The problem of satellite orbital evolution with the combined influence of a distant perturbing body and the planet oblateness is well known (Laplace, 1805; Lidov, 1962, 1973; Kozai, 1963; Kudielka, 1994, 1997). The case of near-circular orbits is investigated in more details in (Sekiguchi, 1961; Allan and Cook, 1964; Vashkovyak, 1974).

We study the normal variational equation of degree five:

where λ is a parameter and C5(t) verifies

The solution C5(t) of (1.2) is the inverse function of

We prove that the pole t5 of C5(t) defined by is given by

where

Scheifele (1970) applied Delaunay-Similar (DS) elliptic Keplerian elements (with the true anomaly as the independent variable) to the J2 Problem in Artificial Satellite Theory, making an element of the true anomaly. Deprit (1981) views Scheifele’s TR-mapping as an extension of Hill’s transformation from a 6-dimensional phase space to an enlarged, 8-dimensional one. To adapt this approach to elliptic-type two-body problems with a time-varying Keplerian parameter μ(t), Floía (1997, §3, §4) treated a Gylden system (Deprit 1983) and derived “Delaunay-Similar” variables via a TR-like transformation. Now we extend our treatment to perturbed Gylden systems, and modify the TR-map to deal with any kind of two-body orbit. We work out our generalization and the resulting variables within a unified pattern whatever the type of motion, in terms of universal functions (Stiefel & Scheifele 1971, §11; Battin 1987, §4.5, §4.6) and auxiliary angle-like parameters.

The set of orbits of the Two Fixed Centres problem has been known for a long time (Chartier, 1902, 1907; Pars, 1965), since it is an integrable Hamiltonian system.

We consider a plane that contains the fixed masses. Denote by φ the angle denned by this plane and the one that contains also the third body. The momentum pφ is a first integral of the system and when pφ is different from zero, the manifold generated by the generalized coordinates and momenta are two copies of the three-dimensional sphere S3. If pφ = 0, that is to say when the planet crosses the line joining both suns, the motion is restricted to a planar one. All the equilibrium points appears in this case and therefore the phase spaces are more complex. We restrict our attention to this case which has two degrees of freedom.

It is again a Bott-integrable Hamiltonian system. The set of periodic orbits of this systems can be studied from a subset of them, the Non-Singular Morse-Smale type orbits (see Casasayas, 1992). It is proved in Campos (1997) that a small perturbation of a Bott-integrable Hamiltonian system transforms it into a Non-Singular Morse-Smale system. The NMS periodic orbits belong to both the NMS system and the Hamiltonian one. Moreover, The NMS p.o. can be continued to nearly Hamiltonian systems. For instance, in our case to the Restricted Three Body Problem and in the study of the motion of a material point moving inside the gravitational field generated by two stars. This approximation is also useful when the motion of an artificial satellite around a spheroidal body is considered.

The temporal structure of chaos in three-body dynamics is analyzed; the emphasis is made on a similarity and difference between three-body chaos and basic patterns of chaotic behaviour known in nonlinear physics.

1. With the use of homology mapping (Agekian and Anosova 1967), we study a set of computer models of thee-body systems in a stationary spherically symmetric potential well (Valtonen et al. (1994); the well confines the bodies, and because of this the system can generate fairly long time series. Typical time series reveal sequences of seemingly periodic motion and short bursts of strong chaos that appear in an irregular manner (Heinämäki et al. 1998). The quasi-ordered states are associated with hierarchical homology, and the quasi-period of the low-amplitude oscillations is very near the period of the temporary close binary in the system. The high-amplitude irregular states are mostly due to active three-body interplay when each of the bodies interacts with the two others with almost equal intensity. In the evolutionary history of most systems, these two extreme kinds of states alternate in an apparently random way producing together a non-stationary pattern of unpredictable behaviour.