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

TL;DR

This paper introduces a duplicate-aware shift-and-lift Carleman linearization method that reduces complexity and improves accuracy in modeling nonlinear dynamics, validated through benchmark comparisons.

Contribution

It proposes a novel shift-and-lift architecture with duplicate-aware coefficient coalescing and moving-center expansion for more efficient Carleman linearization.

Findings

Convergence observed in benchmarks with refinement.

Regime-dependent accuracy improvements over Jacobian linearization.

Reduced index-resolution overhead and write-path irregularity.

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.