GAUSSIAN WHITE NOISE CALCULUS OF GENERALIZED EXPANSION

A new framework of Gaussian white noise calculus is established, in line with generalized expansion in [3, 4, 7]. A suitable frame of Fock expansion is presented on Gaussian generalized expansion functionals being introduced here, which provides the integral kernel operator decomposition of the second quantization of Koopman operators for chaotic dynamical systems, in terms of annihilation operators partial derivative(t) and its dual, creation operators partial derivative(t)*.

Stochastic stability of FitzHugh-Nagumo systems perturbed by Gaussian white noise

The current paper is devoted to the study of the stochastic stability of FitzHugh-Nagumo systems perturbed by Gaussian white noise. First, the dynamics of stochastic FitzHugh-Nagumo systems are studied. Then, the existence and uniqueness of their invariant measures, which mix exponentially are proved. Finally, the asymptotic behaviors of invariant measures when size of noise gets to zero are investigated.

THE REGULARIZED SOLUTION APPROXIMATION OF FORWARD/BACKWARD PROBLEMS FOR A FRACTIONAL PSEUDO-PARABOLIC EQUATION WITH RANDOM NOISE

This paper deals with the forward and backward problems for the nonlinear fractional pseudo-parabolic equation ut+(-Δ)^(s1)ut+β(-Δ)^(s2)u=F(u,x,t)subject o random Gaussian white noise for initial and final data.Under the suitable assumptions s1,s2andβ,we first show the ill-posedness of mild solutions for forward and backward problems in the sense of Hadamard,which are mainly driven by random noise.Moreover,we propose the Fourier truncation method for stabilizing the above ill-posed problems.We derive an error estimate between the exact solution and its regularized solution in an E‖·‖Hs22norm,and give some numerical examples illustrating the effect of above method.

NOISE-INDUCED CHAOTIC MOTIONS IN HAMILTONIAN SYSTEMS WITH SLOW-VARYING PARAMETERS

This paper studies chaotic motions in quasi-integrable Hamiltonian systems with slow-varying parameters under both harmonic and noise excitations. Based on the dynamic theory and some assumptions of excited noises, an extended form of the stochastic Melnikov method is presented. Using this extended method, the homoclinic bifurcations and chaotic behavior of a nonlinear Hamiltonian system with weak feed-back control under both harmonic and Gaussian white noise excitations are analyzed in detail. It is shown that the addition of stochastic excitations can make the parameter threshold value for the occurrence of chaotic motions vary in a wider region. Therefore, chaotic motions may arise easily in the system. By the Monte-Carlo method, the numerical results for the time-history and the maximum Lyapunov exponents of an example system are finally given to illustrate that the presented method is effective.

Slowing down critical transitions via Gaussian white noise and periodic force

Stochastic perturbations and periodic excitations are generally regarded as sources to induce critical transitions in complex systems. However, we find that they are also able to slow down an imminent critical transition. To illustrate this phenomenon, a periodically driven bistable eutrophication model with Gaussian white noise is introduced as a prototype class of real systems.The residence probability(RP) is presented to measure the possibility that the given system stays in the oligotrophic state versus Gaussian white noise and periodic force. Variations in the mean first passage time(MFPT) and the mean velocity(MV) of the first right-crossing process are also calculated respectively. We show that the frequency of the periodic force can increase the MFPT while reduce the MV under different control parameters. Nevertheless, the noise intensity or the amplitude may result in an increase of the RP only in the case of control parameters approaching the critical values. Furthermore, for an impending critical transition, an increase of the RP appears with the interaction between the amplitude and noise intensity or the combination of the noise intensity and frequency, while the interaction of the frequency and amplitude leads to an extension of the MFPT or a decrease of the MV. As a result, an increase of the RP and MFPT, and a decrease of the MVobtained from our results claim that it is possible to slow down an imminent critical transition via Gaussian white noise and periodic force.

Hopf bifurcation of nonlinear system with multisource stochastic factors

The article mainly explores the Hopf bifurcation of a kind of nonlinear system with Gaussian white noise excitation and bounded random parameter.Firstly,the nonlinear system with multisource stochastic fac-tors is reduced to an equivalent deterministic nonlinear system by the sequential orthogonal decomposi-tion method and the Karhunen-Loeve(K-L)decomposition theory.Secondly,the critical conditions about the Hopf bifurcation of the equivalent deterministic system are obtained.At the same time the influence of multisource stochastic factors on the Hopf bifurcation for the proposed system is explored.Finally,the theorical results are verified by the numerical simulations.

Predicting solutions of the stochastic fractional order dynamical system using machine learning

The solution of fractional-order systems has been a complex problem for our research.Traditional methods like the predictor-corrector method and other solution steps are complicated and cumbersome to derive,which makes it more difficult for our solution efficiency.The development of machine learning and nonlinear dynamics has provided us with new ideas to solve some complex problems.Therefore,this study considers how to improve the accuracy and efficiency of the solution based on traditional methods.Finally,we propose an efficient and accurate nonlinear auto-regressive neural network for the fractional order dynamic system prediction model(FODS-NAR).First,we demonstrate by example that the FODS-NAR algorithm can predict the solution of a stochastic fractional order system.Second,we compare the FODS-NAR algorithm with the famous and good reservoir computing(RC)algorithms.We find that FODS-NAR gives more accurate predictions than the traditional RC algorithm with the same system parameters,and the residuals of the FODS-NAR algorithm are closer to 0.Consequently,we conclude that the FODS-NAR algorithm is a method with higher accuracy and prediction results closer to the state of fractional-order stochastic systems.In addition,we analyze the effects of the number of neurons and the order of delays in the FODS-NAR algorithm on the prediction results and derive a range of their optimal values.

Stochastic Stability of FitzHugh-Nagumo Systems in Infinite Lattice Perturbed by Gaussian White Noise

The current paper is devoted to the study of stochastic stability of FitzHugh-Nagumo systems in infinite lattice perturbed by Gaussian white noise. We first study the dynamics of stochastic FitzHugh-Nagumo systems, then prove the existence and uniqueness of their equilibriums, which mix exponentially. Finally, we investigate asymptotic behavior of equilibriums when the size of noise gets to zero.

Damage detection of 3D structures using nearest neighbor search method

An innovative damage identification method using the nearest neighbor search method to assess 3D structures is presented.The frequency response function was employed as the input parameters to detect the severity and place of damage in 3D spaces since it includes the most dynamic characteristics of the structures.Two-dimensional principal component analysis was utilized to reduce the size of the frequency response function data.The nearest neighbor search method was employed to detect the severity and location of damage in different damage scenarios.The accuracy of the approach was verified using measured data from an experimental test;moreover,two asymmetric 3D numerical examples were considered as the numerical study.The superiority of the method was demonstrated through comparison with the results of damage identification by using artificial neural network.Different levels of white Gaussian noise were used for polluting the frequency response function data to investigate the robustness of the methods against noise-polluted data.The results indicate that both methods can efficiently detect the damage properties including its severity and location with high accuracy in the absence of noise,but the nearest neighbor search method is more robust against noisy data than the artificial neural network.

Construction of Regular Rate-Compatible LDPC Convolutional Codes

In this paper, we propose a new method to derive a family of regular rate-compatible low-density parity-check（RC-LDPC） convolutional codes from RC-LDPC block codes. In the RC-LDPC convolutional family, each extended sub-matrix of each extended code is obtained by choosing specified elements from two fixed matrices HE1K and HE1K, which are derived by modifying the extended matrices HE1 and HE2 of a systematic RC-LDPC block code. The proposed method which is based on graph extension simplifies the design, and prevent the defects caused by the puncturing method. It can be used to generate both regular and irregular RC-LDPC convolutional codes. All resulted codes in the family are systematic which simplify the encoder structure and have maximum encoding memories which ensure the property. Simulation results show the family collectively offer a steady improvement in performance with code compatibility over binary-input additive white Gaussian noise channel（BI-AWGNC）.

Stochastic response of fractional-order van der Pol oscillator

We studied the response of fractional-order van de Pol oscillator to Gaussian white noise excitation in this letter. An equivalent integral-order nonlinear stochastic system is obtained to replace the given system based on the principle of minimum mean-square error. Through stochastic averaging, an averaged Ito equation is deduced. We obtained the Fokker–Planck–Kolmogorov equation connected to the averaged Ito equation and solved it to yield the approximate stationary response of the system. The analytical solution is confirmed by using Monte Carlo simulation.

STOCHASTIC HOPF BIFURCATION IN QUASIINTEGRABLE-HAMILTONIAN SYSTEMS

A new procedure is developed to study the stochastic Hopf bifurcation in quasi- integrable-Hamiltonian systems under the Gaussian white noise excitation.Firstly,the singular bound- aries of the first-class and their asymptotic stable conditions in probability are given for the averaged Ito differential equations about all the sub-system's energy levels with respect to the stochastic aver- aging method.Secondly,the stochastic Hopf bifurcation for the coupled sub-systems are discussed by defining a suitable bounded torus region in the space of the energy levels and employing the theory of the torus region when the singular boundaries turn into the unstable ones.Lastly,a quasi-integrable- Hamiltonian system with two degrees of freedom is studied in detail to illustrate the above procedure. Moreover,simulations by the Monte-Carlo method are performed for the illustrative example to verify the proposed procedure.It is shown that the attenuation motions and the stochastic Hopf bifurcation of two oscillators and the stochastic Hopf bifurcation of a single oscillator may occur in the system for some system's parameters.Therefore,one can see that the numerical results are consistent with the theoretical predictions.

Sparse identification method of extracting hybrid energy harvesting system from observed data

Hybrid energy harvesters under external excitation have complex dynamical behavior and the superiority of promoting energy harvesting efficiency.Sometimes,it is difficult to model the governing equations of the hybrid energy harvesting system precisely,especially under external excitation.Accompanied with machine learning,data-driven methods play an important role in discovering the governing equations from massive datasets.Recently,there are many studies of datadriven models done in aspect of ordinary differential equations and stochastic differential equations(SDEs).However,few studies discover the governing equations for the hybrid energy harvesting system under harmonic excitation and Gaussian white noise(GWN).Thus,in this paper,a data-driven approach,with least square and sparse constraint,is devised to discover the governing equations of the systems from observed data.Firstly,the algorithm processing and pseudo code are given.Then,the effectiveness and accuracy of the method are verified by taking two examples with harmonic excitation and GWN,respectively.For harmonic excitation,all coefficients of the system can be simultaneously learned.For GWN,we approximate the drift term and diffusion term by using the Kramers-Moyal formulas,and separately learn the coefficients of the drift term and diffusion term.Cross-validation(CV)and mean-square error(MSE)are utilized to obtain the optimal number of iterations.Finally,the comparisons between true values and learned values are depicted to demonstrate that the approach is well utilized to obtain the governing equations for the hybrid energy harvester under harmonic excitation and GWN.

Multiple Input Multiple Output CDSK Chaotic Communication System and Its Performance Analysis

This paper propose a novel noncoherent chaotic com- munication scheme named multiple-input multiple-output correlation-delay-shill- keying （MIMO-CDSK）. In this scheme, multiple antennas are employed to strengthen the robustness in transmission, and to get more diversity gain. The bit error rate （BER） of the MIMO-CDSK is studied analytically in AWGN channel model and multipath fading channel model. The theory and simulation results show that, the performance gain can be obtained with multiple antennas allocated in the transmitter and receiver. Moreover, it is observed that MIMO-CDSK system can effectively reduce the multipath interference.

A Novel Minimum Output Variance Estimator for Carrier Frequency Offset in OFDM Systems

In orthogonal frequency-division multiplexing （OFDM） systems, the carrier frequency offset （CFO） destroys the orthogonality among subcarriers which degrades system performance. Various CFO estimation methods have been developed to compensate for the CFO at the receiver. This paper describes a novel minimum output variance method for OFDM systems with CFO in additive white Gaussian noise channels. This method utilizes the phase and the amplitude of the received signal and reduces the mean square error of the CFO by about 3 dB compared with the original minimum output variance method.

Stochastic P-bifurcations of a noisy nonlinear system with fractional derivative element

This paper investigates the stochastic P-bifurcation(SPB)of a fractionally damped oscillator subjected to additive and multiplicative Gaussian white noise.Variable transformation and the stochastic averaging technique are applied to derive the expression of probability density function(PDF)of the system response.Critical conditions of the stochastic bifurcation induced by system parameters are presented based on the change in the number of extreme points of the probability density function.Numerical results are given to show the effectiveness of the proposed approach.Stochastic P-bifurcations for additive and multiplicative noise are studied in detail according to the critical conditions.

Low-Drift Closed-Loop Fiber Optic Gyroscope of High Scale Factor Stability Driven by Laser With External Phase Modulation

In view of the poor scale factor stability of the interferometric fiber optic gyroscope(IFOG),it is a creative method to use laser to drive the IFOG for its better frequency stabilization characteristics instead of the broadband light source.As the linewidth of laser is narrow,the errors of coherent backscattering,polarization coupling,and Kerr effect are reintroduced which cause more noise and drift.This paper studies laser spectrum broadening based on external phase modulation of Gaussian white noise(GWN).The theoretical analysis and test results indicate that this method has a good effect on spectrum broadening and can be used to improve the performance of the laser-driven IFOG.In the established closed-loop IFOG,a four-state modulation(FSM)is adopted to avoid temperature instability of the multifunction integrated-optic chip(MIOC)and drift caused by the electronic circuit in demodulation.The experimental results show that the IFOG driven by broadened laser has the angular random walk noise of 0.0038°/√h and the drift of 0.017°/h,which are 62%and 66%better than those without modulation respectively,of which the drift has reached the level of the broadband light source.Although the noise still needs further reduction,its scale factor stability is 0.38 ppm,which has an overwhelming advantage compared with the traditional IFOG.

Oscillating gravity,non-singularity and mass quantization from Moffat stochastic gravity arguments

In Moffat stochastic gravity arguments, the spacetime geometry is assumed to be a fluctuating background and the gravitational constant is a control parameter due to the presence of a timedependent Gaussian white noise ξ(t). In such a surrounding, both the singularities of gravitational collapse and the Big Bang have a zero probability of occurring. In this communication, we generalize Moffat’s arguments by adding a random temporal tiny variable for a smoothing purpose and creating a white Gaussian noise process with a short correlation time. The Universe accordingly is found to be non-singular and is dominated by an oscillating gravity. A connection with a quantum oscillator was established and analyzed. Surprisingly, the Hubble mass which emerges in extended supergravity may be quantized.