Spiral patterns and numerical bifurcation analysis in a three-component Brusselator model for chemical reactions

Abstract

Pattern formation is a natural phenomenon that can be modeled by reaction–diffusion systems. Regular rotating spiral pattern formation and the instability of the spiral dynamics are very closely related to morphogenesis processes, cardiac arrhythmias, and chemical reactions. Since regular rotating spirals can emit periodic traveling waves from their core, hence the analysis of periodic traveling wave solutions of an oscillatory dynamical system is crucial to understand the instability of spiral patterns. To investigate the spiral patterns for chemical reactions, we consider a three-component Brusselator model. We first examine the local behavior of solutions of the model. Then we analyze the occurrence of periodic traveling wave solutions in a two-dimensional parameter plane. Next, we explore the stability outcome of periodic traveling wave solutions. It is observed that periodic traveling wave solutions change their stability, which is the Eckhaus type. The results are justified by a direct numerical simulation in one and two space dimensions. The regular and oscillatory periodic traveling wave solutions are revealed in the one-dimensional space dimension of the model by considering an Eckhaus bifurcation point. Moreover, we illustrate regular rotating spiral waves and the phenomenon for far-field breakup from the spiral waves in the two-dimensional space.