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Hovering insects require a rather large lift coefficient. Many insects hover with a large stroke amplitude (120°-170°), and it has been found that the high lift is mainly produced by the delayed-stall mechanism. However, some insects hover with a small stroke amplitude (e.g., 65°). The delayed-stall mechanism might not work for these insects because the wings travel only a very short distance in a stroke, and other aerodynamic mechanisms must be operating. Here we explore the aerodynamic mechanisms of a hoverfly hovering with an inclined stroke plane and a small stroke amplitude (65.6°). The Navier-Stokes equations are numerically solved to give the flows and forces and the theory of vorticity dynamics used to reveal the aerodynamic mechanisms. The majority of the weight-supporting vertical force is produced in the mid portion of the downstroke, a short period (about 26% of the stroke cycle) in which the vertical force coefficient is larger than 4. The force is produced using a new mechanism, the "paddling mechanism." During the short period, the wing moves rapidly downward and forward at a large angle of attack (about 48°), and strong counter clockwise vorticity is produced continuously at the trailing edge and clockwise vorticity at the leading edge, resulting in a large time rate of change in the first moment of vorticity, hence the large aerodynamic force. It is interesting to note that with the well known delayed stall mechanism, the force is produced by the relative motion of two vortices of opposite sign, while in the "paddling mechanism," it is produced by generating new vortices of opposite sign at different locations. [ABSTRACT FROM AUTHOR]