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.

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.

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 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.

Stochastic resonance induced by Lévy noise in a tumor growth model with periodic treatment

2014 ◽  
Vol 28 (11) ◽  
pp. 1450085 ◽  
Author(s):  
Wei Xu ◽  
Mengli Hao ◽  
Xudong Gu ◽  
Guidong Yang
Keyword(s):  
Tumor Growth ◽  
Stochastic Resonance ◽  
Growth Model ◽  
Tumor Therapy ◽  
Stability Index ◽  
Resonance Phenomenon ◽  
Lévy Noise ◽  
Tumor Growth Model ◽  
Periodic Treatment ◽  
Levy Noise

In this paper, the stochastic resonance phenomenon in a tumor growth model under subthreshold periodic therapy and Lévy noise excitation is investigated. The possible reoccurrence of tumor due to stochastic resonance is discussed. The signal-to-noise ratio (SNR) is calculated numerically to measure the stochastic resonance. It is found that smaller stability index is better for avoiding tumor reappearance. Besides, the effect of the skewness parameter on the tumor regrowth is related to the stability index. Furthermore, increasing the intensity of periodic treatment does not always facilitate tumor therapy. These results are beneficial to the optimization of periodic tumor therapy.

Levy noise-induced inverse stochastic resonance in a single neuron

2019 ◽  
Vol 33 (21) ◽  
pp. 1950252
Author(s):  
Yue Zhao ◽  
Dongxi Li
Keyword(s):  
Stochastic Resonance ◽  
Single Neuron ◽  
Scale Parameter ◽  
Stability Index ◽  
Current Strength ◽  
Lévy Noise ◽  
Critical Threshold ◽  
Interspike Intervals ◽  
Discharge Patterns ◽  
Levy Noise

In this paper, the phenomenon of inverse stochastic resonance (ISR) induced by Levy noise is investigated in a single neuron. Firstly, the discharge of Hodgkin–Huxley neuron at different current strengths without noise is studied. Then we cite the concept of average spiking rate and use it as a scale to measure the discharge of neuron. Under the influence of Levy noise, it is found that the average firing rate can reach a minimum caused by stability index and a scale parameter of Levy noise. The conclusion is that ISR could be induced by Levy noise when the input current strength reaches the critical threshold. Besides, in order to further prove that Levy noise can produce ISR, interspike intervals (ISIs) histogram is used for studying discharge patterns in voltage. Finally, we find that with the increase of stability index and scale parameter of Levy noise, the total number of ISIs decreases first and then increases, which implies the appearance of ISR.

scholarly journals A Stochastic Pitchfork Bifurcation in Most Probable Phase Portraits

2018 ◽  
Vol 28 (01) ◽  
pp. 1850017 ◽  
Author(s):  
Hui Wang ◽  
Xiaoli Chen ◽  
Jinqiao Duan
Keyword(s):  
Gaussian Noise ◽  
Pitchfork Bifurcation ◽  
Phase Portraits ◽  
Lévy Noise ◽  
Stochastic Bifurcation ◽  
Bifurcation Phenomena ◽  
Pitchfork Bifurcations ◽  
Non Gaussian ◽  
Levy Noise ◽  
Qualitative Changes

We study stochastic bifurcation for a system under multiplicative stable Lévy noise (an important class of non-Gaussian noise), by examining the qualitative changes of equilibrium states with its most probable phase portraits. We have found some peculiar bifurcation phenomena in contrast to the deterministic counterpart: (i) When the non-Gaussianity parameter in Lévy noise varies, there is either one, two or no backward pitchfork type bifurcations; (ii) When a parameter in the vector field varies, there are two or three forward pitchfork bifurcations; (iii) The non-Gaussian Lévy noise clearly leads to fundamentally more complex bifurcation scenarios, since in the special case of Gaussian noise, there is only one pitchfork bifurcation which is reminiscent of the deterministic situation.

Controllability for impulsive neutral stochastic delay partial differential equations driven by fBm and Lévy noise

2020 ◽  
Vol 21 (02) ◽  
pp. 2150013
Author(s):  
Diem Dang Huan ◽  
Ravi P. Agarwal
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Sufficient Conditions ◽  
Nonlinear Heat Equation ◽  
Index Formula ◽  
Lévy Noise ◽  
Partial Differential ◽  
Stochastic Delay ◽  
Delay Partial Differential Equations ◽  
Levy Noise

This paper aims to investigate the controllability for impulsive neutral stochastic delay partial differential equations (PDEs) driven by fractional Brownian motion (fBm) with Hurst index [Formula: see text] and Lévy noise in Hilbert spaces. By using a fixed point approach without imposing a severe compactness condition on the semigroup, a new set of sufficient conditions is derived. The results in this paper are generalization and continuation of the recent results on this issue. At the end, an application to the stochastic nonlinear heat equation with delays driven by a fBm and Lévy noise is given.

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