Comments on multiple oscillatory solutions in systems with time-delay feedback

A complex Ginzburg-Landau equation subjected to local and global time-delay feedback terms is considered. In particular, multiple oscillatory solutions and their properties are studied. We present novel results regarding the disappearance of limit cycle solutions, derive analytical criteria for frequency degeneration, amplitude degeneration, frequency extrema. Furthermore, we discuss the influence of the phase shift parameter and show analytically that the stabilization of the steady state and the decay of all oscillations (amplitude death) cannot happen for global feedback only. Finally, we explain the onset of traveling wave patterns close to the regime of amplitude death.