We consider bistable stochastic systems fed with a small periodic signal - two-state Markov chains with discrete and continuous time and diffusions in double-well potentials. In the Markov chain case, one-step or infinitesimal probabilities are periodically modulated. In the diffusion case, depths of the potential wells are periodically changed in time. We introduce several measures of goodness for tuning, the most important one of which is the coefficient of the spectral power amplification (SPA), which describes the spectral energy carried by the averaged random output corresponding to the frequency of a small deterministic periodic perturbation.
For the Markov chains, this coefficient is studied as a function of noise, and the coordinate of its local maximum, the resonance point, is determined asymptotically in the small noise limit. Six more measures of goodness are investigated, including the SPA-to-noise ratio, the energy, the out-of-phase measure, the relative entropy, and the entropy of the invariant measure. Optimal tuning rates are obtained for all these measures.
To investigate optimal tuning for diffusions we study adapted Markov chains whose dynamical properties retain the essentials of the diffusions' hopping properties between the two metastable states. Surprisingly, due to the influence of many small oscillations near the potential valley bottoms, the tuning properties of a diffusion's SPA coefficient do not match those of the adapted Markov chain's coefficient. However, if the fluctuations of the diffusion near the wells' minima are cut off, the modified SPA coefficient has a local maximum with approximately the same coordinates as the adapted Markov chain.
Our methods are based on the study of the equilibrium measures of the considered random processes. To determine the invariant densities of diffusions, we use spectral analysis of the respective infinitesimal generators which describes them in Fourier type series expansions. The analysis of optimal tuning and the comparison to the adapted Markov chains become possible by a precise study of the lowest order eigenvalues and eigenfunctions and the discovery of a spectral gap between the first and second eigenvalue.