Deterministic fluctuation-response relation

Efficient detection of a weak signal in a noisy environment is one of the most intensely studied topics in signal processing. In such research, the concept of a noise-induced giant response, such as occurs in stochastic resonance, has attracted a great deal of attention. In the present study, by analyzing a system subjected to a deterministic fluctuation, we show that the occurrence of a giant response is strongly related to instability in the system. We derive the response function of the system, and obtain its eigenfunctions which determine the response behavior of the system. We find that the maximum eigenvalue of the response function diverges when instability arises in the system, and the infinite eigenvalue gives rise to a giant response. Our approach is analogous to the well-established fluctuation-response theory, and can be easily extended to a system with purely stochastic fluctuations.