Elbaz I. Abouelmagd

Reduction the secular solution to periodic solution in the generalized restricted three-body problem

The aim of the present work is to find the secular solution around the triangular equilibrium points and reduce it to the periodic solution in the frame work of the generalized restricted thee-body problem. This model is generalized in sense that both the primaries are oblate and radiating as well as the gravitational potential from a belt. We show that the linearized equation of motion of the infinitesimal body around the triangular equilibrium points has a secular solution when the value of mass ratio equals the critical mass value. Moreover, we reduce this solution to periodic solution, as well as some numerical and graphical investigations for the effects of the perturbed forces are introduced. This model can be used to examine the existence of a dust particle near the triangular points of an oblate and radiating binary stars system surrounded by a belt.

Dynamics of a dumbbell satellite under the zonal harmonic effect of an oblate body

Abstract The aim of the present paper is to study the dynamics of a dumbbell satellite moving in a gravity field generated by an oblate body considering the effect of the zonal harmonic parameter. We prove that the pass trajectory of the mass center of the system is periodic and different from the classical one when the effect of the zonal harmonic parameter is non zero. Moreover, we complete the classical theory showing that the equations of motion in the satellite approximation can be reduced to Beletsky’s equation when the zonal harmonic parameter is zero. The main tool for proving these results is the Lindstedt–Poincare’s technique.

Mode-mismatched estimator design for Markov jump genetic regulatory networks with random time delays

In this paper, the problem of H ∞ state estimation is investigated for a class of discrete-time Markov jump genetic regulatory networks (GRNs) with random time delays. A mismatching characteristic of modes jumping between GRNs modes and desired mode-dependent estimators is recognized, and a nonstationary mode transition among the estimators is used to model the mismatching characteristic of modes jumping to different degrees. The time delays are supposed to be time-varying and subject to another Markov chain. By using the linear matrix inequality techniques, sufficient conditions on the existence of the estimators with mismatching characteristic of modes jumping are first derived such that the resulting estimation error system is stochastically stable with a prescribed H ∞ performance index. One interesting phenomenon is disclosed, i.e., the optimal performance index varies monotonously as changing the mismatching degrees of modes jumping. A numerical example is exploited to illustrate the effectiveness of the theoretical findings.

Periodic Orbits of the Planar Anisotropic Kepler Problem

In this paper, we prove that at every energy level the anisotropic Kepler problem with small anisotropy has two periodic orbits which bifurcate from elliptic orbits of the Kepler problem with high eccentricity. Moreover we provide approximate analytic expressions for these periodic orbits. The tool for proving this result is the averaging theory.

On the libration collinear points in the restricted three – body problem

Abstract In the restricted problem of three bodies when the primaries are triaxial rigid bodies, the necessary and sufficient conditions to find the locations of the three libration collinear points are stated. In addition, the Linear stability of these points is studied for the case of the Euler angles of rotational motion being θi = 0, ψi + φi = π/2, i = 1, 2 accordingly. We underline that the model studied in this paper has special importance in space dynamics when the third body moves in gravitational fields of planetary systems and particularly in a Jupiter model or a problem including an irregular asteroid.

Numerical integration of a relativistic two-body problem via a multiple scales method

We offer an analytical study on the dynamics of a two-body problem perturbed by small post-Newtonian relativistic term. We prove that, while the angular momentum is not conserved, the motion is planar. We also show that the energy is subject to small changes due to the relativistic effect. We also offer a periodic solution to this problem, obtained by a method based on the separation of time scales. We demonstrate that our solution is more general than the method developed in the book by Brumberg (Essential Relativistic Celestial Mechanics, Hilger, Bristol, 1991). The practical applicability of this model may be in studies of the long-term evolution of relativistic binaries (neutron stars or black holes).

Analytical Study of Periodic Solutions on Perturbed Equatorial Two-Body Problem

This paper presents analytical derivations to study periodic solutions for the two-body problem perturbed by the first zonal harmonic parameter. In particular, three different semianalytical approaches to solve this problem have been studied: (1) the classic perturbation theory, (2) the Lindstedt–Poincare technique, and (3) the Krylov–Bogoliubov–Mitropolsky method. In addition, the numerical integration by Runge–Kutta algorithm is established. However, the numerical comparison tests show that by increasing the value of angular momentum the solutions provided by Lindstedt–Poincare and Krylov–Bogoliubov–Mitropolsky methods become similar, and they provide almost identical results using a smaller value for the perturbed parameter which quantify the dynamical flattening of the main body, the Krylov–Bogoliubov–Mitropolsky provides more accurate results to design elliptical periodic solutions than Lindstedt–Poincare technique when the perturbed parameter has a relatively large value, regardless of the value of angular momentum. This study can be applied to equatorial orbits to obtain closed-form analytical solutions.

A First Order Automated Lie Transform

The objective of the present paper is to focus on the problem of the normalization of a Hamiltonian system via the elimination of angle variables involved using the Lie transform technique. The algorithm that we construct assumes that the Hamiltonian is periodic in n angle variables, with two rates: fast and slow. If the angle variables have the same rate only one transformation is required. The equations needed to evaluate the elements of each transformation and the secular perturbations are constructed.