Reshaping of the probability density function of nonlinear stochastic systems against abrupt changes

The operation of dynamic systems is often accompanied by abrupt and random changes in their configurations, which will dramatically change the stationary probability density function of their response. In this article, an effective procedure is proposed to reshape the stationary probability density function of nonlinear stochastic systems against abrupt changes. Based on the Markov jump theory, such a system is formulated as a continuous system with discrete Markov jump parameters. The limiting averaging principle is then applied to suppress the rapidly varying Markov jump process to generate a probability-weighted system. Then, the approximate expression of the stationary probability density function of the system is obtained, based on which the reshaping control law can be designed, which has two parts: (i) the first part (conservative part) is designed to make the reshaped system and the undisturbed system have the same Hamiltonian; (ii) the second (dissipative part) is designed so that the stationary probability density function of the reshaped system is the same as that of undisturbed system. The proposed law is exactly analytical and no online measurement is required. The application and effectiveness of the proposed procedure are demonstrated by using an example of three degrees-of-freedom nonlinear stochastic system subjected to abrupt changes.

Noise-induced transition in an underdamped asymmetric bistable system driven by Lévy noise

In this paper, the Lévy noise-induced transition in an underdamped asymmetric bistable system is discussed. Lévy noise is generated through the Janicki–Weron algorithm and the numerical solutions of system equation is obtained by the fourth-order Runge–Kutta method. Then the stationary probability density functions are obtained by solving the equation of system. The influence of the damped coefficient [Formula: see text], asymmetric parameter r of system, stability index [Formula: see text], skewness parameters [Formula: see text] and noise intensity D on the stationary probability density are analyzed. The numerical simulation results show that the asymmetric parameter r, stability index [Formula: see text], skewness parameters [Formula: see text] and noise intensity D can induce the phase transition. However, the phase transition cannot be induced by the damped coefficient [Formula: see text].

Bifurcations in tri-stable Duffing–Van der Pol oscillator with recycling noise

Recycling noise is a kind of more common noise. The nonlinear dynamic system can be controlled by adjusting its parameters. However, so far, the effect of recycling noise on tri-stable dynamic system has not been reported. In this paper, stochastic P-bifurcations in tri-stable Duffing–Van der Pol oscillator induced by additive recycling noise are investigated. Firstly, the stationary probability density function is derived using stochastic averaging method. Then, the general expression of the critical parameter conditions of stochastic P-bifurcation is given by the singularity theory. The stationary probability density of response amplitude in different parameter areas are also shown, which is verified by Monte Carlo numerical simulation. Based on these results, the influence of related parameters of recycling noise and damping coefficient on the stochastic P-bifurcation is studied. The result shows that the critical parameter of bifurcation can be changed by adjusting the delay time and fraction coefficient of the recycling noise. It has also been found that the stationary probability density and stochastic bifurcation show a periodic dependence on the delay time.

Response of Duffing Oscillator with Time Delay Subjected to Combined Harmonic and Random Excitations

This paper aims to investigate the stationary probability density functions of the Duffing oscillator with time delay subjected to combined harmonic and white noise excitation by the method of stochastic averaging and equivalent linearization. By the transformation based on the fundamental matrix of the degenerate Duffing system, the paper shows that the displacement and the velocity with time delay in the Duffing oscillator can be computed approximately in non-time delay terms. Hence, the stochastic system with time delay is transformed into the corresponding stochastic non-time delay equation in Ito sense. The approximate stationary probability density function of the original system can be found by combining the stochastic averaging method, the equivalent linearization method, and the technique of auxiliary function. The response of Duffing oscillator is investigated. The analytical results are verified by numerical simulation results.

Stochastic Resonance in a Bacterium Growth System with Time Delay and Colored Noise

The phenomenon of stochastic resonance in a bacterium growth system that is with two different kinds of time delays and is driven by colored noises is investigated. Based on the extended unified colored noise theory and the method of the probability density approximation, the Fokker–Planck equation and the stationary probability density function are derived. Then via the theory of adiabatic limit, the analytical expression of the signal-to-noise ratio (SNR) is obtained. The different effects of the time delays existed in the nonlinear system and the noise correlation times on the stationary probability density and the signal-to-noise rate are discussed respectively. Finally, numerical simulations are offered and are consistent with approximate analytical results.

Dynamics of a Nonlinear Business Cycle Model Under Poisson White Noise Excitation

Several observations in real economic systems have shown the evidence of non-Gaussianity behavior, and one of mathematical models to describe these behaviors is Poisson noise. In this paper, stationary probability density of a nonlinear business cycle model under Poisson white noise excitation has been studied analytically. By using the stochastic averaged method, the approximate stationary probability density of the averaged generalized FPK equations are obtained analytically. The results show that the economic system occurs jump and bifurcation when there is a Poisson impulse existing in the periodic economic system. Furthermore, the numerical solutions are presented to show the effectiveness of the obtained analytical solutions.