The chaotic dynamics of a low-order Galerkin truncation of the two-dimensional magnetohydrodynamic system, which reproduces the dynamics of fluctuations described by nearly incompressible magnetohydrodynamic in the plane perpendicular to a background magnetic field, is investigated by increasing the external forcing terms. Although this is the case closest to two-dimensional hydrodynamics, which shares some aspects with the classical Feigenbaum scenario of transition to chaos, the presence of magnetic fluctuations yields a very complex interesting route to chaos, characterized by the splitting into multiharmonic structures of the field amplitudes, and a mixing of phase-locking and free phase precession acting intermittently. When the background magnetic field lies in the plane, the system supports the presence of Alfvén waves thus lowering the nonlinear interactions. Interestingly enough, the dynamics critically depends on the angle between the direction of the magnetic field and the reference system of the wave vectors. Above a certain critical angle, independently from the external forcing, a breakdown of the phase locking appears, accompanied with a suppression of the chaotic dynamics, replaced by a simple periodic motion.