Bifurcation Analysis of a Self-Sustained Birhythmic Oscillator Under Two Delays and Colored Noises

2020 ◽  
Vol 30 (01) ◽  
pp. 2050013
Author(s):  
Yuanli Sun ◽  
Lijuan Ning
Keyword(s):  
Colored Noise ◽  
Additive Noise ◽  
Numerical Solutions ◽  
Multiple Scale ◽  
Stochastic Averaging ◽  
Practical Applications ◽  
Two Delays ◽  
Stationary Probability Density Function ◽  
Enzyme Molecules ◽  
Stochastic Bifurcations

In this manuscript, an investigation on bifurcations induced by two delays and additive and multiplicative colored noises in a self-sustained birhythmic oscillator is presented, both theoretically and numerically, which serves for the purpose of unveiling extremely complicated nonlinear dynamics in various spheres, especially in biology. By utilizing the multiple scale expansion approach and stochastic averaging technique, the stationary probability density function (SPDF) of the amplitude is obtained for discussing stochastic bifurcations. With time delays, intensities and correlation time of noises regarded as bifurcation parameters, rich bifurcation arises. In the case of additive noise, it is identified that the bifurcations induced by the two delays are entirely distinct and longer velocity delay can accelerate the conversion rate of excited enzyme molecules. A novel type of P-bifurcation emerges from the process in the case of multiplicative colored noise, with the SPDF qualitatively changing between crater-like and bimodal distributions, while it cannot be generated when the multiplicative colored noise is coupled with additive noise. The feasibility and effectiveness of analytical methods are confirmed by the good consistency between theoretical and numerical solutions. This investigation may have practical applications in governing dynamical behaviors of birhythmic systems.

Stochastic Bifurcations of a Fractional-Order Vibro-Impact Oscillator Subjected to Colored Noise Excitation

2021 ◽  
Vol 31 (12) ◽  
pp. 2150177
Author(s):  
Ya-Hui Sun ◽  
Yong-Ge Yang ◽  
Ling Hong ◽  
Wei Xu
Keyword(s):  
Fractional Order ◽  
Colored Noise ◽  
Numerical Solutions ◽  
Gaussian White Noise ◽  
Research Work ◽  
Critical Conditions ◽  
Stochastic Bifurcation ◽  
Response Analysis ◽  
Impact Oscillator ◽  
Stochastic Bifurcations

A stochastic vibro-impact system has triggered a consistent body of research work aimed at understanding its complex dynamics involving noise and nonsmoothness. Among these works, most focus is on integer-order systems with Gaussian white noise. There is no report yet on response analysis for fractional-order vibro-impact systems subject to colored noise, which is presented in this paper. The biggest challenge for analyzing such systems is how to deal with the fractional derivative of absolute value functions after applying nonsmooth transformation. This problem is solved by introducing the Fourier transformation and deriving the approximate probabilistic solution of the fractional-order vibro-impact oscillator subject to colored noise. The reliability of the developed technique is assessed by numerical solutions. Based on the theoretical result, we also present the critical conditions of stochastic bifurcation induced by system parameters and show bifurcation diagrams in two-parameter planes. In addition, we provide a stochastic bifurcation with respect to joint probability density functions. We find that fractional order, coefficient of restitution factor and correlation time of colored noise excitation can induce stochastic bifurcations.

scholarly journals Response of Cantilever Model with Inertia Nonlinearity under Transverse Basal Gaussian Colored Noise Excitation

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Bo Li ◽  
Kai Hu ◽  
Guoguang Jin ◽  
Yanyan Song ◽  
Gen Ge
Keyword(s):  
Differential Equation ◽  
Colored Noise ◽  
Averaging Method ◽  
Stochastic Averaging ◽  
Stochastic Averaging Method ◽  
Beam Model ◽  
Averaging Methods ◽  
Gaussian Colored Noise ◽  
Stationary Probability Density Function ◽  
Probability Density Function Pdf

Considering the curvature nonlinearity and longitudinal inertia nonlinearity caused by geometrical deformations, a slender inextensible cantilever beam model under transverse pedestal motion in the form of Gaussian colored noise excitation was studied. Present stochastic averaging methods cannot solve the equations of random excited oscillators that included both inertia nonlinearity and curvature nonlinearity. In order to solve this kind of equations, a modified stochastic averaging method was proposed. This method can simplify the equation to an Itô differential equation about amplitude and energy. Based on the Itô differential equation, the stationary probability density function (PDF) of the amplitude and energy and the joint PDF of the displacement and velocity were studied. The effectiveness of the proposed method was verified by numerical simulation.

Stochastic Response of SDOF Self-Centering System

2020 ◽  
Vol 20 (05) ◽  
pp. 2050062
Author(s):  
Huiying Hu ◽  
Lincong Chen
Keyword(s):  
Stochastic Averaging ◽  
Nonlinear Stochastic System ◽  
Amplitude Envelope ◽  
Stochastic Response ◽  
Equivalent Damping ◽  
Yield Displacement ◽  
Seismic Loadings ◽  
New Type ◽  
Stationary Probability Density Function ◽  
Energy Dissipation Coefficient

As a new type of seismic resisting device, the self-centering system is attractive due to its excellent re-centering capability, but research on such a system under random seismic loadings is quite limited. In this paper, the stochastic response of a single-degree-of-freedom (SDOF) self-centering system driven by a white noise process is investigated. For this purpose, the original self-centering system is first approximated by an auxiliary nonlinear system, in which the equivalent damping and stiffness coefficients related to the amplitude envelope of the response are determined by a harmonic balance procedure. Subsequently, by the method of stochastic averaging, the amplitude envelope of the response of the equivalent nonlinear stochastic system is approximated by a Markovian process. The associated Fokker–Plank–Kolmogorov (FPK) equation is used to derive the stationary probability density function (PDF) of the amplitude envelope in a closed form. The effects of energy dissipation coefficient and yield displacement on the response of system are examined using the stationary PDF solution. Moreover, Monte Carlo simulations (MCS) are used for ascertaining the accuracy of the analytical solutions.

The Effects of Correlated Noise on Bifurcation in Birhythmicity Driven by Delay

2018 ◽  
Vol 28 (10) ◽  
pp. 1850127 ◽  
Author(s):  
Lijuan Ning ◽  
Zhidan Ma
Keyword(s):  
Averaging Method ◽  
Harmonic Approximation ◽  
Stochastic Averaging ◽  
Stochastic Averaging Method ◽  
Self Control ◽  
Correlated Noise ◽  
State Variables ◽  
Stationary Probability Density Function ◽  
Fpk Equation ◽  
Control Feedback

We consider bifurcation regulations under the effects of correlated noise and delay self-control feedback excitation in a birhythmic model. Firstly, the term of delay self-control feedback is transferred into state variables without delay by harmonic approximation. Secondly, FPK equation and stationary probability density function (SPDF) for amplitude can be theoretically mapped with stochastic averaging method. Thirdly, the intriguing effects on bifurcation regulations in a birhythmic model induced by delay and correlated noise are observed, which suggest the violent dependence of bifurcation in this model on delay and correlated noise. Particularly, the inner limit cycle (LC) is always standing due to noise. Lastly, the validity of analytical results was confirmed by Monte Carlo simulation for the dynamics.

Non-Standard Reduction of Noisy Structural Systems: Shallow Arches

2000 ◽  
Author(s):  
Lalit Vedula ◽  
N. Sri Namachchivaya
Keyword(s):  
Markov Process ◽  
Random Perturbation ◽  
Exit Time ◽  
Stochastic Averaging ◽  
Dimensional System ◽  
Shallow Arches ◽  
Higher Dimensional ◽  
Mean Exit Time ◽  
Stationary Probability Density Function ◽  
Low Dimensional

Abstract The dynamics of a shallow arch subjected to small random external and parametric excitation is invegistated in this work. We develop rigorous methods to replace, in some limiting regime, the original higher dimensional system of equations by a simpler, constructive and rational approximation – a low-dimensional model of the dynamical system. To this end, we study the equations as a random perturbation of a two-dimensional Hamiltonian system. We achieve the model-reduction through stochastic averaging and the reduced Markov process takes its values on a graph with certain glueing conditions at the vertex of the graph. Examination of the reduced Markov process on the graph yields many important results such as mean exit time, stationary probability density function.

scholarly journals A Collocation Approach for Solving Time-Fractional Stochastic Heat Equation Driven by an Additive Noise

Symmetry ◽  
2020 ◽  
Vol 12 (6) ◽  
pp. 904 ◽  
Author(s):  
Afshin Babaei ◽  
Hossein Jafari ◽  
S. Banihashemi
Keyword(s):  
Order Of Convergence ◽  
Additive Noise ◽  
Numerical Solutions ◽  
Numerical Test ◽  
Test Problems ◽  
Heat Equations ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Stochastic Heat Equations ◽  
Collocation Approach

A spectral collocation approach is constructed to solve a class of time-fractional stochastic heat equations (TFSHEs) driven by Brownian motion. Stochastic differential equations with additive noise have an important role in explaining some symmetry phenomena such as symmetry breaking in molecular vibrations. Finding the exact solution of such equations is difficult in many cases. Thus, a collocation method based on sixth-kind Chebyshev polynomials (SKCPs) is introduced to assess their numerical solutions. This collocation approach reduces the considered problem to a system of linear algebraic equations. The convergence and error analysis of the suggested scheme are investigated. In the end, numerical results and the order of convergence are evaluated for some numerical test problems to illustrate the efficiency and robustness of the presented method.

scholarly journals Attention-Based Residual Network with Scattering Transform Features for Hyperspectral Unmixing with Limited Training Samples

2020 ◽  
Vol 12 (3) ◽  
pp. 400 ◽  
Author(s):  
Zeng ◽  
Ritz ◽  
Zhao ◽  
Lan
Keyword(s):  
Neural Network ◽  
Additive Noise ◽  
Hyperspectral Data ◽  
Training Data ◽  
K Nearest Neighbors ◽  
Practical Applications ◽  
Hyperspectral Unmixing ◽  
Scattering Transform ◽  
Training Samples ◽  
Limited Training Samples

This paper proposes a framework for unmixing of hyperspectral data that is based on utilizing the scattering transform to extract deep features that are then used within a neural network. Previous research has shown that using the scattering transform combined with a traditional K-nearest neighbors classifier (STFHU) is able to achieve more accurate unmixing results compared to a convolutional neural network (CNN) applied directly to the hyperspectral images. This paper further explores hyperspectral unmixing in limited training data scenarios, which are likely to occur in practical applications where the access to large amounts of labeled training data is not possible. Here, it is proposed to combine the scattering transform with the attention-based residual neural network (ResNet). Experimental results on three HSI datasets demonstrate that this approach provides at least 40% higher unmixing accuracy compared to the previous STFHU and CNN algorithms when using limited training data, ranging from 5% to 30%, are available. The use of the scattering transform for deriving features within the ResNet unmixing system also leads more than 25% improvement when unmixing hyperspectral data contaminated by additive noise.

Analysis of Heat Conduction in a Heterogeneous Material by a Multiple-Scale Averaging Method

2015 ◽  
Vol 137 (7) ◽  
Author(s):  
James White
Keyword(s):  
Heat Conduction ◽  
Thermal Energy ◽  
Energy Equation ◽  
Numerical Solutions ◽  
Scale Effects ◽  
Three Dimensional ◽  
Heterogeneous Material ◽  
Multiple Scale ◽  
Computational Effort ◽  
Short Scale

In order to better manage computational requirements in the study of thermal conduction with short-scale heterogeneous materials, one is motivated to arrange the thermal energy equation into an accurate and efficient form with averaged properties. This should then allow an averaged temperature solution to be determined with a moderate computational effort. That is the topic of this paper as it describes the development using multiple-scale analysis of an averaged thermal energy equation based on Fourier heat conduction for a heterogeneous material with isotropic properties. The averaged energy equation to be reported is appropriate for a stationary or moving solid and three-dimensional heat flow. Restrictions are that the solid must display its heterogeneous properties over short spatial and time scales that allow averages of its properties to be determined. One distinction of the approach taken is that all short-scale effects, both moving and stationary, are combined into a single function during the analytical development. The result is a self-contained form of the averaged energy equation. By eliminating the need for coupling the averaged energy equation with external local problem solutions, numerical solutions are simplified and made more efficient. Also, as a result of the approach taken, nine effective averaged thermal conductivity terms are identified for three-dimensional conduction (and four effective terms for two-dimensional conduction). These conductivity terms are defined with two types of averaging for the component material conductivities over the short-scales and in terms of the relative proportions of the short-scales. Numerical results are included and discussed.

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