Stated for $du/dt = F_2 u^{\otimes2} + F_1 u + F_0(t)$ with $F_1$ diagonalizable and eigenvalues ordered $Re(\lambda_n) \leq \ldots \leq Re(\lambda_1) < 0$, that is, a strictly dissipative linear part. Liu et al. give a convergence theorem for $R < 1$, where $R = (1/|Re(\lambda_1)|)(||u_in|| ||F_2|| + ||F_0||/||u_in||)$. It does not apply when the linear part has an eigenvalue with non-negative real part.

$F$, $y_{\mathrm{in}}$, $T$, $\varepsilon$, and a truncation or lift parameter (Carleman truncation level $N$, a phase-space grid, the level-set dimension, the homotopy order).

1. Nonlinear initial-value problem → Linear ODE systemstack the tensor powers, cut at level N

   The lift is the tower of tensor powers: $\hat{y}_j = u^{\otimes j}$ for $j = 1, \ldots, N$, started from $\hat{y}_{in} = [u_{in}; u_{in}^{\otimes 2}; \ldots; u_{in}^{\otimes N}]$. Differentiating $u^{\otimes j}$ along $du/dt = F_2 u^{\otimes 2} + F_1 u + F_0(t)$ leaves each level coupled only to its two neighbours, so the tower obeys $d\hat{y}/dt = A(t)\hat{y} + b(t)$ with $\hat{y}(0) = \hat{y}_{in}$ and $A$ block-tridiagonal: $A^j_{j+1} = F_2 \otimes I^{\otimes j-1} + I \otimes F_2 \otimes I^{\otimes j-2} + \cdots + I^{\otimes j-1} \otimes F_2$, $A^j_j$ the same sum built from $F_1$, $A^j_{j-1}$ the same sum built from $F_0(t)$, and $b(t) = [F_0(t); 0; \ldots; 0]$. $A$ is $(3Ns)$-sparse and the lifted system has dimension $\Delta = n + n^2 + \cdots + n^N = O(n^N)$. approximation: The tower is infinite and the truncation is what closes it — the exact level-$N$ equation carries the coupling $A^N_{N+1}u^{\otimes(N+1)}$ to the level above, and cutting at $N$ drops precisely that term, so the residual $\eta_j(t) := u^{\otimes j}(t) - \hat{y}_j(t)$ solves $d\eta/dt = A(t)\eta + \hat{b}(t)$ with $\eta(0) = 0$ and $\hat{b}$ carrying that one dropped block. The lift itself is exact; this is where all of the error comes from. assumption: The bound on that residual is Liu et al.'s Lemma 2, which assumes Problem 1's hypotheses — $F_1$ diagonalizable with $\mathrm{Re}(\lambda_n) \le \cdots \le \mathrm{Re}(\lambda_1) < 0$ — together with $R < 1$ for $R = (1/|\mathrm{Re}(\lambda_1)|)(\lVert u_{in} \rVert \lVert F_2 \rVert + \lVert F_0 \rVert / \lVert u_{in} \rVert)$ and $\lVert F_0 \rVert \le \lVert F_2 \rVert$, and is run after the rescaling $u \to \gamma u$ that leaves $R$ unchanged while giving $\lVert F_2 \rVert + \lVert F_0 \rVert < |\mathrm{Re}(\lambda_1)|$ and $\lVert u_{in} \rVert < 1$. Under those, $\lVert \eta_j(t) \rVert \le \lVert \eta(t) \rVert \le tN\lVert F_2 \rVert \lVert u_{in} \rVert^{N+1}$, and for a homogeneous equation ($F_0(t) = 0$) Corollary 1 gives $\lVert \eta_j(t) \rVert \le \lVert u_{in} \rVert^j R^{N+1-j}$, with the tighter $\lVert \eta_1(t) \rVert \le \lVert u_{in} \rVert R^N (1 - e^{\mathrm{Re}(\lambda_1)t})^N$ at the first level. That is what the dissipativity condition is for — in the paper's own words, "if $R < 1$, both (4.28) and (4.29) decrease exponentially with $N$, making the truncation efficient", whereas for $R \ge 1$ the same bounds force $N = \Omega(e^{|\mathrm{Re}(\lambda_1)|t})$, so "the truncation order given by Lemma 2 must grow exponentially with $t$". It is sufficient and not necessary: the paper's own intractability result begins at $R \ge \sqrt{2}$ and leaves $1 \le R < \sqrt{2}$ open. The level actually taken is $N = \lceil \log(2T\lVert F_2 \rVert/\delta)/\log(1/\lVert u_{in} \rVert) \rceil$ with $\delta = g\varepsilon/(1+\varepsilon)$ and $g = \lVert u(T) \rVert$. Katz, Muraleedharan and Alase reach the same tower from Koopman linearization by taking the polynomials as the observables and the monomials $\Psi_j: x \mapsto x^j$ as the basis: for $dx/dt = -x^2$ that gives $L\Psi_j = -j\Psi_{j+1}$, so truncating at $N$ leaves the $N \times N$ generator carrying $-1, -2, \ldots, -(N-1)$ on its superdiagonal and zero elsewhere.

   approximationassumption

A linear generator with any inhomogeneity, a lift map, a readout map, and an error bound as a function of the truncation parameter.

given  F_2, F_1, F_0(t), u_in, horizon T, error tolerance eps,
       for the quadratic ODE  du/dt = F_2 u^(x)2 + F_1 u + F_0(t)

# conditions, as this record states them: F_1 diagonalizable with eigenvalues
#   ordered Re(lambda_n) <= ... <= Re(lambda_1) < 0, that is, a strictly
#   dissipative linear part, and R < 1 for
#       R = (1/|Re(lambda_1)|)(||u_in|| ||F_2|| + ||F_0||/||u_in||).
#   Liu et al.'s truncation bound also assumes ||F_0|| <= ||F_2||.

rescale  u -> gamma u      # leaves R unchanged, and gives
                           # ||F_2|| + ||F_0|| < |Re(lambda_1)| and ||u_in|| < 1

choose the truncation level
    N = ceil( log(2 T ||F_2|| / delta) / log(1/||u_in||) ),
    with delta = g eps/(1 + eps)  and  g = ||u(T)||

# the lift: the tower of tensor powers.  The lift itself is exact; all of the
# error comes from the truncation
y_j  := u^(x)j                         for j = 1 ... N
y_in  = [ u_in ; u_in^(x)2 ; ... ; u_in^(x)N ]

# each level couples only to its neighbours, so A(t) is block tridiagonal
for j = 1 ... N:
    A^j_{j+1} = F_2 (x) I^(x)(j-1) + I (x) F_2 (x) I^(x)(j-2) + ... + I^(x)(j-1) (x) F_2
    A^j_j     = the same sum built from F_1
    A^j_{j-1} = the same sum built from F_0(t)
# A is (3 N s)-sparse, and the lifted system has dimension
#     Delta = n + n^2 + ... + n^N = O(n^N)

# truncate at level N: the exact level-N equation carries the coupling
# A^N_{N+1} u^(x)(N+1) to the level above, and cutting at N drops that term
emit  dy/dt = A(t) y + b(t),  y(0) = y_in,  b(t) = [ F_0(t) ; 0 ; ... ; 0 ]

# under R < 1 the truncation error decays with N:
#     ||u(t) - y_1(t)|| <= ||eta(t)|| <= t N ||F_2|| ||u_in||^(N+1)
#   and, for F_0 = 0,  ||eta_j(t)|| <= ||u_in||^j R^(N+1-j)
# R < 1 is sufficient, not necessary: the intractability result begins at
# R >= sqrt(2), and 1 <= R < sqrt(2) is open

return y_1 as the approximation of u; the linear system this produces is solved
       by the layer below -- on its own the linearization is a change of
       variables and buys nothing until something solves the system it produces

Liu et al. state the algorithm's complexity as $T^2 q \cdot \mathrm{poly}(\log T, \log n, \log(1/\varepsilon))/\varepsilon$, where $T$ is the evolution time, $n$ the dimension of the nonlinear system, $\varepsilon$ the allowed error, and $q$ a quantity measuring the decay of the solution. It is quoted here as the paper states it: a complexity for the complete algorithm — this embedding together with the linear solver the truncated tower feeds — and not a standalone cost for the linearization, which on its own is a change of variables and buys nothing until something solves the system it produces. The bound holds under the $R < 1$ hypothesis recorded above. How the truncation level $N$ depends on $R$ and $\varepsilon$ is in the paper's full text and is not quoted here.

Liu et al. also prove that the general quadratic ODE problem is intractable for R ≥ √2, so the band 1 ≤ R < √2 is open and $R < 1$ must not be described as necessary. Wu, Wang and Li subsequently prove linear convergence with respect to the truncation level $N$ under a resonance condition instead of a dissipative one, with numerical experiments on Burgers' equation, Fermi-Pasta-Ulam chains and the Korteweg-de Vries equation; that enlarges the set of systems for which the embedding is known to converge and does not overturn the R ≥ √2 result.