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

2018 ◽  
Vol 32 (28) ◽  
pp. 1850313 ◽  
Author(s):  
Yong-Feng Guo ◽  
Fang Wei ◽  
Lin-Jie Wang ◽  
Jian-Guo Tan
Keyword(s):  
Phase Transition ◽  
Probability Density ◽  
Noise Intensity ◽  
Stability Index ◽  
Bistable System ◽  
Index Formula ◽  
Lévy Noise ◽  
Stationary Probability ◽  
Stationary Probability Density ◽  
Levy 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].

Lévy Noise-Induced Effects in Underdamped Asymmetric Bistable System

2019 ◽  
Vol 19 (01) ◽  
pp. 2050007
Author(s):  
Yongfeng Guo ◽  
Fang Wei ◽  
Linjie Wang
Keyword(s):  
Numerical Solutions ◽  
Signal To Noise Ratio ◽  
Stability Index ◽  
Signal Amplitude ◽  
Noise Signal ◽  
Bistable System ◽  
Index Formula ◽  
Lévy Noise ◽  
Steady State Probability ◽  
Levy Noise

This paper aims to explore the Lévy noise-induced effects in underdamped asymmetric bistable system. Lévy noise is generated by Janicki–Weron algorithm which is different from the usual Gaussian noise. The numerical solutions of system equation are obtained by the fourth-order stochastic Runge–Kutta algorithm. Then the quasi-steady-state probability density (QSPD) is obtained by solving the equation of system, and the stochastic resonance (SR) is determined by the classical measure of signal-to-noise ratio (SNR). The influence of various parameters of the Lévy noise and the system parameters on QSPD and SNR is discussed. Noise-induced transitions occur by varying the parameters of the Lévy noise and the driven system. Moreover, within certain limits, the larger value of the stability index [Formula: see text] of Lévy noise, signal amplitude [Formula: see text], and the absolute values of asymmetric parameter [Formula: see text] can give rise to the SR phenomenon. On the contrary, the larger values of skewness parameters [Formula: see text] of Lévy noise and damping parameter [Formula: see text] further weaken the occurrence of the SR phenomenon in the given system.

Chaos and Control of Ships with Water on Deck Under Periodic Excitation with Lévy Noise

2017 ◽  
Vol 16 (03) ◽  
pp. 1750027 ◽  
Author(s):  
Jiaxin Yuan ◽  
Chen Zhang ◽  
Zhan Qiu ◽  
Fuxin Wang
Keyword(s):  
Sliding Mode ◽  
Chaotic Behavior ◽  
Global Bifurcation ◽  
Stability Index ◽  
Phase Portraits ◽  
Index Formula ◽  
Critical Parameters ◽  
Lévy Noise ◽  
Water On Deck ◽  
Levy Noise

Non-linear dynamic and chaotic roll motion response of ships with water on deck induced by uncertain jumps are investigated. The huge wave with random jump can be described as Lévy noise with critical parameters [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text]. Through sensitive study, the stability index [Formula: see text] and the scale parameter [Formula: see text] are specified as two significant parameters in chaotic motion induction. On the same [Formula: see text] condition, the motion histories, phase portraits and Poincare maps are all recorded to highlight the effect of [Formula: see text] upon uncertain jump system, and their global bifurcation characteristics with the fluctuating amplitude [Formula: see text] are analyzed. Results show that the decrement of stability index [Formula: see text] makes the curve much thicker, and leads the acceptable stable [Formula: see text] region becomes smaller. Finally, an adaptive fuzzy sliding mode control is proposed to eliminate the chaotic behavior and stabilize the system. The asymptomatic stability from the perspective of mean square convergence is analyzed and simulated results show the effectiveness of the method.

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

2015 ◽  
Vol 3 (2) ◽  
pp. 176-183 ◽  
Author(s):  
Jiaorui Li ◽  
Shuang Li
Keyword(s):  
Business Cycle ◽  
White Noise ◽  
Economic System ◽  
Probability Density ◽  
Stationary Probability ◽  
Cycle Model ◽  
Poisson White Noise ◽  
Stationary Probability Density ◽  
White Noise Excitation ◽  
Business Cycle Model

AbstractSeveral 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.

Lévy noise-induced inverse stochastic resonance on Newman–Watts networks of Hodgkin–Huxley neurons

2020 ◽  
Vol 34 (19) ◽  
pp. 2050185
Author(s):  
Dongxi Li ◽  
Shuling Song ◽  
Ni Zhang
Keyword(s):  
Firing Rate ◽  
Stochastic Resonance ◽  
Neuronal Network ◽  
Coupling Strength ◽  
Stability Index ◽  
Neuronal Firing ◽  
Lévy Noise ◽  
Neuron Network ◽  
Average Firing Rate ◽  
Levy Noise

This paper primarily investigates the inverse stochastic resonance (ISR) of neuron network driven by Lévy noise with electrical autapse and chemical autapse, respectively. Firstly, the discharge of Hodgkin–Huxley (HH) neuron network under different noise parameters, autapse parameters and network coupling strength is shown. Then, the variation of average firing rate with Lévy noise in the case of electrical autapse and chemical autapse is presented. We find that there exists a minimum value of the average firing rate curve caused by stability index and noise intensity of Lévy noise across the whole network, which is the phenomenon of ISR. With the increase of autaptic intensity and coupling strength, the ISR inhibitory effect of neuron discharge is weakened. In addition, with the increase of coupling strength, the neuron discharge of neural network is more intense and regular. As a consequence, our work suggests that autaptic intensity and coupling efficient of neuronal network can regulate the neuronal firing activities and suppress the effect of ISR, and Lévy noise can induce ISR phenomenon in Newman–Watts neuronal network.

scholarly journals The stationary probability density of a class of bounded Markov processes

2010 ◽  
Vol 42 (04) ◽  
pp. 986-993 ◽  
Author(s):  
Muhamad Azfar Ramli ◽  
Gerard Leng
Keyword(s):  
Differential Equation ◽  
Integral Equation ◽  
Markov Process ◽  
Probability Density ◽  
Markov Processes ◽  
Transition Probability ◽  
Stationary Probability ◽  
Probability Functions ◽  
Stationary Probability Density

In this paper we generalize a bounded Markov process, described by Stoyanov and Pacheco-González for a class of transition probability functions. A recursive integral equation for the probability density of these bounded Markov processes is derived and the stationary probability density is obtained by solving an equivalent differential equation. Examples of stationary densities for different transition probability functions are given and an application for designing a robotic coverage algorithm with specific emphasis on particular regions is discussed.

Sign in / Sign up

Export Citation Format

Share Document