  Multimode fibers (MMFs) support abundant spatial modes and involve rich
spatiotemporal dynamics, yielding many promising applications. Here, we
investigate the influences of the number and initial energy of high-order modes
(HOMs) on the energy flow from the intermediate modes (IMs) to the fundamental
mode (FM) and HOMs. It is quite surprising that random distribution of
high-order modes evolves to a stationary one, indicating the asymptotic
behavior of orbits in the same attraction domain. By employing the Lyapunov
exponent, we prove that the threshold of the HOMs-attractor is consistent with
the transition point of the energy flow which indiactes the HOMs-attracotr acts
as a "valve" in the modal energy flow. Our results provide a new perspective to
explore the nonlinear phenomena in MMFs, such as Kerr self-cleaning, and may
pave the way to some potential applications, such as secure communications in
MMFs.
