The propagation of intense electromagnetic waves in cold magnetized plasma is tackled through a relativistic hydrodynamic approach. The analysis of coupled transverse-longitudinal plasma oscillations is performed for traveling plane waves. When these waves propagate perpendicularly to a static magnetic field, the model is describable in terms of a nonlinear dynamical system with 2 degrees of freedom. A constant of motion is obtained and the powerful classical mechanics methods can be used. A new class of solutions, i.e., the chaotic solutions, is discovered by the Poincaré surface of sections. As a result, coupled transverse-longitudinal plasma oscillations become aperiodically modulated.