System Of Deterministic Equations Research Articles

Chaos synchronization of two stochastic Duffing oscillators by feedback control

This paper addresses chaos synchronization of two identical stochastic Duffing oscillators with bounded random parameters subject to harmonic excitations. In the analysis the stochastic Duffing oscillator is first transformed into an equivalent deterministic nonlinear system by Gegenbauer polynomial approximation, so that the chaos synchronization problem of stochastic Duffing oscillators can be reduced into that of the equivalent deterministic systems. Then a feedback control strategy is adopted to synchronize chaotic responses of two identical equivalent deterministic systems under different initial conditions. The feedback parameters are determined through analysis of the top Lyapunov exponent of the variational equation of the controlled responding system. Numerical analysis shows that the feedback control strategy is an effective way to synchronize two identical stochastic Duffing systems.

Stochastic chaos in stochastic Duffing systems and its control by delayed feedback

The problem of stochastic chaos and its control by delayed feedback in a Duffing system with bounded random parameters (a stochastic Duffing system in short) under harmonic excitations is considered in detail. At first, the stochastic Duffing system is transformed into its equivalent deterministic nonlinear system by the Gegenbauer polynomial approximation. Thus, the problem of chaotic response and its control in stochastic Duffing system can be reduced to that in an equivalent deterministic system. So the available effective mathematical methods and control strategies can be applied to the latter. Then, the main feature of stochastic chaos is fully explored, where the top Lyapunov exponent of the equivalent system obtained by Wolf's algorithm is used to identify the dynamic behavior of stochastic Duffing system. Finally, the control strategy of delayed feedback is applied to suppress or to induce chaotic response in the system. The results of numerical simulation show that by proper choice of feedback intensity and time delay, either suppressing or inducing stochastic chaos can be achieved. Hence, the strategy of delayed feedback control is also effective to stochastic chaos.

An efficient computational method for statistical moments of Burger's equation with random initial conditions

The paper is concerned with efficient computation of numerical solutions to Burger's equation with random initial conditions. When the Lax-Wendroff scheme (LW) is expanded using the Wiener chaos expansion (WCE), random and deterministic effects can be separated and we obtain a system of deterministic equations with respect to Hermite-Fourier coefficients. One important property of the system is that all the statistical moments of the solution to the Burger's equation can be computed using the solution of the system only. Thus LW with WCE presents an alternative to computing moments by the Monte Carlo method (MC). It has been numerically demonstrated that LW with WCE approach is equally accurate but substantially faster than MC at least for certain classes of initial conditions.

Stochastic chaos in a Duffing oscillator and its control

Stochastic chaos discussed here means a kind of chaotic responses in a Duffing oscillator with bounded random parameters under harmonic excitations. A system with random parameters is usually called a stochastic system. The modifier ‘stochastic’ here implies dependent on some random parameter. As the system itself is stochastic, so is the response, even under harmonic excitations alone. In this paper stochastic chaos and its control are verified by the top Lyapunov exponent of the system. A non-feedback control strategy is adopted here by adding an adjustable noisy phase to the harmonic excitation, so that the control can be realized by adjusting the noise level. It is found that by this control strategy stochastic chaos can be tamed down to the small neighborhood of a periodic trajectory or an equilibrium state. In the analysis the stochastic Duffing oscillator is first transformed into an equivalent deterministic nonlinear system by the Gegenbauer polynomial approximation, so that the problem of controlling stochastic chaos can be reduced into the problem of controlling deterministic chaos in the equivalent system. Then the top Lyapunov exponent of the equivalent system is obtained by Wolf’s method to examine the chaotic behavior of the response. Numerical simulations show that the random phase control strategy is an effective way to control stochastic chaos.

Symmetry-breaking bifurcation analysis of stochastic van der pol system via Chebyshev polynomial approximation

Chebyshev polynomial approximation is applied to the symmetry-breaking bifurcation problem of a stochastic van der Pol system with bounded random parameter subjected to harmonic excitation. The stochastic system is reduced into an equivalent deterministic system, of which the responses can be obtained by numerical methods. Nonlinear dynamical behaviors related to various forms of stochastic bifurcations in stochastic system are explored and studied numerically.

The use of the moment method to derive the gas and plasma transport equations with transport coefficients in higher-order approximations

Grad's moment method is used to derive the linear equations of mass momentum and energy transfer of the components and to obtain all the transport coefficients (kinetic coefficients) for a multicomponent mixture of monatomic gases. A system of equations for the expansion coefficients of the non-equilibrium correction to the distribution function using a system of irreducible tensorial Hermite polynomials (the equations of moments) is obtained on the basis of the linearized Boltzmann equations for the components of the mixture. The assumptions under which these equations reduce to a system of algebraic equations for determining of the mass diffusion fluxes, the heat fluxes of the components and the partial viscous stress tensors are analysed. The advantage in writing the transport equation in a “forces in terms of fluxes” representation for solving actual problems of the flows of multicomponent mixtures as compared with the classical “fluxes in terms of forces” representation in the standard Chapman-Enskog method [1–3] is demonstrated. Different ways of representing the transport equations and expressions for the transport coefficients are considered in an arbitrary order of approximation with respect to the number of Sonine polynomials which are retained in the expansion of the distribution function (the Chapman-Cowling method). This enables one, in particular, to establish a direct link between results obtained by different methods and to track more clearly the constraints which are actually used when employing classical method [1–3] and the modified method [4, 5] of deriving the transport equations and calculating the transport coefficients in the Chapman-Enskog scheme

Feedback linearization control design for systems with fuzzy uncertainty

We derive a state feedback control design for nonlinear systems with fuzzy uncertainty. We use a fuzzy version of the Kolmogorov forward equation and an equivalent deterministic system to represent the effects of the fuzzy uncertainty. Feedback linearization of the equivalent deterministic system cancels both the system nonlinearities and the effects of the fuzzy uncertainty on the state membership function. Our controller obtains asymptotic stability and an approximation to our controller drives the system state to a small bounded set.

A stochastic control algorithm for systems with control dependent plant and measurement noise

The problem of control of a stochastic system with control dependent noise is investigated. Both the system driving noise and system measurement noise have control dependent terms. The approximation of neglecting higher order terms in the control is introduced in the derivation of the a posteriori conditional covariance matrices of the system state. These approximate conditional covariance matrices are used in a stochastic dynamic programming algorithm to obtain a linear feedback law. The control feedback matrices multiplying the estimates for system state are not those of the equivalent deterministic system and the separation theorem in its usual form does not apply to this class of problems. The control policy given by the algorithm reduces to the optimal control policy in two cases in which the optimal control is known. These are the case of no control dependent noise and the case of no measurement noise. A third order numerical example is investigated to illustrate the nature of the algorithm.

Continuous flow systems with stochastic transfer functions

Continuous flow systems which produce a random output signal from a deterministic tracer signal at the input require a stochastic transfer function for their description. In many cases stochastic differential equations are appropriate for the description of these systems. The parameters in these equations are random variables or stochastic processes rather than deterministic constants. The behaviour of the continuous stirred tank is analysed on this basis with both white noise and first order equilibrium Markov processes perturbing the system. Expressions are derived for the mean response and the autocorrelation function of the output signal. The mean response does not degenerate into the response of the equivalent deterministic system but depends strongly on the statistical properties of the perturbing signal.

Minimum Effort Control Of Several Terminal Components

The stochastic control problem of minimizing the total average velocity correction with several prescribed terminal variances in the presence of random injection and measurement errors is considered. It is shown that, for the case of linear feedback, this can be formulated as an optimization problem for an equivalent deterministic system whose states are the covariances of the predicted miss. However, the deterministic optimization problem is “degenerate” causing some difficulty in the computation of the feedback gain. It is shown that the optimum linear corrective strategy is, in general, discontinuous and consists of an initial period of no control, followed by a period of continuous control and finally a period of no control and possibly an impulse at the end. Equations are derived from which the variable feedback gain and the various time intervals can be computed. Two simple examples involving (1) the control of two terminal position components, and (2) the control of both the terminal position and t...