We propose the nullspace-preserving high-index saddle dynamics (NPHiSD) method for degenerating multiple solution systems in constrained and unconstrained settings. The NPHiSD efficiently locates high-index saddle points and provides parent states for downward searches of lower-index saddles, thereby constructing the solution landscape systematically. The NPHiSD method searches along multiple efficient ascent directions by excluding the nullspace, which is the key for upward searches in degenerate problems. To reduce the cost of frequent nullspace updates, the search is divided into segments, within which the ascent directions remain orthogonal to the nullspace of the initial state of each segment. A sufficient and necessary condition for characterizing the segment that admits efficient ascent directions is proved. Extensive numerical experiments for typical problems such as Lifshitz-Petrich, Gross-Pitaevskii, and Lennard-Jones models are performed to show the universality and effectiveness of the NPHiSD method.

翻译：本文针对约束与非约束条件下的退化多解系统，提出了零空间保持高阶鞍点动力学方法。该方法能高效定位高阶鞍点，并为向下搜索低阶鞍点提供父状态，从而系统性地构建解景观。通过排除零空间，NPHiSD方法沿多个高效上升方向进行搜索，这是处理退化问题中向上搜索的关键。为降低频繁更新零空间的计算成本，搜索过程被划分为若干区段，在每个区段内上升方向始终保持与该区段初始状态零空间的正交性。文中证明了一个充分必要条件，用于表征允许高效上升方向的搜索区段特征。通过对Lifshitz-Petrich模型、Gross-Pitaevskii方程及Lennard-Jones势等典型问题开展大量数值实验，验证了NPHiSD方法的普适性与有效性。