Various applications of volume dielectric barrier discharges (DBDs) with airflows have attracted significant attention such as in the fields of plasma medicine,surface modification,ozone synthesis,etc.In this work,the nonlinearity characteristics of DBDs in initiating and extinguishing boundaries with airflows are experimentally investigated.It is found that the difference between initiating pulse repetition frequencies (PRFs) and extinguishing PRFs is affected by the addition of airflows.A hysteresis region is produced between these two PRFs.A topological rule of Thom's classification theorem is proposed to investigate the hysteresis phenomenon of discharges with airflows.It is concluded that the discharge state is dependent on the operation route.The discharge state would transit from initiating to extinguishing,or in the opposite direction,while passing along a specific operation route.Based on the topological method,two nonlinear laws of discharge structure transition under the typical operation routes are predicted and verified in the discharge experiments.