Study On Stochastic Resonance In Nonlinear Systems Driven By Stable Distribution Noise

In nonlinear systems, the signal-to-noise ratio (SNR) may not be reduced and sometimes even be enhanced by the assistance of noise when the synchronization of nonlinear system with ordered weak stimulus and disordered random perturbation occurs. This kind of abnormal phenomenon with the name ’stochastic resonance’ was firstly been put forward by R. Benzi. Stochastic resonance has been a new focus of academic research in the subject of random vibration, and the regarding theories and experiments have been developed for decades. Up to now, most of studies on stochastic resonance have focused on the ideal Gaussian noise, and few studies have attempted to analyse the situation when the background noise has stable distribution. In fact, stable distribution noise which is a very flexible modeling tool has wider applicability than Gaussian noise. Therefore, on the basis of predecessor’s works, this dissertation will focus on the studies involving stochastic resonance phenomena of various nonlinear systems in stable distribution noise environments.First, we introduce the definition and basic properties of stable distribution, and describe the methods of generating the random numbers with stable distribution. Based on the most typical stochastic resonance model, the movement of an over-damped particle in a double-well potential, we derived the theoretical dynamical probability density of system response by solving the corresponding space-fractional Fokker-Planck equation (FFPE) via Gruwald-Letnikov fractional difference method. The output SNR is defined to measure the system performance when the system is modulated by a constant signal. When the system is modulated by a periodic square wave signal, the system response will enter the cyclostationary state after a period of time. The system detection probability is used to measure the performance. The theoretical calculation shows that the traditional stochastic resonance phenomena have emerged in above systems.If the delays in bistable systems are small in comparison with other timescales of the system, one can approximate the delay-differential systems by a system of ordinary stochastic differential equations. According to the approximate drift and diffusion terms of the resulting approximating non-delayed stochastic differential equations, the equivalent potential function can also be obtained. Based on the adiabatic approximation theory, the effect of delay on the system output SNR is considered. When the input signal is aperiodic binary pulse amplitude modulation signal and the background noise has Gaussian distribution, we use the eigenfunction expansion method to solve the Fokker-Planck function corresponding to the approximate delay-differential equation. When the input signal is aperiodic binary pulse amplitude modulation signal and the background noise has Gaussian distribution, the eigenfunction expansion method is used to solve the Fokker-Planck function corresponding to the approximate delay-differential equation. Both the dynamical probability density and the system response speed are obtained, and we find that the increase of delayed time will reduce the system response speed. As the noise becomes stable distribution noise, the response’s dynamical probability density of the delayed system can be calculated via the difference method. The probability of system detection error is adopted to measure the performance, and the existence of stochastic resonance phenomena is demonstrated by theoretical calculation and simulation experiments.Considering that the potential functions have a direct influence on the system performance, we evaluate the difference of nonlinear systems with diverse potential functions in signal processing. Due to external interference or system distortion, it is inevitable to meet bistable systems with asymmetric potentials. The results indicate that the effects of asymmetry on the system performance can be reduced by tuning the system parameters. Studies involving the application of parameter-tuning stochastic resonance in signal processing show that the bistable system with optimal parameters is often close to monostable systems. So, the performance difference between bistable systems and monostable systems is also investigated. Besides, as the distribution of the stable noise becomes asymmetric, the detection threshold has great influence on the system performance.The traditional stochastic resonance phenomenon is widespread in sub-threshold systems, and the optimal noise intensity can be theoretical calculated when the distribution of noise can be written analytically. As the signal is interfered by stable distribution noise, an absolutely converged algorithm can be developed to search the optimal noise intensity. The Monte-Carlo simulation experiments are carried out to verify the accuracy of the results. The simulation results also indicate that the system performance can be improved by increasing the units of the threshold systems.