An oscillator with two degrees of freedom can, under certain conditions, oscillate simultaneously at two different frequencies. The ratio of the two frequencies may be rational (synchronous oscillations) or irrational (asynchronous oscillations). The conditions necessary for 'simultaneous oscillations are discussed for these two cases. The problem is essentially non-linear and cannot be solved by the methods of linear analysis. The method of equivalent linearization, developed by Kryloff and Bogoliuboff, is shown to give a good correlation with experimental results.