  Different models of self-excited oscillators which are four-dimensional
extensions of the van der Pol system are reported. Their symmetries are
analyzed. Three of them were introduced to model the release of vortices behind
circular cylinders with a possible transition from a symmetric to an
antisymmetric Benard-von Karman vortex street. The fourth reported self-excited
oscillator is a new model which implements the breaking of the inversion
symmetry. It presents the phenomenon of second harmonic generation in a natural
way. The parallelism with second harmonic generation in nonlinear optics is
discussed. There is also a small region in the parameter space where the
dynamics of this system is quasiperiodic or chaotic.
