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.

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