Two groups of frequency sweeping modes are observed and interpreted in the HL-2 A plasmas with qmin ∼ 1. The tokamak simulation code calculations indicate the presence of a reversed shear q-profile during the existence of these modes. The mode frequencies lie in between TAE and BAE frequencies, i.e. ωBAE < ω < ωTAE, and these modes are highly localized near qmin, i.e. r/a ∼ 0.25. A group of modes characterized by down-sweeping frequency with qmin decrease due to qmin > 1 and nqmin − m > 0, and another group of modes characterized by up-sweeping frequency with qmin drop, owing to qmin < 1 and nqmin − m < 0 before sawtooth crash. The kinetic Alfvén eigenmode code analysis supports that the down-sweeping modes are kinetic reverse shear Alfvén eigenmodes (KRSAEs), and the up-sweeping modes are RSAEs, which exist in the ideal or kinetic MHD limit. In addition, the down- and up-sweeping RSAEs both have fast nonlinear frequency behaviour in the process of slow frequency sweeping, i.e. producing pitch-fork phenomena. These studies provide valuable constraint conditions for the q-profile measurements.