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 $N+1\mathrm{}$ 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.