Ergodic behaviour of nonlinear hydromagnetic waves in a cold collisionless plasma

Abstract

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 parameterb00is a rational number, the wave is periodic, and that, ifb00is 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.