Ergodic behaviour of nonlinear hydromagnetic waves in a cold collisionless plasma

Nonlinear hydromagnetic waves propagating along a magnetic field in a cold collisionless plasma are investigated. A criterion for ergodicity of the waves is obtained using the usual method of analytical dynamics. According to this criterion, it is found that, if a parameter b 00 is a rational number, the wave is periodic, and that, if b 00 is an irrational number, the wave is ergodic. Therefore, the waves are almost always ergodic, i.e. the trajectory of the wave fills up a region in the phase plane of the magnetic field. On the other hand, periodic solutions can exist only with measure zero.