Classification of instabilities for the nonideal Brusselator model

TL;DR

This work classifies pattern-forming instabilities in a thermodynamically consistent, nonideal Brusselator reaction-diffusion system by applying the Cross-Hohenberg framework. Through linear stability analysis, the authors show that only type I and type III instabilities can occur, with energetic (nonideal) contributions not by themselves generating instabilities but modulating the steady state and hence the onset. Oscillatory instabilities arise only as Type III-o, while stationary patterns emerge as Type I-s or Type III-s depending on diffusion and interaction parameters; Type II and Type IV instabilities are ruled out. Numerical simulations validate the analytical predictions, revealing how diffusion coefficients tune pattern selection and yield diverse structures such as labyrinths, droplets, and zigzags. The approach provides a general methodological pathway to predict pattern formation in nonideal RD systems with molecular interactions, with potential extensions to more complex reaction networks and active media.

Abstract

We investigate a nonideal, thermodynamically consistent Brusselator reaction-diffusion (RD) system that explicitly incorporates molecular interactions among species in both the diffusion process and the underlying chemical reaction network. Within this framework, we systematically revisit the Cross-Hohenberg classification of instabilities to assess the feasibility and characteristics of the various types of instability arising from the interplay between entropic and energetic contributions. Our analysis demonstrates that only type I and type III instabilities (the Cross-Hohenberg classification) can occur in this system; Energetic contributions do not explicitly generate instabilities, but may implicitly control their occurrence through their influence on the fixed-point (steady-state) concentrations. In cases where instabilities of different types coexist, we show that the resulting patterns are highly sensitive to the relative strengths of the competing instabilities.

Figures (6)

• Figure 1: (a) The two branches ($\mathscr{B}_1$ and $\mathscr{B}_2$) of the homogeneous steady state for different interaction parameter $\chi$ that exist above the critical point $\chi_c \approx -0.1298$. Phase portrait of the homogeneous dynamics (\ref{['eq:RD']} without the diffusion term) for (b) $\chi \approx-0.1296$ and (c) $\chi=-0.1270$. The two steady state solutions are located at the intersections of the two nullclines, $\mathcal{J}_1=0$ and $\mathcal{J}_2=0$. They depart from each other from (b) to (c) as $\chi$ moves away from the critical point $\chi_c$. The other parameters are $a=2.50$, $b=8$, $\mu_1^0=0$ and $\mu_2^0=0$.
• Figure 2: A schematic representation of dispersion curves, $\Omega(q)$, corresponding to three types of instabilities: (a) type I, (b) type II, and (c) type III. Here, $\Omega$ denotes the eigenvalue with the largest real part, and $q \equiv \sqrt{\theta}$ is the wavenumber.
• Figure 3: Type III-s instability in the dispersion curve. Parameters are specified as $a=2.50$, $b=8$, $\chi= -0.11$, $D_1=1$, $D_2=2$, $\kappa_1=0.1$, and $\kappa_2=0.05$.
• Figure 4: The sketch of the type IV dispersion curve.
• Figure 5: Dispersion curves, and snapshots of $x_1$ concentration field at different time points for three different inhibitors' diffusion coefficients: (a) $D_2=1.84$, dominating type III-o instability, (b) $D_2=2$, comparable type I-s and III-o, (c) $D_2=2.30$, stronger type I-s instability. Stationary spatial patterns emerge in (b) and (c). The remaining simulation parameter values are $x_1^*=1.72$, $x_2^*=2.47$, $a=2.50$, $b=8$, $\mu_1^0=0$, $\mu_2^0=0$, $D_1=1$, $\chi=0.15$, $\kappa_1=0.10$, and $\kappa_2=0.05$.