Stochastic Stability And Vibrational Multiresonance In Several Different Systems

Stochastic stability is one of the most important issues in the field of stochastic dynamics. It characterizes the stability of a random system in the vicinity of the steady state. The measure indicators are maximal Lyapunov exponent and moment Lyapunov exponent, and they determine the almost assure asymptotically stability and pth moment asymptotically stability respectively. Due to the universal existence of noise in the natural world, the problem of stochastic stability has significant research value in natural science and engineering fields.In some nonlinear systems that excited simultaneously by high- and low-frequency signals, the response amplitude at the low-frequency is a nonlinear function of the amplitude of the high-frequency signal, and it presents"resonance"by adjusting the high-frequency signal. This phenomenon is defined as vibrational resonance (VR). Based on the mechanism of VR, the weak low-frequency signal can be amplified in the system. Biharmonic signals are widely applied in communication technology, laser physics, ion physics, acoustics, neuroscience, etc. Hence, the investigation of VR and its control have important and potential values in a wide range of fields. In the present paper, besides the stochastic stability of some systems that are parametrically excited by different kinds of noise, the vibrational multiresonance and its control in nonlinear systems are also investigated. The main works and results are as following:(1) The stochastic stability of a co-dimensional two-bifurcation system driven by a white noise is studied. Based on the theory of singular boundary, all possible singularities that exist in the one-dimensional phase diffusion process are detailed discussed. For all the cases, the different conditions of the matrix that included in the noise excitation term are deduced, and the analytical expressions of the invariant measure are obtained. In addition, the P-bifurcation for the invariant measure is researched completely. Finally, according to the results of the maximal Lyapunov exponent, the regions of the almost assure asymptotically stability are given for all the cases.(2) The stochastic stability of a co-dimensional two-bifurcation system subjected to parametric excitation by a real noise that is assumed to be an integrable function of an n-dimensional Ornstein-Uhlenbeck vector process is investigated. Via the spectral analysis for an n-dimensional linear filter system, the spectral density functions for the real noise are given. Then, the standard FPK equation of the invariant measure and the calculation expression of the maximal Lyapunov exponent are obtained. Through discussing all the possible singular boundaries that exist in the one-dimensional phase diffusion process, the corresponding conditions of the matrix that included in the noise term, the expressions of the invariant measure and the maximal Lyapunov exponents are given for all the cases. In addition, the P-bifurcation of the invariant measure is investigated. And finally, the regions for the almost assure asymptotically stability of the system are obtained.(3) The stochastic stability of a van der Pol-Duffing oscillator that under the parametric excitation by a non-Gaussian colored noise is studied. After the simplification of the noise, through scale transformation and linear stochastic transformation, the second-order solution of the moment Lyapunov is obtained, and the conditions for the pth moment asymptotically stability and almost assure asymptotically are given finally.(4) A novel and much more regular vibrational multiresonance in a multistable system that driven by biharmonic signals is reported. For the underdamped and overdamped cases, the mechanism of the sequential vibrational multiresonance is discovered by both theoretical and numerical methods. The effect of the damping coefficient on the multiresonance is a focus in the investigation. The potential application values of vibrational multiresonance are discussed.(5) Under the excitations of both high- and weak low-frequency signals, the effects of linear time delay feedback on the VR in different kinds of systems are investigated. The results show that the vibrational multiresonance appears through adjusting the delay parameter, and then the VR phenomenon can be effectively controlled. Via modulating the delay parameter, the VR phenomenon not only can occur or vanish but also can be enhanced and then improve the weak low-frequency signal further. With the variation of the delay parameter, the response amplitude of the system to the weak low-frequency signal presents vibrational multiresonance, and it is periodic in the delay with two different periods, i.e., the periods of the two-frequency exciting signals. The investigation also shows that the linear time delay feedback can induce VR even in the nonlinear system in which there is no classical VR existing.