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Exact solutions of cyclically symmetric oscillator equations with non-linear coupling. Part II: Coupling with phase shift

Several authors have found very simple exact limit cycle solutions in cyclically symmetric systems of N oscillator equations with linear coupling in zero order of a perturbation parameter and non-linear coupling in first order. In contrast with such solutions in most other non-linear systems, each of these limit cycles is an exact normal mode of the unperturbed equations with no change in frequency or addition of higher harmonics. 
 
In Part I of this paper it was shown that the construction and analysis of such systems of equations are substantially simplified if the equations are expressed in terms of the normal mode co-ordinates of the unperturbed system. the effects of the cyclic symmetry, as well as those of a higher symmetry shared by previous authors' models, were studied. 
 
It is shown here that similar results can be obtained in systems if the coupling involves a phase shift. the phase shift places added conditions on the systems, so that some sets of equations, shown to have the simple limit cycle solutions, no longer have them after shift is introduced. the methods of the earlier paper, however, can be used to find families of systems with phase shifts which have such solutions. 
 
A result in Part I, that frequencies in a system with the higher symmetry mentioned above are unchanged from those of the unperturbed system, is not valid if phase shifts are introduced.