The Theory Of Aperiodic Stochastic Resonance In Anomalous Processes

Stochastic resonance (SR) and anomalous processes are ubiquitous in different areas. SR can be regarded as the result of the cooperation between the nonlinear system, the noise and the external input signal. Anomalous processes, which are non-Markovian in time or nonlocal in space, attract more and more attention recently. In this thesis, the phenomenon of aperiodic stochastic resonance (ASR) in bistable anomalous system is investigated in the view of binary signal processing. The anomalous temporal and spatial properties are discussed respectively. The parameter-induced stochastic resonance (PSR) in the normal bistable system can be seen as one special case of PSR in anomalous systems. These works are important and significative for the development of PSR theory.In the study of subdiffusive bistable system, we start from the solution of the time-fractional Fokker-Planck equation (TFFPE), which describes the probability of the subdiffusive system output. Compared to the exponential relaxation mode of the normal system, the subdiffusive system relaxes in Mittag-Leffler mode. It is shown that, with the lower subdiffusive index, the system relaxes more slowly. When the system is modulated by constant signal, we investigate the residual probability of the system output. It is shown that, under the condition that the system response speed keeps constant, the lower subdiffusive index leads to higher minimum residual probability. We also investigate the binary signal processing problem in subdiffusive bistable system. Because of the non-Markovian property of the subdiffusive system, it is hard to detect the input signal from the current system output. The performance of the system is heavily reduced by the lower subdiffusive index. There are also large deviations between the theoretical bit error rate (BER) and the simulated results.In the study of superdiffusive system (i.e., system excited by Lévy noise), we solve the corresponding space-fractional Fokker-Planck equation (SFFPE) by Grunwald-Letnikov fractional difference method. To measure the system performance, BER is defined based on the solution of SFFPE.We discuss noised-induced ASR and PSR. It is shown that, noise-induced ASR, which includes subthreshold ASR and residual ASR, still exists in superdiffusive bistable system, but is reduced by the lower Lévy index. Lower Lévy index leads to higher minimum BER. When the noise intensity is fixed, we find that BER varies with the system parameter non-monotonically, which means PSR also exists in superdiffusive bistable system. By choosing the optimum system parameter, the lower Lévy index leads to lower minimum BER. Therefore the performance of optimal bistable system in Lévy noise is better than its performance in Gaussian noise with the same intensity. In the potential order higher than quartic, it is shown that the order effect on the minimum BER is very limited. Finally, we investigate the system performance in asymmetric Lévy noise. Because of the asymmetric distribution of system output, the optimal detection threshold is chosen based on the minimum BER. It is shown that, by choosing the optimal detection threshold, the effects of noise asymmetry on minimum BER are not great.