  Dynamic encircling a second-order exception point (EP) exhibit chiral state
transfer, while there is few research on dynamic encircling multiple and
higher-order EPs. Here, we study proximity-encirclement of the EPs in a
multimode optomechanical system to understand the closed path evolution of
high-order non-Hermitian systems. The optomechanical system has three types of
situations about EPs: the system has no EP, a pair of second-order EPs, and a
third-order EP. The dynamical behavior of the system's dependence on the
initial state, orientation, and velocity of the loop, the variance in the
starting point of the loop, as well as the number and order of EPs encircled by
the loop have been investigated in the process of state transfer. The results
show that chiral or non-reciprocal state transfer can be realized when the loop
encircling a second-order EP with different radius. Only chiral state transfer
occurs when encircling two second-order EPs. Moreover, chiral and
non-reciprocal state transfer can happen in a single loop encircling a
third-order EP. The phenomena about encircling the EPs in a multimode
optomechanical system provides another means for manipulating state transfer in
higher-order non-Hermitian systems.
