AI 中文总结
该研究针对一维布鲁塞尔子，开发基于特征的延拓框架，结合模拟与模式分类绘制参数空间中波、螺旋、靶状等模式的转变边界，获取了可靠的转变曲线并验证了方法有效性。
AI 中文摘要
在反应-扩散系统的参数空间不同区域，可存在不同的长时间模式。我们研究二维参数平面(σ,b)上的一维布鲁塞尔子这类模式之间的转变曲线，重点关注波/条纹状、螺旋/源缺陷状以及靶状状态。我们开发了基于特征的延拓框架，该框架构建于随时间变化的偏微分方程(PDE)模拟之上。从后期解数据中提取的标量可观测量可区分不同区域，并定义特征交叉为正则的阈值水平集。我们使用正割预测器和局部一维扫描校正器来追踪这些水平集。对于螺旋转变，我们引入了分支适配的时空对称性缺陷特征，该特征可将非对称源状模式与更对称的波模式区分开；对于靶转变，我们使用核心空间方差得分的最小值和尾部时间方差得分的最小值来检测半靶状态的特征结构。该方法可恢复两条转变曲线的可靠侧段。在参数较低区域，混合模式和不规则模式使单个标量特征的特异性降低，我们转而报告从垂直参数扫描和时空图直接检查得到的转变估计值。这些结果表明，如何将基于模拟的延拓与直接模式分类相结合，以绘制区域边界，同时保留不同水平的数值证据。
英文摘要
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 $(σ,b)$, 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.
Comments65 pages, 16 figures. MATLAB code is available at https://github.com/Jominemyqs/Brusselator_continuation
