System Of Deterministic Equations Research Articles

Solution of the Stochastic Heat Equation with Nonlinear Losses Using Wiener-Hermite Expansion

In the current work, the Wiener-Hermite expansion (WHE) is used to solve the stochastic heat equation with nonlinear losses. WHE is used to deduce the equivalent deterministic system up to third order accuracy. The solution of the equivalent deterministic system is obtained using different techniques numerically and analytically. The finite-volume method (FVM) with Pickard iteration is used to solve the equivalent system iteratively. The WHE with perturbation technique (WHEP) is applied to deduce more simple and decoupled equivalent deterministic system that can be solved numerically without iterations. The system resulting from WHEP technique is solved also analytically using the eigenfunction expansion technique. The Monte-Carlo simulations (MCS) are performed to get the statistical properties of the stochastic solution and to verify other solution techniques. The results show that higher-order solutions are essential especially in case of nonlinearities where non-Gaussian effects cannot be neglected. The comparisons show the efficiency of the numerical WHE and WHEP techniques in solving stochastic nonlinear PDEs compared with the analytical solution and MCS.

Dynamic Analysis of the Nonlinear Chaotic System with Multistochastic Disturbances

The nonlinear chaotic system with multistochastic disturbances is investigated. Based on the orthogonal polynomial approximation, the method of transforming the system into an equivalent deterministic system is given. Then dynamic analysis of the nonlinear chaotic system with multistochastic disturbances can be reduced into that of its equivalent deterministic system. Especially, the Lorenz system with multistochastic disturbances is studied to demonstrate the feasibility of the given method. And its dynamic behaviors are gained including the phase portrait, the bifurcation diagram, the Poincaré section, and the maximum Lyapunov exponent.

Switched Projective Synchronization of the Stochastic Newton-Leipnik System

The paper is involved with switched projective synchronization of two identical chaotic systems with random parameter using adaptive control method. Based on the orthogonal polynomial expansion of the Hilbert spaces, the Newton-Leipnik system with random parameter is transformed as the equivalent deterministic system. At last, an adaptive controller can be designed by the Lyapunov stability theorem for achieving switched projective synchronization of the equivalent deterministic system with different initial values. Corresponding numerical simulations are performed to verify the effectiveness of presented schemes for synchronizing the stochastic Newton-Leipnik system.

Chaos Control of the Fractional Order Stochastic Chen System

In this paper, we study chaos control of the fractional order Chen system with the bounded random parameter. Firstly, we transform the fractional order Chen system with random parameter into an equivalent deterministic system by the orthogonal polynomial approximation. Secondly, based on Routh-Hrwitz criterion, the derivative feedback control laws are applied to fractional order equivalent deterministic Chen system. Lastly, numerical simulations show that the control method is effective and feasible.

Modified projective synchronization of stochastic fractional order chaotic systems with uncertain parameters

In this paper, the synchronization of fractional order chaotic systems with random and uncertain parameters is analyzed. Firstly, based on the orthogonal polynomial expansion, the fractional order Lu and Lorenz systems with random and uncertain parameters are reduced into the equivalent deterministic systems. Secondly, modified projective synchronization of equivalent deterministic Lu and Lorenz systems is explored. Lastly, the theoretical results are verified by the numerical simulations.

Generalized Synchronization of Stochastic Discrete Chaotic System with Poisson Distribution Coefficient

This paper addresses the generalized synchronization of stochastic discrete chaotic systems with Poisson distribution coefficient. Firstly, based on the orthogonal polynomial approximation theory of discrete random function in Hilbert spaces, the discrete chaotic system with random parameter is transformed into its equivalent deterministic system. Secondly, a general method for the generalized synchronization of discrete chaotic system with random parameter is presented by Lyapunov stability theory and contraction theorem. Finally, two synchronization examples numerically illustrated that the proposed control scheme is effective for any stochastic discrete system.

Chaos Control in the Fractional-Order Lorenz System with Random Parameter

In this paper, we study the chaos control in the fractional-order Lorenz system with random parameter. Firstly, according to orthogonal polynomial approximation principle of the Functional analysis, the fractional-order Lorenz system with random parameter is reduced to its equivalent deterministic one. Secondly, chaos control equivalent deterministic system research using the linear feedback method. Finally, though numerical results show the effective and feasible of this method.

Chaos in a Circuit System with Ferroelectric Capacitor

Chaos depends on the random parameters are explored in circuit system with ferroelectric capacitor in this article. A random parameter is considered to characterize the random factors in this kind of capacitor. After we reduce the nonlinear circuit system with random parameter into the equivalent deterministic system by orthogonal polynomial approximation, the available effective numerical methods is applied to obtain the responses of the equivalent deterministic circuit system. Then by numerical simulations, we systematically explore the effect of the random parameter on chaos, and find the intensity of random parameter can present the rich and colorful structures of chaotic attractors.

The effect of the random parameter on the basins and attractors of the elastic impact system

The global dynamic behavior of the elastic impact system with one random parameter is researched in this paper. First, based on the orthogonal polynomial approximation, the stochastic system is transformed into an equivalent deterministic system. Then the composite cell coordinate system method is used to derive the basins and attractors of the equivalent system. It is found that the random parameter affects the global dynamics of the system tremendously. Under the same conditions, the basins and attractors of the stochastic system are entirely different from that of the deterministic system.

Chance Constrained Linear Plus Linear Fractional Bi-level Programming Problem

We present fuzzy goal programming approach to solve chance constrained linear plus linear fractional bi-level programming problem. The chance constraints with right hand parameters as random variables of prescribed probability distribution functions are transformed into equivalent deterministic system constraints. We construct nonlinear membership functions based on deterministic system constraints. The nonlinear membership functions are transformed into linear membership functions by using first order Taylor’s series approximation. In the bi-level decision making context, decision deadlock may arise due to the dissatisfaction of the lower level decision maker with the decision of upper level decision maker. To overcome this problem, decision maker of each level gives his preference bounds on decision variables under his/her control to provide some relaxation on their decisions. Fuzzy goal programming model is used to achieve highest membership goals by minimizing negative deviational variables. Euclidean distance function is used in order to find out the most satisfactory solution. We solve a chance constrained linear plus linear fractional bi-level programming problem to illustrate the proposed approach. General Terms Bi-level programming, linear plus linear fractional programming.

GOAL PROGRAMMING APPROACH TO CHANCE CONSTRAINED MULTI-OBJECTIVE LINEAR FRACTIONAL PROGRAMMING PROBLEM BASED ON TAYLOR’S SERIES APPROXIMATION

This paper deals with goal programming approach to chance constrained multi-objective linear fractional programming problem based on Taylor’s series approximation. We consider the constraints with right hand parameters as the random variables of known mean and variance. The random variables are transformed into standard normal variables with zero mean and unit variance. We convert the chance constraints with known confidence level into equivalent deterministic constraints. The goals of linear fractional objective functions are determined by optimizing it subject to the equivalent deterministic system constraints. Then the fractional objective functions are transformed into equivalent linear functions at the optimal solution point by using first order Taylor polynomial series. In the solution process, we use three minsum goal programming models and identify the most compromise optimal solution by using Euclidean distance function.

GOAL PROGRAMMING APPROACH TO CHANCE CONSTRAINED MULTI-OBJECTIVE LINEAR FRACTIONAL PROGRAMMING PROBLEM BASED ON TAYLOR’S SERIES APPROXIMATION

This paper deals with goal programming approach to chance constrained multi-objective linear fractional programming problem based on Taylor’s series approximation. We consider the constraints with right hand parameters as the random variables of known mean and variance. The random variables are transformed into standard normal variables with zero mean and unit variance. We convert the chance constraints with known confidence level into equivalent deterministic constraints. The goals of linear fractional objective functions are determined by optimizing it subject to the equivalent deterministic system constraints. Then the fractional objective functions are transformed into equivalent linear functions at the optimal solution point by using first order Taylor polynomial series. In the solution process, we use three minsum goal programming models and identify the most compromise optimal solution by using Euclidean distance function.

Hopf bifurcation in a kind of stochastic van der Pol system

The Hopf bifurcation of van der Pol system with random parameter is studied. Firstly according to the orthogonal polynomial approximation in Hilbert space, the van der Pol system with random parameter can be reduced into the equivalent deterministic system. Then the Hopf bifurcation can be explored by the traditional methods in deterministic bifurcation theory. After the critical point of Hopf bifurcation in stochastic van der Pol system is obtained, the influence of the random parameter on Hopf bifurcation in stochastic van der Pol system is analyzed. At last we verified these results by numerical simulations.

Stochastic period-doubling bifurcation analysis of a Rössler system with a bounded random parameter

This paper aims to study the stochastic period-doubling bifurcation of the three-dimensional Rössler system with an arch-like bounded random parameter. First, we transform the stochastic Rössler system into its equivalent deterministic one in the sense of minimal residual error by the Chebyshev polynomial approximation method. Then, we explore the dynamical behaviour of the stochastic Rössler system through its equivalent deterministic system by numerical simulations. The numerical results show that some stochastic period-doubling bifurcation, akin to the conventional one in the deterministic case, may also appear in the stochastic Rössler system. In addition, we also examine the influence of the random parameter intensity on bifurcation phenomena in the stochastic Rössler system.

Polynomial Chaos-Based Analysis of Probabilistic Uncertainty in Hypersonic Flight Dynamics

In this paper, we present a novel computational framework for analyzing the evolution of the uncertainty in state trajectories of a hypersonic air vehicle due to the uncertainty in initial conditions and other system parameters. The framework is built on the so-called generalized polynomial chaos expansions. In this framework, stochastic dynamical systems are transformed into equivalent deterministic dynamical systems in higher dimensional space. Here, the evolution of uncertainty due to initial condition, ballistic coefficient, lift over drag ratio, and atmospheric density is analyzed. The problem studied here is related to the Mars entry, descent, and landing problems. We demonstrate that the polynomial chaos framework is able to predict evolution of uncertainty, in hypersonic flight, with the same order of accuracy as the Monte-Carlo methods but with more computational efficiency.

Positivity and stabilisation for nonlinear stochastic delay differential equations

We use state dependent Gaussian perturbations to stabilise the solutions of differential equations with coefficients that take, as arguments, averaged sets of information from the history of the solution, as well as isolated past and present states. The properties that guarantee stability also guarantee positivity of solutions as long as the initial value is nonzero. We do not require that any component of the coefficients of the equations satisfy Lipschitz conditions. Instead, we require that the functional part of each coefficient which feeds back the present state of the process admit to bounds imposed by a member of a particular class of concave functions. Lipschitz conditions are included as a special case of these bounds. We generalise these results to the finite dimensional case, also constructing perturbations that can destabilise the otherwise stable solutions of a deterministic system of equations.

Stochastic period-doubling bifurcation analysis of stochastic Bonhoeffer–van der Pol system

In this paper, the Chebyshev polynomial approximation is applied to the problem of stochastic period-doubling bifurcation of a stochastic Bonhoeffer?van der Pol (BVP for short) system with a bounded random parameter. In the analysis, the stochastic BVP system is transformed by the Chebyshev polynomial approximation into an equivalent deterministic system, whose response can be readily obtained by conventional numerical methods. In this way we have explored plenty of stochastic period-doubling bifurcation phenomena of the stochastic BVP system. The numerical simulations show that the behaviour of the stochastic period-doubling bifurcation in the stochastic BVP system is by and large similar to that in the deterministic mean-parameter BVP system, but there are still some featured differences between them. For example, in the stochastic dynamic system the period-doubling bifurcation point diffuses into a critical interval and the location of the critical interval shifts with the variation of intensity of the random parameter. The obtained results show that Chebyshev polynomial approximation is an effective approach to dynamical problems in some typical nonlinear systems with a bounded random parameter of an arch-like probability density function.

Flutter analysis of an airfoil with bounded random parameters in incompressible flow via Gegenbauer polynomial approximation

The effects of parameter uncertainty on the flutter characteristics of a two-dimensional airfoil in an incompressible flow were investigated through Gegenbauer polynomial approximation. The uncertain parameters, such as the linear and cubic pitch stiffness coefficients are modeled as bounded random variables with λ-PDFs (probability density functions). With the aid of Gegenbauer polynomial approximation, the two-dimensional stochastic airfoil system is transformed at first into its equivalent deterministic one. Then the Hopf-bifurcation point is determined through the equivalent deterministic system, and the onset of the flutter, together with the flutter frequency against the probability density distribution parameter and the intensity of the random variable is explored. In addition, the ranges of the peak pitch and plunge responses against the flight speed and the PDFs of the peak pitch response are obtained. Numerical results show how the probability density distribution parameters and the intensities of random parameters affect the flutter onset speed, flutter angular frequency, the upper and lower boundaries of peak responses, and the PDFs of peak responses.

Stochastic Hopf bifurcation and chaos of stochastic Bonhoeffer–van der Pol system via Chebyshev polynomial approximation

In this paper, we analyzed stochastic chaos and Hopf bifurcation of stochastic Bonhoeffer–van der Pol (SBVP for short) system with bounded random parameter of an arch-like probability density function. The modifier ‘stochastic’ here implies dependent on some random parameter. In order to study the dynamical behavior of the SBVP system, Chebyshev polynomial approximation is applied to transform the SBVP system into its equivalent deterministic system, whose response can be readily obtained by conventional numerical methods. Thus, we can further explore the nonlinear phenomena in SBVP system. Stochastic chaos and Hopf bifurcation analyzed here are by and large similar to those in the deterministic mean-parameter Bonhoeffer–van der Pol system (DM–BVP for short) but there are also some featuring differences between them shown by numerical results. For example, in the SBVP system the parameter interval matching chaotic responses diffuses into a wider one, which further grows wider with increasing of intensity of the random variable. The shapes of limit cycles in the SBVP system are some different from that in the DM–BVP system, and the sizes of limit cycles become smaller with the increasing of intensity of the random variable. And some biological explanations are given.

Stochastic chaos and its control in the stochastic Bonhoeffer-Van der Pol system

In this paper, the stochastic chaos of the stochastic Bonhoeffer-Van der Pol system with bounded random parameter and its control by noise are investigated. The stochastic Bonhoeffer-Van der Pol system is first transformed into an equivalent deterministic system using the Chebyshev polynomial approximation, so that the problem of controlling stochastic chaos is reduced to the problem of controlling deterministic chaos in the equivalent system. Thus, the top Lyapunov exponent of the equivalent system can be used to examine the chaotic behavior and its control of the responses. The numerical simulations show that the chaos behavior in the stochastic Bonhoeffer-Van der Pol system is by and large similar to that in the equivalent deterministic Bonhoeffer-Van der Pol system. The chaos behavior can be controlled to the periodic states by noise, but under the effect of the random parameter and its intensity, it has cortain specific features.