Separating the real and imaging parts, the roots can be obtained.

Lemma 3.3.  Re(𝑑𝜆/𝑑𝜏)|_{𝜏=𝜏𝑗}^(𝑘)_,𝑤=𝑤0^(𝑘)>0.

Proof. Differentiating both sides of (3.13) with respect to 𝜏 gives 𝑑𝜆𝑑𝜏⁻¹=2𝜆+𝑎+𝑏−𝑢_𝑘𝑒^−𝜆𝜏𝑢_𝑘𝜆𝑒^−𝜆𝜏, (3.17) that is, Re𝑑𝜆𝑑𝜏⁻¹_{𝜏=𝜏𝑗}^(𝑘)_,𝑤=𝑤0^(𝑘)=𝑤₀^(𝑘)2Δ𝑎²+𝑏²+2𝑤₀^(𝑘)2>0, (3.18) where Δ=(𝑎+𝑏)²[𝑤₀^(𝑘)]²+[𝑤₀^(𝑘)]²[[𝑤₀^(𝑘)]²−𝑎𝑏]².

Theorem 3.4. Let 𝜏₀=min{𝜏₀⁽¹⁾,𝜏₀⁽²⁾,…,𝜏₀^(𝑛)}.(1)If max_{1≤𝑘≤𝑛}|𝑢_𝑘|<𝑎𝑏, then the zero solution of (3.7) is absolutely stable.(2)If min_{1≤𝑘≤𝑛}|𝑢_𝑘|≥𝑎𝑏,max_{1≤𝑘≤𝑛}Re𝑢_𝑘<𝑎𝑏, then the zero solution of (3.7) is asymptotically stable when 𝜏∈[0,𝜏₀) and unstable when 𝜏>𝜏₀, and (3.7) undergoes a Hopf bifurcation at the origin 0 when 𝜏=𝜏_𝑗^(𝑘),𝑗=0,1,…,𝑘=1,2,…,𝑛, where 𝜏=𝜏_𝑗^(𝑘) is defined in Lemma 3.2.(3)If min_{1≤𝑘≤𝑛}Re𝑢_𝑘≥𝑎𝑏, then the zero solution of (3.7) is unstable for all 𝜏≥0.

Proof. For 𝜏=0, (3.13) becomes 𝜆²+(𝑎+𝑏)𝜆+𝑎𝑏−𝑐_𝑘+𝑑_𝑘𝑖=0. (3.19)
Let 𝜆=𝛼+𝑖𝛽 be a root of (3.19). Separating the real and imaginary parts, we can obtain 𝛼²−𝛽²+(𝑎+𝑏)𝛼+𝑎𝑏−𝑐_𝑘()=0,2𝛼𝛽+𝑎+𝑏𝛽−𝑑_𝑘=0. (3.20) Hence, 𝛼=(−(𝑎+𝑏)±(𝑎+𝑏)²−4(𝑎𝑏−𝑐_𝑘))/2. If 𝑐_𝑘<0, then 𝛼<0. Using Lemmas 3.2 and 3.3 the conclusions follow.

4. Algebraic Criterion of Zero for Some Real Polynomial

Theorem 4.1. Let 𝐴,𝐵 be (𝑘,𝑙) concordant. If 𝑓(𝑧) be a polynomial about 𝑧, 𝑧₀ is a root of 𝑧^𝑚𝑓(𝑧)𝐼−𝐴−𝐵=0, (4.1) then 𝑢=𝑓𝑧₀−𝑎𝑧₀^−𝑚 (4.2) is an eigenvalue of 𝐴, or 𝑣=𝑧^𝑚₀𝑓𝑧₀−𝑎 (4.3) is an eigenvalue of 𝐵, where 𝑎=𝑘 or 𝑙.

Proof. Let us assume that 𝑃𝐴𝑃⁻¹=𝐴₁₁𝐴₁₂𝑂𝑘𝐼_𝑛2,𝑃𝐵𝑃⁻¹=𝑙𝐼_𝑛1𝐵₁₂𝑂𝐵₂₂. (4.4) Because 𝑧₀ is a root of 𝑧^𝑚𝑓(𝑧)𝑧𝐼−𝐴−𝐵=0,_𝑚0𝑓𝑧₀𝐼_𝑛1−𝐴₁₁−𝑙𝐼_𝑛1∗𝑂𝑧^𝑚₀𝑓𝑧₀𝐼_𝑛2−𝑘𝐼_𝑛2−𝐵₂₂=0, (4.5) then 𝑧^𝑚₀𝑓𝑧₀𝐼_𝑛1−𝐴₁₁−𝑙𝐼_𝑛1=0or𝑧^𝑚₀𝑓𝑧₀𝐼_𝑛2−𝑘𝐼_𝑛2−𝐵₂₂=0. (4.6)  If |𝑧^𝑚₀[𝑓(𝑧₀)𝐼_𝑛1−𝐴₁₁]−𝑙𝐼_𝑛1|=0, then 𝑢=𝑓(𝑧₀)−𝑎𝑧₀^−𝑚 is an eigenvalue of 𝐴.  If |𝑧^𝑚₀[𝑓(𝑧₀)𝐼_𝑛2−𝑘𝐼_𝑛2]−𝐵₂₂|=0, then 𝑣=𝑧^𝑚₀[𝑓(𝑧₀)−𝑎] is an eigenvalue of 𝐵.
For the application of Theorem 4.1, consider the discretization of BAM neural network (3.7). Let 𝑈(𝑡)=𝑋(𝑡−𝜏₂),𝑉(𝑡)=𝑌(𝑡),𝜏=(𝜏₁+𝜏₂), then (3.7) can be rewritten as 𝑈̇(𝑡)(𝑡)(𝑉(,𝑉̇(𝑡)(𝑡)(𝑈(𝑡.=−𝑎𝑈+𝑓𝑡−𝜏))=−𝑏𝑉+𝑔)) (4.7) Let 𝑀(𝑡)=𝑈(𝑡𝜏),𝑁(𝑡)=𝑉(𝑡𝜏), then (4.7) can be rewritten as 𝑀̇(𝑡)(𝑡)(𝑁(,𝑁̇(𝑡)(𝑡)(𝑀(𝑡.=−𝑎𝜏𝑀+𝜏𝑓𝑡−1))=−𝑏𝜏𝑁+𝜏𝑔)) (4.8) Let ℎ=1/𝑚,𝑚∈𝑁₊, using an Euler method to (4.8), we obtain 𝑀_𝑛+1=𝑀_𝑛−ℎ𝑎𝜏𝑀_𝑛+ℎ𝜏𝑓𝑁_𝑛−𝑚,𝑁_𝑛+1=𝑁_𝑛−ℎ𝑏𝜏𝑁_𝑛+ℎ𝜏𝑔𝑀_𝑛. (4.9) Let 𝑍_𝑛=(𝑀^𝑇_𝑛,𝑁^𝑇_𝑛,𝑁^𝑇_𝑛−1,…,𝑁^𝑇_𝑛−𝑚). Using the notation, (4.9) can be expressed as 𝑍_𝑛+1=𝐹𝑍_𝑛,𝜏, (4.10) where 𝐹=(𝐹^𝑇₋₁,𝐹^𝑇₀,𝐹^𝑇₁,…,𝐹^𝑇_𝑚) and 𝐹_𝑘=(1−ℎ𝑎𝜏)𝑀_𝑛+ℎ𝜏𝑓𝑁_𝑛−𝑚()𝑁,𝑘=−1,1−ℎ𝑏𝜏_𝑛+ℎ𝜏𝑔𝑀_𝑛𝑁,𝑘=0,_𝑛−𝑘,1≤𝑘≤𝑚. (4.11) It is clear that 0 is also an equilibrium of (4.10). The linearization of (4.10) at the origin 0 is 𝑍_𝑛+1=𝐴(𝜏)𝑧_𝑛,𝐴(𝜏)=(1−ℎ𝑎𝜏)𝐼_𝑛00⋯0ℎ𝜏𝐵₁ℎ𝜏𝐵₂(1−ℎ𝑏𝜏)𝐼_𝑛0⋯000𝐼_𝑛0⋯00⋯⋯⋯⋯⋯⋯000⋯𝐼_𝑛0. (4.12)(4.13) The characteristic equation of (4.12) is 𝑧𝐼−𝐴(𝜏)=0. (4.14) Since |1−ℎ𝑎𝜏|<1,|1−ℎ𝑏𝜏|<1, hence there is no influence on the stability of (4.10). Let 𝑧≠1−ℎ𝑎𝜏,𝑧≠1−ℎ𝑏𝜏, then (4.14) can be rewritten as 𝑧^𝑚(𝑧−1+ℎ𝑎𝜏)𝑧^𝑚(𝑧−1+ℎ𝑏𝜏)𝐼_𝑛−(ℎ𝜏)²𝐵₁𝐵₂=0. (4.15) Let 𝑢_𝑘(1≤𝑘≤𝑛) be an eigenvalue of 𝐵₁𝐵₂. Using Theorem 4.1, we have 𝑧^𝑚(𝑧−1+ℎ𝑎𝜏)𝑧^𝑚(𝑧−1+ℎ𝑏𝜏)𝐼_𝑟−(ℎ𝜏)²𝑢_𝑘=0. (4.16) 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) 𝑋̇=−𝑎𝑋+𝑓(𝑋(𝑡−𝜏)), (4.17) where 𝑋∈𝐑^𝑛, which is a Hopfiled neural network with delay. Its discrete system is 𝑀_𝑛+1=(1−ℎ𝑎𝜏)𝑀_𝑛+ℎ𝜏𝑓𝑀_𝑛−𝑚. (4.18) The linear part is 𝑀_𝑛+1=(1−ℎ𝑎𝜏)𝑀_𝑛+ℎ𝜏𝐵𝑀_𝑛−𝑚. (4.19) It is clear that we only need to discuss the distribution of the roots 𝑧^𝑚(𝑧−1+ℎ𝑎𝜏)=ℎ𝜏𝑢_𝑘, (4.20) where 𝑢_𝑘(1≤𝑘≤𝑛) is an eigenvalue of 𝐵.

Lemma 4.2. Let max_{1≤𝑘≤𝑛}Re𝑢_𝑘<𝑎. There exists a 𝜏>0 such that for 0<𝜏<𝜏, all roots of (4.20) have moduli less than one.

Proof. When 𝜏=0, (4.20) has an m-fold root 𝑧=0 and a simple root 𝑧=1 at 𝜏=0. Consider the root 𝑧(𝑡) such that 𝑧(0)=1. This root depends continuously on 𝜏 and is a differential function of 𝜏: 𝑑|𝑧|²𝑑𝜏_{𝑧=1,𝜏=0}=𝑧𝑑𝑧+𝑧𝑑𝜏𝑑𝑧𝑑𝜏_{𝑧=1,𝜏=0}=2−𝑎+Re𝑢_𝑘ℎ<0. (4.21) Hence, |𝑧(𝑡)|<1 for all sufficiently small 𝜏>0. Thus, all roots of (4.20) lie in |𝑧(𝑡)|<1 for sufficiently small positive 𝜏 and existence of the maximal 𝜏 follows.

Lemma 4.3. Assume that the seep size h is sufficiently small. Let max_{1≤𝑘≤𝑛}|𝑢_𝑘|<1, then there are no roots of (4.20) with moduli one for all 𝜏>0.

Proof. Let 𝑧=𝑒^{𝑖𝑤∗/𝑚} be a root of (4.20) when 𝜏=𝜏^∗, then cos𝑚+1𝑚𝑤^∗−1−ℎ𝑎𝜏^∗_𝑘cos𝑤^∗=ℎ𝜏^∗_𝑘𝑐_𝑘,sin𝑚+1𝑚𝑤^∗−1−ℎ𝑏𝜏^∗_𝑘sin𝑤^∗=ℎ𝜏^∗_𝑘𝑑_𝑘, (4.22) where 𝑢_𝑘=𝑐_𝑘+𝑖𝑑_𝑘, hence 𝑤cos^∗𝑚𝑎=1+²−𝑢_𝑘²ℎ²𝜏^∗22(1−ℎ𝑎𝜏^∗). (4.23) Let |𝑢_𝑘|<𝑎, for sufficiently small ℎ>0, |cos(𝑤^∗/𝑚)|>1, which yields a contradiction. The proof is complete.

Lemma 4.4. If the seep size ℎ is sufficiently small, then 𝑑|𝑧|²𝑑𝜏_{𝜏=𝜏∗𝑘,𝑤=𝑤∗}>0, (4.24) where 𝜏^∗, 𝑤^∗satisfy (4.22).

Proof. Consider the following equation: 𝑑|𝑧|²𝑑𝜏_{𝜏=𝜏∗𝑘,𝑤=𝑤∗}=𝑧𝑑𝑧+𝑧𝑑𝜏𝑑𝑧𝑑𝜏_𝜏=𝜏^∗_{𝑘,𝑤=𝑤}^∗=2𝜏×(𝑚+1)1−cos𝑤^∗/𝑚+𝑚1−𝜏^∗_𝑘ℎ𝑎1−cos𝑤^∗/𝑚(𝑚+1)²+1−ℎ𝑎𝜏^∗_𝑘²+2𝑚(𝑚+1)1−ℎ𝑎𝜏^∗_𝑘cos(𝑤^∗/𝑚)>0. (4.25)

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

Theorem 4.5. (1) If max_{1≤𝑘≤𝑛}|𝑢_𝑘|<𝑎, then the zero solution of (4.10) is unstable for 𝜏≥0.
(2) If min_{1≤𝑘≤𝑛}|𝑢_𝑘|≥𝑎, max_{1≤𝑘≤𝑛}𝑢_𝑘<𝑎, then the zero solution of (4.18) is asymptotically stable when 𝜏∈(0,𝜏^∗), where 𝜏^∗=min{𝜏^∗₁,𝜏^∗₂,…,𝜏^∗_𝑛}, and (4.10) undergoes a Naimark-Sacker bifurcation.
(3) If min_{1≤𝑘≤𝑛}Re𝑢_𝑘≥𝑎, then the zero solution of (4.10) is unstable for all 𝜏≥0.

5. Computer Simulation

To illustrate the analytical results found, let us consider the following particular case of (3.7): 𝑥̇₁=−𝑥₁+2tanh𝑦₂𝑡−𝜏₁,𝑥̇₂=−𝑥₂+2tanh𝑦₃𝑡−𝜏₁,𝑥̇₃=−𝑥₃+2tanh𝑦₁𝑡−𝜏₁,𝑦̇₁=−2𝑦₁−1.5tanh𝑥₂𝑡−𝜏₂,𝑦̇₂=−2𝑦₂−1.5tanh𝑥₃𝑡−𝜏₂,𝑦̇₃=−2𝑦₃−1.5tanh𝑥₁𝑡−𝜏₂, (5.1) where 𝜏₁,𝜏₂ are parameters.

Using the conclusions of Theorem 3.4, the phase-locked periodic solutions appear with 𝜏₁+𝜏₂=1.731. See Figure 1.

(a)

(b)

Figure 1 

Two phase-locked oscillations: 𝑥1(𝑡)=𝑥2(𝑡+𝑃𝑇/3)=𝑥3(𝑡+2𝑃𝑇/3), 𝑦1(𝑡)=𝑦2(𝑡+𝑃𝑇/3)=𝑦3(𝑡+2𝑃𝑇/3).

For the special case of (4.8), we have the following equations: 𝑥_1,𝑛+1=(1−𝜏ℎ)𝑥_1,𝑛+2𝜏ℎtanh𝑦_{2,𝑛−𝑚},𝑥_2,𝑛+1=(1−𝜏ℎ)𝑥_2,𝑛+2𝜏ℎtanh𝑦_{3,𝑛−𝑚},𝑥_3,𝑛+1=(1−𝜏ℎ)𝑥_3,𝑛+2𝜏ℎtanh𝑦_{1,𝑛−𝑚},𝑦_1,𝑛+1=(1−2𝜏ℎ)𝑦_1,𝑛−1.5𝜏ℎtanh𝑥_{2,𝑛−𝑚},𝑦_2,𝑛+1=(1−2𝜏ℎ)𝑦_2,𝑛−1.5𝜏ℎtanh𝑥_{3,𝑛−𝑚},𝑦_3,𝑛+1=(1−2𝜏ℎ)𝑦_3,𝑛−1.5𝜏ℎtanh𝑥_{1,𝑛−𝑚}, (5.2) where 𝜏=1.731,𝑚=3,ℎ=1/𝑚. The phase-locked periodic solutions appear. See Figure 2.

(a)

(b)

Figure 2 

Two phase-locked oscillations: 𝑥1(𝑛)=𝑥2(𝑛+𝑃𝑇/3)=𝑥3(𝑛+2𝑃𝑇/3), 𝑦1(𝑛)=𝑦2(𝑛+𝑃𝑇/3)=𝑦3(𝑛+𝑃𝑇/3).

Acknowledgment

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