Feature-Based Continuation of Pattern Transitions in a One-Dimensional Brusselator

Abstract

Different long-time patterns can prevail in different regions of a reaction--diffusion system's parameter space. We study the transition curves between such regimes for a one-dimensional Brusselator in the two-parameter plane $(\sigma,b)$, focusing on wave/stripe-like, spiral/source-defect-like, and target-like states. We develop a feature-based continuation framework built on time-dependent PDE simulations. Scalar observables extracted from late-time solution data distinguish the regimes and define threshold level sets where the feature crossings are regular. A secant predictor and a local one-dimensional sweep corrector are used to trace these level sets. For the spiral transition, we introduce a branch-adapted spacetime symmetry-defect feature that separates asymmetric source-like patterns from more symmetric wave patterns. For the target transition, we use the minimum of a core spatial-variance score and a tail temporal-variance score to detect the characteristic structure of half-target states. The method recovers robust side portions of both transition curves. In lower parameter regions, where mixed and irregular patterns make a single scalar feature less specific, we instead report transition estimates obtained from vertical parameter sweeps and direct inspection of spacetime plots. These results show how simulation-based continuation and direct pattern classification can be combined to map regime boundaries while preserving the different levels of numerical evidence.