Dynamic transitions and Turing patterns of the Brusselator model

Abstract

The dynamic transitions of the Brusselator model has been recently analyzed in Choi et al. (2021) and Ma and Wang (2011). Our aim in this paper is to address the relation between the pattern formation and dynamic transition results left open in those papers. We consider the problem in the setting of a 2D rectangular box where an instability of the homogeneous steady state occurs due to the perturbations in the direction of several modes becoming critical simultaneously. Our main results are twofold: (1) a rigorous characterization of the types and structure of the dynamic transitions of the model from basic homogeneous states and (2) the relation between the dynamic transitions and the pattern formations.We observe that the Brusselator model exhibits different transition types and patterns depending on the nonlinear interactions of the pattern of the critical modes.

Hopf Bifurcation and Self-Organization Pattern of a Modified Brusselator Model

“…[21][22][23][24]. In what follows, we focus on the spatial dynamics of the diffusive Brusselator model (3). By direct analysis, we give the occurrence conditions of the Turing instability and respectively.…”

“…Chen and Wang [2] reported the boundedness of the solution and studied the properties of the Hopf bifurcation for a generalized Lengyel-Epstein model. Muntari and Sengül [3] performed a rigorous characterization of the types and structure of the dynamic transitions and showed the relation between the dynamic transitions and the pattern formations of a Brusselator model in a 2D rectangular box. Wong and Ward [4] developed a hybrid asymptotic-numerical theory with respect to the effect of different types of localized heterogeneities on the existence, stability, and slow dynamics of spot patterns for the Schnakenberg reaction-diffusion model in a 2-D domain.…”