  In this work, we have systematically investigated the Krylov complexity of
curvature perturbation for the modified dispersion relation in inflation, using
the algorithm in closed system and open system. Our analysis could be applied
to the most inflationary models. Following the Lanczos algorithm, we find the
very early universe is an infinite, many-body, and maximal chaotic system. Our
numerics shows that the Lanczos coefficient and Lyapunov index of the standard
dispersion relation are mainly determined by the scale factor. As for the
modified case, it is nearly determined by the momentum. In a method of the
closed system, we discover that the Krylov complexity will show irregular
oscillation before the horizon exits. The modified case will present faster
growth after the horizon exists. Since the whole universe is an open system,
the approach of an open system is more realistic and reliable. Then, we
construct the exact wave function which is very robust only requiring the
Lanczos coefficient proportional to $n$ (main quantum number). Based on it, we
find the Krylov complexity and Krylov entropy could nicely recover in the case
of a closed system under the weak dissipative approximation, in which our
analysis shows that the evolution of Krylov complexity will not be the same
with the original situation. We also find the inflationary period is a strong
dissipative system. Meanwhile, our numerics clearly shows the Krylov complexity
will grow during the whole inflationary period. But for the small scales, there
will be a peak after the horizon exits. Our analysis reveals that the dramatic
change in background (inflation) will significantly impact the evolution of
Krylov complexity. Since the curvature perturbation will transit from the
quantum level to the classical level.
