We study periodic orbits of Hamiltonian differential systems with three degrees of freedom using the averaging theory. We have chosen the classical integrable Hamiltonian system with the Hooke potential and we study periodic orbits which bifurcate from the periodic orbits of the integrable system perturbed with a non–autonomous potential.

In recent work ([KW1],[KW2]), Kostant and Wallach construct an action of a simply connected Lie group A ≃ C( n 2 ) on gl(n) using a completely integrable system derived from the Poisson analogue of the Gelfand-Zeitlin subalgebra of the enveloping algebra. In [KW1], the authors show that A-orbits of dimension ( n 2 ) form Lagrangian submanifolds of regular adjoint orbits in gl(n). They describe ...

It is often assumed that a warped galaxy can be modeled by a set of rings. This paper verifies numerically the validity of this assumption by the study of periodic orbits populating a heavy self-gravitating warped disk. The phase space structure of a warped model reveals that the circular periodic orbits of a flat disk are transformed in quasi annular periodic orbits which conserve their stabil...

Recent Cassini discoveries have quickly thrust Enceladus into the science spotlight. Unfortunately, the low Enceladus mass and its proximity to Saturn lead to extreme instabilities when considering typical near polar science orbits. Here we find that the usual doubly averaged techniques are invalid for realistic Enceladus orbits. Further, it is shown that a dependence on the low order averaging...

A benchmark database of very high-precision numerical and validated initial conditions of periodic orbits for the Lorenz model is presented. This database is a “computational challenge” and it provides the initial conditions of all periodic orbits of the Lorenz model up to multiplicity 10 and guarantees their existence via computer-assisted proofs methods. The orbits are computed using high-pre...

In recent work ([9],[10]), Kostant and Wallach construct an action of a simply connected Lie group A ≃ C( n 2 ) on gl(n) using a completely integrable system derived from the Poisson analogue of the Gelfand-Zeitlin subalgebra of the enveloping algebra. In [9], the authors show that A-orbits of dimension ( n 2 ) form Lagrangian submanifolds of regular adjoint orbits in gl(n). They describe the o...

Highly elliptical orbits can be used to provide targeted satellite coverage of locations at high latitudes. We review the history of use of these orbits for communication. How elliptical orbits can be used for broadband communication is outlined. We propose an addition of known elliptical orbits to the new equatorial O3b satellite constellation, extending O3b to cover high latitudes and the Ear...

We develop and illustrate techniques to obtain periodic orbits around the second Lagrangian point L2 in the Sun-Earth system based on the Restricted Three-Body Problem. In the case of Lyapunov (planar) orbits, the solutions to the linearized equations of motion allow the generation of the entire family of orbits. For Halo orbits, however, the method of strained coordinates is applied to generat...

Powerful computational techniques have been developed to study chaotic attractors that are generated by stretching and folding processes. These include relative rotation rates for determining the organization of unstable periodic orbits and simplex distortion procedures for estimating the topological entropy of these orbits. These methods are useful for attractors contained in a genus-one torus...

The topic of autonomous close-range rendezvous is discussed in this paper. When designing a controller for an autonomous system, it is important to model the environment of the system as precisely as possible. This paper discusses the disturbances encountered in space and their effects on the rendezvous controller with particular focus on the effects in fuel usage and mission times. Simulations...