This article presents numerical results of flow-induced rotary oscillation of a circular cylinder with rigid splitter plate in steady flow. Different from the previous examinations with freely rotatable assembly which mainly considered linear restoring force, the rotary oscillation of the structure in this work is modelled by a Duffing oscillator with both linear and nonlinear restoring force, denoted by dimensional k and ε, respectively. Numerical simulations were carried out for various reduced velocities Ur ∈ [9 to 15] and ε ∈ [0 to 20] at a relatively low Reynolds number. Our previous investigations of a purely linear oscillator (i.e., ε = 0) show that the equilibrium position of the rotary oscillation is not parallel to the free stream as the reduced velocity exceeds a critical value, that is, bifurcation occurs. The present numerical studies suggest that, for a specific reduced velocity Ur, the increase in the nonlinear stiffness ε can eliminate the undesirable bifurcation. The numerical results also suggest that both odd and even-number lift frequency components appear for bifurcate cases, while only odd-number lift frequencies are observed for non-bifurcate cases. The dynamic mode decompositions for the wake flow corresponding to each lift frequency are presented.