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A review of the basic Hansen's ideal frame algorithms for accurate numerical integration of perturbed elliptic motion is carried out. The fundamental approaches rely on the use of nonsingular variables and differ in the ways in which the ellipse in the orbital plane is determined. It is well known that the accuracy of the propagation of the orbit geometry is notably increased when using time-regularization techniques to transform the independent variable. However, this is at the expense of adding a differential equation to compute the time, which gathers the Lyapunov-type instabilities that are removed from the coordinates. The asynchronism resulting from errors in the numerical integration of the time may be palliated with the use of time elements, to which end a constant and a linear nonsingular time element are presented, which are new to our knowledge. [ABSTRACT FROM AUTHOR]

This article is devoted to study the cluster synchronization for a kind of complex dynamical networks consisting of nonidentical nonlinear Lur’e systems. Different from general time delays, a proportional delay is taken into consideration in this article, which is a kind of unbounded time-varying delays and thus largely increases the difficulty in network synchronization. In consideration of the topology structures of the Lur’e networks, an effective impulsive pinning controller is proposed, which will be placed on the Lur’e systems having directed paths with those in the other clusters. Considering different functional roles that the impulsive effects play, sufficient criteria for the cluster synchronization of the nonidentically coupled Lur’e dynamical networks are obtained by applying the proportionally delayed impulsive comparison principle, the concept of average impulsive interval, and the extended parameters variation formula. Simultaneously, the exponential convergence rates are successfully estimated with respect to different functions of the impulsive effects. In the end of this article, three numerical simulations are proposed to denote the effectiveness of the control protocols and the theoretical results. [ABSTRACT FROM AUTHOR]

This paper investigates the exponential synchronization of coupled Lur’e dynamical networks with multiple time-varying delays and derivative coupling. In order to synchronize the Lur’e dynamical networks to the corresponding Lur’e systems, we propose a kind of impulsive pinning control strategy, where different functions of impulsive effects are taken into account. Sufficient conditions are derived for the exponential synchronization of the derivative coupled Lur’e dynamical networks by jointly applying the contradiction proof method, the concept of an average impulsive interval, comparison principle, and the extended parameters variation formula. Simultaneously, the convergence rates of exponential synchronization are obtained according to the definition of the impulsive solution equation on different functions of impulsive effects. Furthermore, three numerical examples are presented to demonstrate the validity of the theoretical analysis and the control protocol. [ABSTRACT FROM AUTHOR]

Nonlinear problems arise in many heat transfer applications, and several analytical and numerical methods for solving these problems are described in the literature. Here, the method of variation of parameters is shown to be a relatively simple method for obtaining solutions to four specific heat transfer problems: 1. a radiating annular fin, 2. conduction-radiation in a plane-parallel medium, 3. convective and radiative exchange between the surface of a continuously moving strip and its surroundings, and 4. convection from a fin with temperature-dependent thermal conductivity and variable cross-sectional area. The results for each of these examples are compared to those obtained using other analytical and numerical methods. The accuracy of the method is limited only by the accuracy with which the numerical integration is performed. The method of variation of parameters is less complex and relatively easy to implement compared to other analytical methods and some numerical methods. It is slightly more computationally expensive than traditional numerical approaches. The method presented may be used to verify numerical solutions to nonlinear heat transfer problems. [ABSTRACT FROM AUTHOR]

Abstract: Given a basis of solutions to k ordinary linear differential equations , we show how Green''s functions can be used to construct a basis of solutions to the homogeneous differential equation , where ℓ is the composite product . The construction of these solutions is elementary and classical. In particular, we consider the special case when . Remarkably, in this case, if is a basis of , then our method produces a basis of for any . We illustrate our results with several classical differential equations and their special function solutions. [Copyright &y& Elsevier]

Abstract: The problem that motivates the considerations here is the construction of mathematical models of natural phenomena that depend upon past states. The paper divides naturally into two parts: in the first, we expound the inter-connection between ordinary differential equations, delay-differential equations, neutral delay-differential equations and integral equations (with emphasis on certain linear cases). As we show, this leads to a natural hierarchy of model complexity when such equations are used in mathematical and computational modelling, and to the possibility of reformulating problems either to facilitate their numerical solution or to provide mathematical insight, or both. Volterra integral equations include as special cases the others we consider. In the second part, we develop some practical and theoretical consequences of results given in the first part. In particular, we consider various approaches to the definition of an adjoint, we establish (notably, in the context of sensitivity analysis for neutral delay-differential equations) rôles for well-defined adjoints and ‘quasi-adjoints’, and we explore relationships between sensitivity analysis, the variation of parameters formulae, the fundamental solution and adjoints. [Copyright &y& Elsevier]

This paper investigates the problem of estimating synchronization errors and its application to uniform synchronization with an error bound for the general master-slave chaos synchronization scheme via feedback control, which is subjected to disturbances by unknown but bounded channel noise. Based on the Lyapunov function and nonlinear parametric variation techniques, estimation formulae for synchronization errors are derived. It is possible to synchronize two master-slave chaotic systems with a relatively small error bound, even in the case with unknown but bounded noisy disturbances. After the theoretical analysis, some representative examples and their numerical simulations are given for illustration. [ABSTRACT FROM AUTHOR]

We define the principal matrix solution Z(t, s) of the linear Volterra vector integro-differential equation x′(t) = A(t)x(t) + ∫st B(t, u)x(u)du in the same way that it is defined for x′ = A(t)x and prove that it is the unique matrix solution of ∂/∂t Z(t, s) = A(t)Z(t, s) + ∫st B(t, u)Z(u, s)du, Z(s, s)=I. Furthermore, we prove that the solution of x′(t) = A(t)x(t) + ∫τt B(t, u)x(u)du + f(t), x(τ) = x0 is unique and given by the variation of parameters formula x(t) = Z(t, τ)x0 + ∫τt Z(t, s)f(s)ds. We also define the principal matrix solution R(t, s) of the adjoint equation r′(s) = -r(s)A(s) - ∫st r(u)B(u, s)du and prove that it is identical to Grossman and Miller's resolvent, which is the unique matrix solution of ∂/∂t R(t, s) = -R(t, s)A(s) - ∫st R(t, u)B(u, s)du, R(t, t)=I. Finally, we prove that despite the difference in their definitions R(t, s) and Z(t, s) are in fact identical. [ABSTRACT FROM AUTHOR]

Abstract: In spite of being a classical method for solving differential equations, the method of variation of parameters continues having a great interest in theoretical and practical applications, as in astrodynamics. In this paper we analyse this method providing some modifications and generalised theoretical results. Finally, we present an application to the determination of the ephemeris of an artificial satellite, showing the benefits of the method of variation of parameters for this kind of problems. [Copyright &y& Elsevier]

This paper studies absolute stability of Lurie systems with impulsive effects. Using the method of Lyapunov functions and the variation of parameters technique, we establish sufficient conditions for absolute stability. [Copyright &y& Elsevier]