The existence of peculiar peaks in the fluctuation spectrum is predicted for a broad class of nonlinear systems, namely for such systems where the dependence of the natural oscillation frequency on energy has an extremum. An asymptotically accurate (in the low-friction limit) shape of such spectral peaks is found, based on the solution of the Fokker-Planck equation. Examples are given of physical systems where such spectral peaks can be observed. A major attention is given to the model of a periodic potential having several different-height barriers over a period. Such a model can describe a hindered rotation of either an impurity with several positions of equilibrium in a crystal cell or a chain of a polymetric molecule relative to its axis. It is also shown that even at a statistical straggling of the potential barrier heights (e.g., due to defects in the crystal) the fluctuation spectrum has a peak stemming from the existence of an extremum in the energy dependence of the natural oscillation frequency. The criteria of applicability of the obtained results are discussed in detail.