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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]