Multiple Periodic Orbits in the Hill Problem with Oblate Secondary

Frozen orbits of the Hill problem are determined in the double-averaged problem, where short and long-period terms are removed by means of Lie transforms. Due to the perturbation method we use, the initial conditions of corresponding quasi-periodic solutions in the nonaveraged problem are computed straightforwardly. Moreover, the method provides the explicit equations of the transformation that connects the averaged and nonaveraged models. A fourth-order analytical theory is necessary for the accurate computation of quasi-periodic frozen orbits.

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On a One-Parameter Discrete-Time Z4-Equivariant Cubic Dynamical System

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127419500524 ◽

2019 ◽

Vol 29 (04) ◽

pp. 1950052

Author(s):

Armands Gritsans

Keyword(s):

Dynamical System ◽

Periodic Orbits ◽

Discrete Time ◽

Chaotic Behavior ◽

Invariant Curves ◽

Discrete Time System ◽

Time System ◽

Multiple Periodic Orbits ◽

The One ◽

Planar Hamiltonian System

The lemniscate sine and cosine are solutions of a [Formula: see text]-equivariant planar Hamiltonian system for all of which nontrivial solutions are nonhyperbolic periodic orbits. The forward Euler scheme is applied to this system and the one-parameter discrete-time [Formula: see text]-equivariant cubic dynamical system is obtained. The discrete-time system depending upon a parameter exhibits rich dynamics: numerical simulation shows that the system has attracting closed invariant curves, multiple periodic orbits and attracting sets exhibiting chaotic behavior. The approximating system of ordinary differential equations is constructed. We discuss the existence of closed invariant curves for the discrete-time system.

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Numerical Investigation for Periodic Orbits in the Hill Three-Body Problem

Universe ◽

10.3390/universe6060072 ◽

2020 ◽

Vol 6 (6) ◽

pp. 72 ◽

Cited By ~ 1

Author(s):

Vassilis S. Kalantonis

Keyword(s):

Periodic Orbits ◽

Body Problem ◽

Numerical Study ◽

Three Dimensional ◽

Earth Satellite ◽

Mission Design ◽

Special Importance ◽

Three Body Problem ◽

Three Body ◽

The Hill

The current work performs a numerical study on periodic motions of the Hill three-body problem. In particular, by computing the stability of its basic planar families we determine vertical self-resonant (VSR) periodic orbits at which families of three-dimensional periodic orbits bifurcate. It is found that each VSR orbit generates two such families where the multiplicity and symmetry of their member orbits depend on certain property characteristics of the corresponding VSR orbit’s stability. We trace twenty four bifurcated families which are computed and continued up to their natural termination forming thus a manifold of three-dimensional solutions. These solutions are of special importance in the Sun-Earth-Satellite system since they may serve as reference orbits for observations or space mission design.

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