Fractal Spectroscopy by Noise-Free Stochastic Multiresonance at Higher Harmonics

Noise-free stochastic resonance is investigated in two chaotic maps with periodically modulated control parameter close to a boundary crisis: the Hénon map and the kicked spin model. Response of the maps to the periodic signal at the fundamental frequency and its higher harmonics is examined. The systems show noise-free stochastic multiresonance, i.e. multiple maxima of the signal-to-noise ratio at the fundamental frequency as a function of the control parameter. The maxima are directly related to the fractal structure of the attractors and basins of attraction colliding at the crisis point. The signal-to-noise ratios at higher harmonics show more maxima, as well as dips where the signal-to-noise ratio is zero. This opens a way to use noise-free stochastic resonance to probe the fractal structure of colliding sets by a method which can be called “fractal spectroscopy”. Using stochastic resonance at higher harmonics can reveal smaller details of the fractal structures, but the interpretation of results becomes more difficult. Quantitative theory based on a model of a colliding fractal attractor and a fractal basin of attraction is derived which agrees with numerical results for the signal-to-noise ratio at the fundamental frequency and at the first two harmonics, quantitatively for the Hénon map, and qualitatively for the kicked spin model. It is also argued that the maps under study belong to a more general class of thresholdcrossing stochastic resonators with a modulated control parameter, and qualitative discussion of conditions under which stochastic multiresonance appears in such systems is given.