The stochastic stability of the harmonically and randomly excited Duffing oscillator with damping modeled by a fractional derivative of Caputo's definition is analyzed.First,the system state is approximately described by It equations through the stochastic averaging method based on the generalized harmonic function.Then,the associated expression for the largest Lyapunov exponent of the linearized averaged It is derived,and the necessary and sufficient condition for the asymptotic stability with probability one of the trivial solution of the original system is obtained approximately by letting the largest Lyapunov exponent be negative.The effects of fractional orders and random excitation intensities on the asymptotic stability with probability one determined by the largest Lyapunov exponent are shown graphically.
Over exposure is rather annoying in photo taking. However, in some severe light conditions over exposure is inevitable using conventional cameras due to the limitation of dynamic range of the image sensor. The over exposed information would be completely lost and unrecoverable. In order to cope with this problem, we propose a novel technique in which the noise is used to enlarge the dynamic range of the image sensor. The essential mechanism that noise contributes to the information recovery is investigated. It is also proved that the visibility of regained information can reach the peak when specifically added noise is synchronized with the image sensor, thus activating the phenomenon of stochastic resonance （SR）. Four different types of noises are investigated to show the effects of variant distributions on the quality of recovered information. The experimental outcomes are consistent with our theoretical results, which indicates that the SR-based lost information recovery is quite promising.
Stochastic bifurcations of the SD （smooth and discontinuous） oscillator with additive and/or multiplicative bounded noises are studied by the generalized cell mapping method using digraph analysis algorithm. From the global viewpoint, stochastic bifur- cation can be described as a sudden change in shape and size of a random attractor as the system parameter valies. The evolu- tionary process of stochastic bifurcation in the SD oscillator is shown in detail. Meanwhile, we show the phenomenon that un- der stochastic excitation the shape and size of random attractor and random saddle change along with the direction of unstable manifold. A plane stochastic bifurcation diagram is included.
Based on the Grammian and Pfaffian derivative formulae, Grammian and Pfaffian solutions are obtained for a （3＋1）-dimensional generalized shallow water equation in the Hirota bilinear form. Moreover, a Pfaffian extension is made for the equation by means of the Pfaffianization procedure, the Wronski-type and Gramm-type Pfaffian solutions of the resulting coupled system are presented.
