  We consider the existence of invariant curves of real analytic reversible
mappings which are quasi-periodic in the angle variables. By the normal form
theorem, we prove that under some assumptions, the original mapping is changed
into its linear part via an analytic convergent transformation, so that
invariants curves are obtained. In the iterative process, by solving the
modified homological equations, we ensure that the transformed mapping is still
reversible. As an application, we investigate the invariant curves of a class
of nonlinear resonant oscillators, with the Birkhoff constants of the
corresponding Poincar$\acute{e}$ mapping all zeros or not.

Dynamical Systems