An Algebraic Method on the Eigenvalues and Stability of Delayed Reaction-Diffusion Systems

The reaction-diffusion system is a semilinear partial differential equation, which has been used for the study of morphogenesis, population dynamics, and autocatalytic oxidation reactions. The early reaction-diffusion models were constructed to describe the process of chemical reaction. For example, the Brusselator reaction-diffusion system and its improved systems had been researched by many scholars [1, 2]. Besides, during the past years, many results about the stability, steady state bifurcation, and the Hopf bifurcation on the reaction-diffusion systems had been derived [3–8]. From those results, we see that most of them had beenmainly focused on two methods: the analytical methods and the numerical methods.The analytical methods, such as the center manifold theorem, the normal form theory, and the Laplace transformation, were mainly used to research the analytic solutions and dynamical property. The numerical methods, such as the Runge-Kutta methods and linear multistep methods, paid close attention to checking the changes and convergence of solutions. For example, Wu determined the direction and stability of periodic solutions occurring through the Hopf bifurcation by the center manifold theory and the normal form theory, which are the classic analytical methods in functional differential systems [1]. Wei et al. in papers [3–5] demonstrated the bifurcation and stability of different reaction-diffusion systems. In this paper, we will introduce a new algebraic method. In fact, in the past recent years, many algebraic methods were applied to the functional differential systems, such as matrix theory, polynomial theory, the Lie group, and algebraic system. In particular for the study on the more complex systems, such as the high dimensional functional differential systems, the algebraicmethods have their advancement. So in the following, we will discuss the delayed reaction-diffusion systems by the algebraic methods. We will study a general delayed reaction-diffusion system with spatial domain Ω in the following form:


Introduction
The reaction-diffusion system is a semilinear partial differential equation, which has been used for the study of morphogenesis, population dynamics, and autocatalytic oxidation reactions.The early reaction-diffusion models were constructed to describe the process of chemical reaction.For example, the Brusselator reaction-diffusion system and its improved systems had been researched by many scholars [1,2].Besides, during the past years, many results about the stability, steady state bifurcation, and the Hopf bifurcation on the reaction-diffusion systems had been derived [3][4][5][6][7][8].From those results, we see that most of them had been mainly focused on two methods: the analytical methods and the numerical methods.The analytical methods, such as the center manifold theorem, the normal form theory, and the Laplace transformation, were mainly used to research the analytic solutions and dynamical property.The numerical methods, such as the Runge-Kutta methods and linear multistep methods, paid close attention to checking the changes and convergence of solutions.For example, Wu determined the direction and stability of periodic solutions occurring through the Hopf bifurcation by the center manifold theory and the normal form theory, which are the classic analytical methods in functional differential systems [1].Wei et al. in papers [3][4][5] demonstrated the bifurcation and stability of different reaction-diffusion systems.
In this paper, we will introduce a new algebraic method.In fact, in the past recent years, many algebraic methods were applied to the functional differential systems, such as matrix theory, polynomial theory, the Lie group, and algebraic system.In particular for the study on the more complex systems, such as the high dimensional functional differential systems, the algebraic methods have their advancement.So in the following, we will discuss the delayed reaction-diffusion systems by the algebraic methods.We will study a general delayed reaction-diffusion system with spatial domain Ω in the following form: . . .
That is to say, the origin point (0, 0, . . ., 0) is an equilibrium of system (1) and (2).So the matrix form of system (1) can be written as where
The remaining parts of the paper are structured in the following way.In Section 2, we demonstrated the critical condition on the delay  of the system (1) and got the algebraic criteria for determining the pure imaginary eigenvalues.In Section 3, we researched the stability and the Hopf bifurcation of the delayed reaction-diffusion equation (39) with Neumann boundary condition and derived the corresponding algebraic criteria.At last, we described a specific reaction-diffusion equation and simulated the results by MATLAB.
Proof .For (8), let   +  =   ,  = 0, 1, 2, . . . .Then we have Let  =  be a pure imaginary eigenvalue of the system (5), let  be associated eigenvector, and let |||| = 1.We have   (,  − ) = 0.By conjugating and transforming, we can get Via the elementary transform , we get That is Λ 0 () = 0,  = ( * ).We know that det Λ 0 () = 0, and so, det From Theorem 2, we know that all of the pure imaginary eigenvalues of the system (1) are zero points of the algebraic equation det [( −   ) ⊗ ( +   ) +  ⊗ ] = 0. (31) So the pure imaginary eigenvalues of the system (1) or ( 5) can be computed via the algebraic equation (31).In fact, (31) is usually called a polynomial eigenvalue problem.The classical and most widely used approach to research the polynomial eigenvalue problems is linearization, where the polynomial is converted into a larger matrix pencil with the same eigenvalues.There are many forms for the linearization, but the companion form is most typically commission [9].So we have the following results.
Suppose that the system (1) is stable at the initial time.Then the stability of the system will change as the positive real part root of characteristic equation emerged for the different parameter values of system (1).So the first pure imaginary eigenvalue is the critical condition and plays an important role in researching the stability and the Hopf bifurcation of the delayed reaction-diffusion equation.From the previous results, we can get all of the pure imaginary eigenvalues for a delayed reaction-diffusion equation.Here let the time delay  be the parameter.Then we can get the stable or unstable condition of the delayed reaction-diffusion equation for different parameter values.

The Stability and the Hopf Bifurcation of a Delayed Reaction-Diffusion Equation
In the recent years, scholars discussed many different reaction-diffusion systems [10][11][12].For different systems, they expounded different results to justify the stability.Here we will study a general high-dimensional reaction-diffusion equation with single delay.The considered reaction-diffusion system is ) , where   =   (, ) and   =   (,  − ).For simplicity we suppose that the eigenvalues of the Laplace operator Δ by the Neumann boundary condition are simple.From Section 1, we know that the eigenvalues of Δ in (0, ) with the Neumann boundary condition are   = − 2 ,  = 0, 1, 2, . . . .Let  denote an eigenvalue of system (39).Then there exists a nonzero vector  ∈ Dom( Δ) such that The stability of the system (39) is determined by det  (, , ) = 0.
We know that the system (39) is asymptotically stable when all of the eigenvalues of (42) are in the left part of the imaginary axis.Apparently, for nondiffusive case, that is,  1 =  2 = ⋅ ⋅ ⋅ =   = 0, the system (39) becomes a delay differential equation, and the stability has been studied by many scholars.Here we mainly discuss the diffusive system.Let Firstly, when  = 0, the system (39) becomes a partial differential equation with no delay.So we can get the stability in the following.
Next we research the delay-independent and the delaydependent stabilities of the system (39).Let (, ) denote the set which contains all the general eigenvalues of matrix pencil (, ).From (38) in Section 2, we have the following results.
Theorem 5.For any  ≥ 0 and a fixed eigen-mode  ∈ , if the coefficient matrices of the system (39) satisfy the following conditions: (i) Res < 0,  ∈ (  +  + ), (ii) Res ̸ = 0,  ∈ (, ), then the system (39) is asymptotically stable for all  ≥ 0 with fixed eigen-mode ; for example, the stability is delayindependent and eigenmode -dependent.On the other hand, if, for all eigen-mode , the stability is delay independent, then stability of system (39) is delay independent and eigenmode independent.
Lemma 6.Given a fixed eigen-mode  ∈ , let Res < 0,  ∈ (  +  + ).For all the pure imaginary roots  =  of (31) (or general pure imaginary eigenvalues of matrix pencil (, )), one can get the critical value of delay parameter  *  for the fixed eigen-mode .
From the above theorems, we can discuss the stability and the Hopf bifurcation of system (39).So the dynamical character of a generic class of nonlinear reaction-diffusion system with the Neumann boundary condition can be studied by the algebraic methods.It is well known that reaction and diffusion of physical chemistry and chemical or biochemical species can produce all kinds of spatial patterns.Next we will give a general example.

Conclusion
In this paper, we consider a general high-dimensional delayed reaction-diffusion system.By the algebraic methods, such as the matrix pencil and the linear operator, we discussed the eigenvalues and the stability of the delayed reaction-diffusion system (1) and (39).In fact, we only find the pure imaginary  eigenvalues, which are a few parts of the infinite eigenvalues.So we can easily compute pure imaginary eigenvalues from the algebraic equation (31) by MATLAB.Certainly, applying the algebraic methods to analyze the dynamical properties of the reaction-diffusion system with delays is still a new and immature field, so we believe that the algebraic methods used to research the stability of the delayed reaction-diffusion systems would be of more interest in the future.