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Chaotic thermal convection in a rapidly rotating cylindrical annulus is investigated numerically and the relaxation oscillation state is obtained under the no-slip boundary condition. The dominant frequency of the oscillation is inherited directly from a vacillating mode, whose nonlinear interaction with another high-frequency vacillating mode leads to the chaotic state at high Rayleigh numbers through an RTN-type route. Furthermore, the effects of Coriolis parameter and Rayleigh number on the quasi-periodic burst of kinetic energy are discussed as well.