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One of the classical ways to describe the dynamics of nonlinear systems is to analyze theur Fourier spectra. For periodic and quasiperiodic processes the Fourier spectrum consists purely of discrete delta-functions. On the contrary, the spectrum of a chaotic motion is marked by the presence of the continuous component. In this work, we describe the peculiar, neither regular nor completely chaotic state with so called singular-continuous power spectrum. Our investigations concern various cases from most different fields, where one meets the singular continuous (fractal) spectra. The examples include both the physical processes which can be reduced to iterated discrete mappings or even symbolic sequences, and the processes whose description is based on the ordinary or partial differential equations.
We have recently reported the phenomenon of doubly stochastic resonance [Phys. Rev. Lett. 85, 227 (2000)], a synthesis of noise-induced transition and stochastic resonance. The essential feature of this phenomenon is that multiplicative noise induces a bimodality and additive noise causes stochastic resonance behavior in the induced structure. In the present paper we outline possible applications of this effect and design a simple lattice of electronic circuits for the experimental realization of doubly stochastic resonance.