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In our study, based on the normalized linear Schrödinger equation, we have analytically and numerically investigated the propagation dynamics of chirped cosh-Gaussian beams in a medium with parabolic potential. The obtained results show that cosh-Gaussian beams perform a periodic auto-focusing behavior and the parabolic potential determines the focusing ability, including the focal distance as well as peak intensity at the focus. Especially, the intensity distributions and waveform of cosh-Gaussian beam are related to the initial parameter of cosh function. Furthermore, we also demonstrate the effect of chirp factors on the beam and find that the periodic oscillating behavior caused by linear chirp can be used to manipulated the propagation trajectory of beam, but linear chirp does not affect the focal intensity. While the quadratic chirp can enhance the focusing ability and peak intensity of beam on the axis, which indicates that the quadratic chirp factor plays a significant role in the modulation of the energy localization.

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