  In this paper, we explore the embedding of nonlinear dynamical systems into
linear ordinary differential equations (ODEs) via the Carleman linearization
method. Under dissipative conditions, numerous previous works have established
rigorous error bounds and linear convergence for Carleman linearization, which
have facilitated the identification of quantum advantages in simulating
large-scale dynamical systems. Our analysis extends these findings by exploring
error bounds beyond the traditional dissipative condition, thereby broadening
the scope of quantum computational benefits to a new class of dynamical
regimes. This novel regime is defined by a resonance condition, and we prove
how this resonance condition leads to a linear convergence with respect to the
truncation level $N$ in Carleman linearization. We support our theoretical
advancements with numerical experiments on a variety of models, including the
Burgers' equation, Fermi-Pasta-Ulam (FPU) chains, and the Korteweg-de Vries
(KdV) equations, to validate our analysis and demonstrate the practical
implications.
