Self-oscillations in ring Toda chains with negative friction

We study here the different modes of self-oscillations in ring Toda chains with Rayleigh-type negative friction. Assuming that at small friction the shape of self-oscillations is close to one of the known Toda solitonlike solutions we use analytical methods in combination with numerical ones for study of the self-oscillations. We calculate explicitly for a Toda chain consisting of N elements the N+1 different modes of self-oscillations. Among them two modes correspond to left and right rotations of the chain as a whole with a constant velocity Each of the other N-1 modes represents a combination of solitonlike oscillations and a rotation with a velocity depending on the mode number. Only for the mode corresponding to antiphase oscillations of the chain neighboring elements (such oscillations are possible for an even N) the constant component of the velocity is equal to zero.