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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]