Volume 2025, Issue 1 5551426

Research Article

Open Access

The Matrix Pencil Method for Determining Imaginary Axis Eigenvalues and Stability of Neutral Delay Reaction–Diffusion Systems

Corresponding Author

Jian Ma

[email protected]

orcid.org/0000-0003-1161-1256

Department of Mathematics , School of Mathematics and Statistics , Hainan Normal University , Haikou , 571158 , China , hainnu.edu.cn

Key Laboratory of Data Science and Intelligence Education of Ministry of Education , Hainan Normal University , Haikou , 571158 , China , hainnu.edu.cn

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Rui Liu,

Rui Liu

Department of Mathematics , School of Mathematics and Statistics , Hainan Normal University , Haikou , 571158 , China , hainnu.edu.cn

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Jian Ma,

Corresponding Author

Jian Ma

[email protected]

orcid.org/0000-0003-1161-1256

Department of Mathematics , School of Mathematics and Statistics , Hainan Normal University , Haikou , 571158 , China , hainnu.edu.cn

Key Laboratory of Data Science and Intelligence Education of Ministry of Education , Hainan Normal University , Haikou , 571158 , China , hainnu.edu.cn

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Rui Liu,

Rui Liu

Department of Mathematics , School of Mathematics and Statistics , Hainan Normal University , Haikou , 571158 , China , hainnu.edu.cn

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First published: 02 January 2025

https://doi.org/10.1155/ddns/5551426

Academic Editor: Luisa Di Paola

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Abstract

In this note, the stability of neutral delay reaction–diffusion systems (NDRDS) was concerned by applying the matrix pencil and the Kronecker product. A new computing method for the distribution of imaginary axis eigenvalues on general n-dimensional NDRDS will be introduced. A practical, checkable criterion for the asymptotic stability will be derived. The main contribution of this paper is that we provide a computational method for determining imaginary axis eigenvalues and minimal delay margin on general NDRDS.

1. Introduction

Nowadays, delay reaction–diffusion systems have become important natural models in physics, engineering, multibody mechanics, computer-aided design, and economic systems. The research studies on the stability of delay reaction–diffusion systems have received widespread concern [1–5]. However, for more sophisticated delay reaction–diffusion systems, the results are very few because of the large computational complexity. As a new and effective tool, matrix algebra has been applied to this field [6–8]. But the results are very few. In this paper, we will introduce matrix pencil method to study the stability of general neutral delay reaction–diffusion systems (NDRDS).

The general n-dimensional NDRDS can be expressed in matrix form with

()

where the time variable t > 0, the spatial variable x ∈ S is restricted to one-dimensional spatial domain S = (0, π), U_t = ∂U/∂t,

()

As components of vector U, u_i = u_i(x, t) > 0, i = 1, 2, …, n stand for the concentrations of n reactants or the densities of n species. B₁ is a real constant matrix. Δ denotes the Laplace operator. F is a vector function consist of n analytic functions

()

First, if system equation (1) has a positive equilibrium

, then the linearization of system equation (1) at positive equilibrium

is given by

()

The sequence of characteristic equations can be denoted as

()

where A, B₂ are Jacobi matrices of function F with respect to u_i(x, t) and u_i(x, t − τ) at the positive equilibrium

()

As we all know, the distribution of all eigenvalues of system equation (4) can be determined from equation (5). The local stability of system equation (1) can be described by studying the linear system equation (4).

In the sequel, we denote the imaginary axis by ∂C. λ(A), Reλ(A), σ(A), and ρ(A), respectively, denote some eigenvalue, the real part of some eigenvalue, the spectrum, and the spectral radius of the matrix A. Eig(P, Q), ImEig(P, Q), and σ(P, Q), respectively, denote general eigenvalues, imaginary axis general eigenvalues, and the spectrum of the matrix pencil (P, Q).

The remaining parts of the paper are structured in the following way: In Section 2, we will get the algebraic criteria for determining the imaginary axis eigenvalues of system equation (4). In Sections 3 and 4, we will discuss the stability of system equation (4) and present a computational method to determine the minimal delay margin.

2. The Distribution of Imaginary Axis Eigenvalues

For B₁, B₂, D_k + A = A_k ∈ R^n×n, s = iw ∈ ∂C, w > 0, let

()

Theorem 1. Any imaginary axis eigenvalue of system equation (4) is a zero point of det(Λ_k(s)), and also one of the eigenvalues of the matrix pencil (E, F_k).

Proof 1. Let D_k + A = A_k, v ∈ Rⁿ, v ≠ 0, the sequence of characteristic equation (5) can be rewritten as

()

Via the elementary transform φ and the Kronecker product, one can get

()

Corollary 1. If E is nonsingular, imaginary axis eigenvalues of system equation (4) can be calculated from the matrix E⁻¹F_k.

Example 1. Consider system equation (4) with d₁ = d₂ = 1,

()

First, λ(B₁) ≈ {0.655, 0.095}, so matrix E is invertible. By MATLAB, imaginary axis generalized eigenvalues of the matrix pencil (E, F₁), or imaginary axis eigenvalues of the matrix E⁻¹F₁ can be presented as

()

We get all the imaginary axis eigenvalues of system equation (4). For another expression,

()

which is exactly a quadratic eigenvalue problem.

Theorem 2. All imaginary axis roots of equation (14) are general eigenvalues of the matrix pencil (P, Q_k), where

()

Proof 2. Equation (9) can be rewritten as

()

As a quadratic eigenvalue problem, let

()

then Λ_k(s) = a₀s² − a₁ · s − a₂. Denote

()

and then, [sP − Q_k] · u = 0 has nonzero solution if and only if

()

So s is a general eigenvalue of the matrix pencil (P, Q_k), that is,

()

Corollary 2. If B₁ is nonsingular, all generalized imaginary axis eigenvalues of the matrix pencil (P, Q_k) can be calculated from the matrix G_k, where

()

Proof 3. B₁ is nonsingular, then

()

Similarly, all imaginary axis roots of the equation det(sP − Q) = 0 can be found from the matrix

()

By Example 1, we can get

()

By the matrix pencil method, imaginary axis eigenvalues of system equation (4) were simply determined.

3. The Stable Analysis

In this section, the local stability of system equation (1) at positive equilibrium

will be determined. Denote the spectrum of system equation (4) as

()

It is well known that the solution of the neutral system equation (4) is asymptotically stable if all elements of the sets S_k(τ) have a negative real part bounded away from 0. That is, there exists a number r > 0 such that Re(λ) ≤ −r < 0 for any element λ of the sets S_k(τ)(k = 1, 2, …). Because even though characteristic roots satisfy Re(λ) < 0, it is also possible that the solution of system equation (4) will be unbounded as t⟶∞.

In 1993, Hale introduced a stable criterion for neutral systems in Reference [9], which can be extended to the neutral system equation (4).

Lemma 1. Let γ_k = sup{Re(λ)|λ ∈ S_k(τ)}, k = 1, 2, …. If γ_k < 0, then the neutral system equation (4) is asymptotically stable.

From [10], it can be shown that

()

and there must exist a number r > 0, such that Re(λ) ≤ −r < 0.

Lemma 2. Suppose the spectrum radius ρ(B₁) < 1 or ‖B₁‖ < 1. The neutral system equation (4) is asymptotically stable if all elements of the S_k(τ)(k = 1, 2, …) only have negative real parts.

In the following, stable conditions of system equation (4) with respect to the delay parameter τ will be discussed. First, for τ = 0, equation (5) can be written as

()

Lemma 3. System equation (4) is asymptotically stable at τ = 0 if

()

Meantime, the dependence of stability for system equation (4) with respect to parameter τ can be derived.

Theorem 3. Suppose that system equation (4) is asymptotically stable at τ = 0 and ρ(B₁) < 1 or ‖B₁‖ < 1. System equation (4) is asymptotically stable for any τ ≥ 0 if σ(P, Q_k)∩∂C⁺ = ∅ for k = 1, 2, …. Otherwise, the stability of system equation (4) may change at some τ^∗ when σ(P, Q_k)∩∂C⁺ ≠ ∅ for some fixed k.

4. The Computational of Minimal Delay Margin

In this section, the minimal critical delay τ^∗ will be found such that system equation (4) is asymptotically stable for τ ∈ (0, τ^∗). Let z = e^−λτ, λ = ±iw, ∂D = {z||z| = 1, z ∈ C}. Denote

()

Let iw_j ∈ ∂C⁺, z_j ∈ ∂D such that iw_j ∈ σ(P_k(z_j)), that is,

()

Let

()

For some fixed k, let , λ = iw_j ∈ σ(P, Q_k), and w_j ∈ R⁺. The minimal critical delay τ^∗ = min{τ_j} can be obtained from .

Theorem 4. Assuming that system equation (4) is asymptotically stable at τ = 0, and ρ(B₁) < 1 or ‖B₁‖ < 1. System equation (4) is asymptotically stable for all τ ≥ 0 if σ(P, Q_k)∩∂C⁺ = ∅, or σ(P, Q_k)∩∂C⁺ ≠ ∅ and σ(M_k, N_k)∩∂D = ∅ for k = 1, 2, …. Otherwise, system equation (4) must be asymptotically stable for τ ∈ [0, τ^∗). The stability may change when τ > τ^∗, and some bifurcation will appear at τ^∗.

Proof 4. Analogously as Section 2, Equation (30) is equivalent to

()

By direct unfolding,

()

Similarly, [zM_k − N_k]u = 0 has nonzero solution if and only if

()

So z is a generalized eigenvalue of the matrix pencil (M_k, N_k).

By Example 1, λ(B₁) = {0.655, 0.095}, so ρ(B₁) < 1. When τ = 0, k = 1,

()

Next for τ > 0, we have obtained all imaginary axis eigenvalues of system equation (4)

()

When , |z| ≈ 1.000, τ₁ ≈ 0.476. When . The minimal critical delay τ^∗ = min{τ₁, τ₂} = 0.374. The system is stable for τ ∈ (0, τ^∗).

5. Conclusions

In this paper, we have studied the stable properties of general NDRDS using the matrix pencil method. In a similar manner, these results can be extended directly to lagged delay reaction–diffusion systems, neutral or lagged delay systems, and so on.

Conflicts of Interest

The authors declare no conflicts of interest.

Funding

This work was supported by the Hainan Provincial Natural Science Foundation of China (Grant Nos. 121MS030 and 623RC482) and the specific research fund of the Innovation Platform for Academicians of Hainan Province.

Open Research

Data Availability Statement

Data availability is not applicable to this article as no new data were created or analyzed in this study.

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The Matrix Pencil Method for Determining Imaginary Axis Eigenvalues and Stability of Neutral Delay Reaction–Diffusion Systems

Abstract

1. Introduction

2. The Distribution of Imaginary Axis Eigenvalues

3. The Stable Analysis

4. The Computational of Minimal Delay Margin

5. Conclusions

Conflicts of Interest

Funding

Open Research

Data Availability Statement

References

References

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The Matrix Pencil Method for Determining Imaginary Axis Eigenvalues and Stability of Neutral Delay Reaction–Diffusion Systems

Abstract

1. Introduction

2. The Distribution of Imaginary Axis Eigenvalues

3. The Stable Analysis

4. The Computational of Minimal Delay Margin

5. Conclusions

Conflicts of Interest

Funding

Open Research

Data Availability Statement

References

References

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