Duplicate-Aware Shift-and-Lift Carleman Linearization:Structure, Complexity, and Comparative Evaluation

Abstract: The primary objective of this study is to remove duplicated monomial contributions that proliferate in Carleman linearization as state dimension and truncation order increase. To do so, we adopt a shift-and-lift architecture, since it exposes repeated exponent targets and allows duplicate-aware coefficient coalescing during lifted-operator assembly. This architecture also makes high-order truncation practical, but that regime intensifies local convergence and closure sensitivity for higher-order nonlinearities. We therefore pair shift-and-lift with a moving-center expansion so that shift and lift are updated jointly around evolving local centers, improving validity of the truncated model along the trajectory. The resulting workflow combines symmetry-reduced monomial bases, packed exponent-key indexing, and sparse triplet coalescing to preserve truncated affine dynamics while reducing index-resolution overhead and write-path irregularity. We analyze variable growth, preprocessing complexity, and truncation-induced error mechanisms, and we compare against Jacobian linearization through fixed-step error, admissible step size, and cost-at-target-accuracy criteria. Two benchmarks (bilinear driver and logistic interaction) show convergence under refinement for both approaches, with regime-dependent accuracy gains for the proposed method rather than universal superiority.