Spectral Power Amplification Research Articles

Stochastic multiresonance due to interplay between noise and fractals.

Stochastic multiresonance is shown to occur in a general class of threshold-crossing systems, in which a derivative of the threshold-crossing probability with respect to a system parameter is a nonmonotonic function of the noise intensity. As an example, a two-dimensional chaotic map is considered, where the threshold-crossing probability follows the overlap of the fractal structures of chaotic saddles and the basins of escape in noise-induced crisis. The analytic theory is in reasonable agreement with the numerical results for spectral power amplification.

Log-periodic oscillations and noise-free stochastic multiresonance due to self-similarity of fractals

The origin of log-periodic oscillations around the power-law trend of the escape probability from a precritical attractor and of the noise-free stochastic multiresonance, found in numerical simulations in chaotic systems close to crises is discussed. It is shown that multiple maxima of the spectral power amplification vs. the control parameter result from a fractal structure of a precritical attractor colliding with a possibly fractal basin of attraction at the crisis point. Qualitative explanation of the multiresonance, based on a concept of fractal self-similarity, or discrete-scale invariance, is given and compared with numerical results and analytic theory using a simple geometric models of the colliding fractal sets.

MODEL REDUCTION AND STOCHASTIC RESONANCE

We provide a mathematical underpinning of the physically widely known phenomenon of stochastic resonance, i.e. the optimal noise-induced increase of a dynamical system's sensitivity and ability to amplify small periodic signals. The effect was first discovered in energy-balance models designed for a qualitative understanding of global glacial cycles. More recently, stochastic resonance has been rediscovered in more subtle and realistic simulations interpreting paleoclimatic data: the Dansgaard–Oeschger and Heinrich events. The underlying mathematical model is a diffusion in a periodically changing potential landscape with large forcing period. We study optimal tuning of the diffusion trajectories with the deterministic input forcing by means of the spectral power amplification measure. Our results contain a surprise: due to small fluctuations in the potential valley bottoms the diffusion — contrary to physical folklore — does not show tuning patterns corresponding to continuous time Markov chains which describe the reduced motion on the metastable states. This discrepancy can only be avoided for more robust notions of tuning, e.g. spectral amplification after elimination of the small fluctuations.

Noisefree stochastic multiresonance near chaotic crises.

We report on the phenomenon of noisefree stochastic multiresonance that appears in a natural way in systems where the threshold crossing probability has a nonmonotonous derivative with respect to the control parameter. In particular, we consider periodically driven chaotic dynamical systems above crisis threshold where the nonmonotonicity is caused by the fractal structure of precritical attractors and, possibly, their basins of attraction. The spectral power amplification as a function of the control parameter can be easily obtained from the postcritical average transient times, and the heights of its multiple maxima can be estimated on the basis of simple geometric models.

Stochastic resonance enhanced by dichotomic noise in a bistable system

We study linear responses of a stochastic bistable system driven by dichotomic noise to a weak periodic signal. We show that the effect of stochastic resonance can be greatly enhanced in comparison with the conventional case when dichotomic forcing is absent, that is, both the signal-to-noise ratio and the spectral power amplification reach much greater values than in the standard stochastic resonance setup.

Coherence and stochastic resonance in a two-state system

The subject of our study is a two-state dynamics driven by Gaussian white noise and a weak harmonic signal. The system resulting from a piecewise linear FizHugh-Nagumo model in the case of perfect time scale separation between fast and slow variables shows either bistable, excitable, or oscillatory behavior. Its output spectra as well as the spectral power amplification of the signal can be calculated for arbitrary noise strength and frequency, allowing characterization of the coherence resonance in the bistable and excitable regimes as well as quantification of nonadiabatic resonances with respect to the external signal in all regimes.

Array-enhanced stochastic resonance in finite systems

A finite system of three coupled stochastic resonators is investigated with respect to the coupling constant. The consideration allows the analytical calculation of the spectral power amplification and of the signal-to-noise ratio. Whereas the local signal-to-noise ratio (SNR) of a single spin exhibits a maximum with respect to the coupling, the SNR of the global summed output of all spins decreases monotonically.

Quantum stochastic resonance in symmetric systems.

We investigate the low-temperature quantum stochastic resonance (QSR) phenomenon in a two-level system (TLS) which is coupled to an Ohmic heat bath. In contrast to common belief we find that QSR occurs also for symmetric (i.e., unbiased) TLS's if the viscous friction parameter alpha exceeds a critical value: We demonstrate that with respect to the spectral power amplification measure QSR always occurs for alpha>1; in contrast, the output signal-to-noise ratio exhibits an amplification only for alpha>3/2.

Analytical study of coupled two-state stochastic resonators

The two-state model of stochastic resonance is extended to a chain of coupled two-state elements governed by the dynamics of Glauber's stochastic Ising model. Appropriate assumptions on the model parameters turn the chain into a prototype system of coupled stochastic resonators. In a weak-signal limit analytical expressions are derived for the spectral power amplification and the signal-to-noise ratio of a two-state element embedded into the chain. The effect of the coupling between the elements on both quantities is analysed and array-enhanced stochastic resonance is established for pure as well as noisy periodic signals. The coupling-induced improvement of the SNR compared to an uncoupled element is shown to be limited by a factor four which is only reached for vanishing input noise.