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.

65 pages, 16 figures. MATLAB code is available at https://github.com/Jominemyqs/Brusselator_continuation