A systematic numerical analysis is performed for the quasi-periodicity in the wake where a circular cylinder is rotationally oscillated in time. The main emphasis is placed on the identification of frequency selection subjected to the controlled perturbations in the vicinity of lock-on. The frequency responses are scrutinized by measuring the lift coefficient (CL). A direct numerical simulation is made to portray the unsteady dynamics of wake flows at Re = 110. It is found that, after the shedding frequency is bifurcated at the boundary of lock-on, one frequency follows the forcing frequency and the other gradually converges to the natural shedding frequency. The asymptotic convergence phenomena are observed by solving the Van der Pol equation and the circle map. A new frequency selection formula is proposed. The quasi-periodic states are interpreted in terms of the forcing frequency, shedding frequency and modulated frequencies by employing the torus concept and the CL(t) diagram. In the quasi-periodic state, the variation of magnitudes and relevant phase changes of CL with forcing phase are examined.
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