Stochastic bifurcation of one flexible beam subject to axial Gauss white noise excitation

A stochastic nonlinear dynamical model is proposed to describe the vibration of rectangular thin plate under axial inplane excitation considering the influence of random environment factors. Firstly, the model is simplified by applying the stochastic averaging method of quasi-nonintegrable Hamilton system. Secondly, the methods of Lyapunov exponent and boundary classification associated with diffusion process are utilized to analyze the stochastic stability of the trivial solution of the system. Thirdly, the stochastic Hopf bifurcation of the vibration model is explored according to the qualitative changes in stationary probability density of system response, showing that the stochastic Hopf bifurcation occurs at two critical parametric values. Finally, some explanations are given in a simple way on the potential applications of stochastic stability and bifurcation analysis.

Stochastic stability and bifurcation in a macroeconomic model

Nonlinear dynamic characteristics and optimal control of magnetic shape memory alloy (MSMA) cantilever beam subjected to axial stochastic excitation is studied in this paper. Nonlinear difference item is introduced to interpret the hysteretic phenomena of MSMA, and the nonlinear dynamic model of MSMA cantilever beam subjected to axial stochastic excitation is developed. The steady-state probability density function of the dynamic response of the system is obtained, and the conditions of stochastic Hopf bifurcation are analyzed. The reliability function of the system is solved, and then the probability density of the first-passage time is obtained. Finally, the optimal control strategy is proposed in stochastic dynamic programming method. Numerical simulations show that the stability of the trivial solution varies with bifurcation parameter, and stochastic Hopf bifurcation appears in the process; the reliability of the system is proved by stochastic optimal control, and the first-passage time is delayed. The results are helpful to engineering application of MSMA cantilever beam.

In this paper, nonlinear dynamic characteristics of giant magnetostructive nanofilm-shape memory alloy (SMA) composite beam in axial stochastic excitation were studied. Von del Pol nonlinear difference item was introduced to interpret the hysteresis phenomenon of the strain-stress curve of SMA, and the hysteretic nonlinear dynamic model of giant magnetostructive nanofilm-SMA composite beam in axial stochastic excitation was developed. The steady-state probability density function and the joint probability density function of the system were obtained in quasi-nonintegrable Hamiltonian system theory. The result of simulation shows that the stability of the trivial solution varies with bifurcation parameter, and stochastic Hopf bifurcation appears in the process. The result is helpful to stochastic bifurcation control to giant magnetostructive nanofilm-SMA composite beam.

Stochastic bifurcation characteristics and optimal control of giant magnetostrictive film (GMF)-shape memory alloy (SMA) composite plate subjected to in-plane stochastic excitation were studied in this paper. Nonlinear difference item was introduced to interpret the hysteretic phenomena of both GMF and SMA, and then the nonlinear dynamic model of GMF-SMA composite plate subjected to in-plane stochastic excitation was developed. The stochastic stability of the system was analyzed, and the condition of stochastic Hopf bifurcation was discussed. The reliability function of the system was solved from backward Kolmogorov equation, and then the probability density of the first-passage time was obtained. Finally, the stochastic optimal control strategy was obtained in stochastic dynamic programming method. Numerical simulation shows that the stability of the trivial solution varies with bifurcation parameters, and stochastic Hopf bifurcation appears in the process; the reliability of the system is improved by stochastic optimal control, and the first-passage time is delayed. GMF-SMA composite plate combines the advantages of both GMF and SMA, and can reduce vibration through passive control and active control effectively. The results of this paper are helpful to application of GMF-SMA composite plate in engineering fields.

A kind of shape memory alloy (SMA) hysteretic nonlinear model is developed, and the stochastic bifurcation characteristics of SMA intravascular stents subjected to radial and axial excitations are studied in this paper. A new nonlinear differential item is introduced to interpret the hysteretic phenomena of SMA strain-stress curves, and the dynamic model of SMA intravascular stent subjected to radial and axial stochastic excitations is established. The conditions of the system's stochastic stability are determined, and the probability density function of the system response is obtained. Finally, the stochastic Hopf bifurcation characteristics of the system are analyzed. Theoretical analysis and numerical simulation show that the system stability varies with bifurcation parameters, and stochastic Hopf bifurcation occurs in the process; there are two limit cycles in the stationary probability density of the system response in some cases, which means that there are two vibration amplitudes whose probability are both very high; jumping phenomena between the two vibration amplitudes appears with the change of conditions, which may cause stent fracture or loss. The results of this paper are helpful for application of SMA intravascular stent in biomedical engineering fields.

Stochastic bifurcation and fractal and chaos control of a giant magnetostrictive film–shape memory alloy (GMF–SMA) composite cantilever plate subjected to in-plane harmonic and stochastic excitation were studied. Van der Pol items were improved to interpret the hysteretic phenomena of both GMF and SMA, and the nonlinear dynamic model of a GMF–SMA composite cantilever plate subjected to in-plane harmonic and stochastic excitation was developed. The probability density function of the dynamic response of the system was obtained, and the conditions of stochastic Hopf bifurcation were analyzed. The conditions of noise-induced chaotic response were obtained in the stochastic Melnikov integral method, and the fractal boundary of the safe basin of the system was provided. Finally, the chaos control strategy was proposed in the stochastic dynamic programming method. Numerical simulation shows that stochastic Hopf bifurcation and chaos appear in the parameter variation process. The boundary of the safe basin of the system has fractal characteristics, and its area decreases when the noise intensifies. The system reliability was improved through stochastic optimal control, and the safe basin area of the system increased.

Hysteretic nonlinear characteristics and stochastic bifurcation of cantilevered piezoelectric energy harvester was studied in this paper. Piezoelectric ceramics was adhesively bonded on the substrate of cantilever beam to make piezoelectric cantilever beam. Von de Pol difference item was introduced to interpret the hysteretic phenomena of piezoelectric ceramics, and then the nonlinear dynamic model of piezoelectric cantilever beam subjected to axial stochastic excitation was developed. The stochastic stability of the system was analyzed, and the steady-state probability density function and the joint probability density function of the dynamic response of the system were obtained. Finally, the conditions of stochastic Hopf bifurcation were determined. Numerical simulation shows that stochastic Hopf bifurcation appears when bifurcation parameter varies, which can increase vibration amplitude of cantilever beam system and improve the efficiency of piezoelectric energy harvester. The results of this paper are helpful to application of cantilevered piezoelectric energy harvester in engineering fields.

In this paper, we describe a general variational Bayesian approach for approximate inference on nonlinear stochastic dynamic models. This scheme extends established approximate inference on hidden-states to cover: (i) nonlinear evolution and observation functions, (ii) unknown parameters and (precision) hyperparameters and (iii) model comparison and prediction under uncertainty. Model identification or inversion entails the estimation of the marginal likelihood or evidence of a model. This difficult integration problem can be finessed by optimising a free-energy bound on the evidence using results from variational calculus. This yields a deterministic update scheme that optimises an approximation to the posterior density on the unknown model variables. We derive such a variational Bayesian scheme in the context of nonlinear stochastic dynamic hierarchical models, for both model identification and time-series prediction. The computational complexity of the scheme is comparable to that of an extended Kalman filter, which is critical when inverting high dimensional models or long time-series. Using Monte-Carlo simulations, we assess the estimation efficiency of this variational Bayesian approach using three stochastic variants of chaotic dynamic systems. We also demonstrate the model comparison capabilities of the method, its self-consistency and its predictive power.

Stochastic stability and bifurcation characteristics of multiwalled carbon nanotubes-absorbing hydrogen atoms subjected to thermal perturbation

A Limit Theorem for the Solutions of Differential Equations with Random Right-Hand Sides

Considering the dynamic behavior of doubly‐fed induction generators (DFIGs) under the influence of random factors, this paper not only analyzes the phenomenon of stochastic instability and bifurcation of a DFIG dynamic variable in its random space when they are affected by environmental noise, but also proposes a method based on the Tkagi–Sugneo (T–S) fuzzy control strategy to control its stochastic bifurcation. First, a four‐dimensional stochastic dynamic DFIG model is established by using multiplicative white noise to simulate the influence of environmental noise on electrical variables, and stochastic central manifold theory is used to reduce the dimensionality of a planar model in the bifurcation neighborhood. Then, the stochastic stability of the model is investigated based on singular boundary theory, while the steady‐state probability density of the stochastic DFIG is determined using the Fokker–Plank–Kolmogorov equation to obtain the location and probability density of its stochastic P‐bifurcation. Finally, the influence of stochastic bifurcation behavior is eliminated by H∞‐fuzzy output feedback control. The numerical simulation results indicate that the location and probability of stochastic bifurcation in a DFIG will vary with the change in noise intensity, and the bifurcation parameter values and stochastic stability domain are obtained. The harm caused by random factors can be solved based on H∞‐fuzzy output feedback, which provides a theoretical basis for the stable operation of the DFIG system.

In this paper asymptotic and numerical methods are used to study the phenomenon of stochastic Hopf bifurcation. The analysis is carried out through studying a noisy Duffing-van der Pol oscillator which exhibits a Hopf bifurcation in the absence of noise as one of the parameters is varied. In the first part of this paper, we present briefly various concepts that are essential to describe stochastic bifurcations. We also present the definitions of P-Bifurcation and D-Bifurcation and illustrate these concepts through a one-dimensional example. In the second part of this paper, we construct an asymptotic expansion for the maximal Lyapunov exponent, the exponential growth rate of solutions to a linear stochastic system, and the moment Lyapunov exponents for a two-dimensional dynamical system driven by a small intensity real noise process. The nonlinear analysis is performed using both the method of stochastic averaging and stochastic normal forms. In this study, stochastic bifurcation implies either qualitative changes to the invariant measures which can be observed by examining the Fokker-Planck equation, or the appearance of a new invariant measure which is, at present, generated numerically through the forward and backward solutions of the stochastic differential equations. For a detailed study of this topic the reader is referred to Arnold et al. [11].

Price system was analyzed by a nonlinear stochastic dynamic model with the supply and demand factors. This model converged to a process in probability. Then the stability of the system was analyzed by the result of drift coefficient and diffusion coefficient of the process. Numerical simulation results of the stationary probability density with different conditions were shown. It can be concluded that the stability of the system varied as the parameters varied. Finally, the influence of parameters on the stochastic bifurcation was analyzed.

Nonlinear dynamics and bifurcation characteristics of shape memory alloy thin films subjected to in-plane stochastic excitation