<p>In this paper, we investigate the bifurcation and control of the Duffing mapping. Initially, based on the theory of the <i>polynomial complete discriminant system</i>, a topological classification of the system at fixed points is conducted, with all non-hyperbolic cases systematically enumerated. Subsequently, employing the center manifold theorem, we rigorously demonstrate the occurrence of a pitchfork bifurcation at the origin and comprehensively analyze flip bifurcations for all fixed points under non-hyperbolic conditions. Moreover, a polynomial controller is designed to explicitly induce a Neimark-Sacker bifurcation near the fixed points for the controlled system by systematically adjusting critical parameters in the controller. Furthermore, by using hybrid and exponential controls, we theoretically establish the asymptotic stability regions for fixed points. To verify the correctness of the theoretical results, we finally present numerical simulations, including bifurcation diagrams, Lyapunov exponent diagrams, and invariant circles.</p>