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

Jian Ma,Baodong Zheng

doi:10.1155/2013/412343

Abstract

The eigenvalues and stability of the delayed reaction-diffusion systems are considered using the algebraic methods. Firstly, new algebraic criteria to determine the pure imaginary eigenvalues are derived by applying the matrix pencil and the linear operator methods. Secondly, a practical checkable criteria for the asymptotic stability are introduced.

Highlights

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
We see that most of them had been mainly focused on two methods: the analytical methods and the numerical methods
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]

Summary

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. 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]. 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]. We described a specific reaction-diffusion equation and simulated the results by MATLAB

Algebraic Criteria for Determining the Pure Imaginary Eigenvalues

Conclusion