Dynamical behavior and bifurcation analysis of a homogeneous reaction\u2013diffusion Atkinson system

Wei Wang,Yanyou Chai,Xuguang Yang,Changjun Yu

doi:10.1186/s13661-018-0939-5

Abstract

In this paper, we are concerned with a homogeneous reaction–diffusion Atkinson oscillator system subject to homogeneous Neumann boundary conditions on a bounded spatial domain. Using the comparison principle and the techniques of invariant rectangle, we prove the existence of the attraction region of the solutions. We thus prove that under certain conditions, the solutions of the PDE system converge to the unique positive equilibrium solutions. We also derive precise conditions such that the system does not have nonconstant positive steady-state solutions. Finally, we use the bifurcation technique to show the existence of Turing patterns. The results provide a clearer understanding of the mechanism of formations of patterns.

Highlights

In 1952, Turing [1] proposed a famous idea of “diffusion-induced instability,” which says that the destabilization of otherwise stable constant steady state will lead to the emergence of stable nonuniform spatial structures, which are usually called Turing patterns
We are concerned with the pattern formations of a kind of diffusive Atkinson oscillator model, which is used to characterize the mechanism of pattern formations
3 Nonexistence of Turing patterns for the particular case where both n1 and n2 equal 1, we show the nonexistence of nonconstant positive steady-state solutions of the system

Summary

Introduction

In 1952, Turing [1] proposed a famous idea of “diffusion-induced instability,” which says that the destabilization of otherwise stable constant steady state will lead to the emergence of stable nonuniform spatial structures, which are usually called Turing patterns. We have the following results on the global existence and boundedness of the timedependent solutions of system (1.2). Proof The existence and uniqueness of local-in-time solutions to the initial-boundary value problem (1.2) is well known [15].

Results

Conclusion