We study the spectra of fluctuations in linear and nonlinear vibrational systems. Fluctuations play a major role in mesoscopic systems explored in nanomechanics, cavity and circuit quantum electrodynamics, and Josephson junction based systems to mention but a few. We find that important insights into the nature of the fluctuations can be gained by investigating the system dynamics in the presence of periodic driving. This is because the interplay of the driving and fluctuations leads to specific pronounced spectral features. Our predictions are corrobarated by measurements on a carbon nanotube resonator which show that the theory allows one both to reveal and to characterize frequency fluctuations in a vibrational system, as well as to determine the decay rate without ring-down measurements. Our results bear on the general area of decoherence of mesoscopic oscillators and also on the classical problems of resonance fluorescence and light scattering by oscillators. An important and poorly understood mechanism of fluctuations in mesoscopic systems is the dispersive mode coupling. This coupling is inherent essentially to all mesoscopic systems. It comes from the nonlinear interaction between vibrational modes with non-resonating frequencies. We consider the power spectrum of one of these modes. Thermal fluctuations of the modes nonlinearly coupled to it lead to fluctuations of the mode frequency and thus to the broadening of its spectrum. However, the coupling-induced broadening is partly masked by the spectral broadening due to the mode decay. We show that the effect of the mode coupling can be identified and characterized using the change of the spectrum by resonant driving. The theoretical analysis is complicated by the fact that the dispersive-coupling induced fluctuations are non-Gaussian. We develop a path-integral method of averaging over the fluctuations and obtain the power spectrum in an explicit form. The shape of the spectrum depends on the interrelation between the coupling strength and the decay rates of the modes involved, providing a means of characterizing these modes even where they cannot be directly accessed. The analysis is extended to the case of coupling to many modes which, because of the cumulative effect, can become effectively strong. We also find the power spectrum of a driven mode where the mode has internal nonlinearity. Unexpectedly, for a driven mode, the power spectra dominated by the intra- and inter-mode nonlinearities are qualitatively different. The analytical results are in excellent agreement with the numerical simulations.Of significant interest for physics and biophysics are overdamped mesoscopic and microscopic systems. Inertial effects play no role in their dynamics. We show that where such systems are periodically driven, along with the conventional delta-peak at the driving frequency their power spectra display extra features. These can be peaks or dips with height quadratic in the driving amplitude, for weak driving. The peaks/dips are generally located at zero frequency and at the driving frequency. The shape and intensity of the spectra sensitively depend on the parameters of the system dynamics. To illustrate this sensitivity and the generality of the effect, we study three types of systems: an overdamped Brownian particle (e.g., an optically trapped particle), a two-state system that switches between the states at random, and a noisy threshold detector. The analytical results are in excellent agreement with numerical simulations.
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