基于切比雪夫级数展开的高效严格延拓 I

中文摘要 AI 辅助

我们研究动力系统中出现的解流形的全局延拓。我们提出了一种基于解流形的切比雪夫级数展开的严格延拓方法。首先用高阶切比雪夫插值多项式逼近分支，然后通过验证该近似附近一个拟牛顿算子的收缩性来获得显式误差界。该收缩性在加权ℓ¹空间上表述，比从一致收缩定理得到的典型C⁰误差界具有更精细的控制。事实上，后者直接从我们的收缩算子得出。此外，我们讨论了我们的策略如何自然地适用于伪弧长延拓，其中延拓参数无法提供有效的局部坐标，并扩展到多参数延拓。最后，我们详细介绍了两个应用：我们计算了Cahn-Hilliard方程的一个双参数稳态族，以及Shigesada-Kawasaki-Teramoto系统的一个经历鞍结分岔的单参数稳态族。

英文摘要

We study the global continuation of solution manifolds arising in dynamical systems. We present a rigorous continuation method based on a Chebyshev series expansion of the solution manifold. The branch is first approximated by a high-order Chebyshev interpolation polynomial, and an explicit error bound is then obtained by verifying the contraction of a quasi-Newton operator near this approximation. The contraction is formulated on a weighted $\ell^1$ space, giving a finer control than the typical $C^0$-error bound obtained from the uniform contraction theorem. In fact, the latter follows directly from our contraction operator. Furthermore, we discuss how our strategy applies naturally to pseudo-arclength continuation, where the continuation parameter fails to provide a valid local coordinate, and extends to multi-parameter continuation. Lastly, we detail two applications in which we compute a two-parameter family of steady-states for the Cahn--Hilliard equation, and a one-parameter family of steady-states undergoing saddle-node bifurcations for the Shigesada--Kawasaki--Teramoto system.

发表机构

• CMAP, CNRS, École polytechnique, Institut Polytechnique de Paris（巴黎综合理工学院，巴黎理工学院）
• National Taiwan University, Department of Mathematics（台湾大学数学系）

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