Tranversality of the invariant manifolds associated to the Lyapunov family of periodic orbits near L2 in the restricted three-body problem

Abstract The restricted three-body problem is considered for values of the Jacobi constant C near the value C 2 associated to the Euler critical point L 2 . A Lyapunov family of periodic orbits near L 2 , the so-called family ( c ), is born for C = C 2 and exists for values of C less than C 2 . These periodic orbits are hyperbolic. The corresponding invariant manifolds meet transversally along homoclinic orbits. In this paper the variation of the transversality is analyzed as a function of the Jacobi constant C and of the mass parameter μ. Asymptotical expressions of the invariant manifolds for C ≲ C 2 and μ ≳ 0 are found. Several numerical experiments provide accurate information for the manifolds and a good agreement is found with the asymptotical expressions. Symbolic dynamic techniques are used to show the existence of a large class of motions. In particular the existence of orbits passing in a random way (in a given sense) from the region near one primary to the region near the other is proved.

[1]  Jaume Llibre,et al.  Some homoclinic phenomena in the three-body problem , 1980 .

[2]  Jaume Llibre,et al.  Oscillatory solutions in the planar restricted three-body problem , 1980 .

[3]  S. Smale.  Diffeomorphisms with Many Periodic Points , 1965 .

[4]  G. M.,et al.  Partial Differential Equations I , 2023, Applied Mathematical Sciences.

[5]  D. Saari,et al.  Stable and Random Motions in Dynamical Systems , 1975 .

[6]  R. Churchill,et al.  A survey of the Hénon-Heiles Hamiltonian with applications to related examples , 1979 .

[7]  V. Alekseev,et al.  QUASIRANDOM DYNAMICAL SYSTEMS. III QUASIRANDOM OSCILLATIONS OF ONE-DIMENSIONAL OSCILLATORS , 1969 .

[8]  C. Conley,et al.  On the ultimate behavior of orbits with respect to an unstable critical point I. Oscillating, asymptotic, and capture orbits , 1969 .

[9]  V. Alekseev,et al.  QUASIRANDOM DYNAMICAL SYSTEMS. II. ONE-DIMENSIONAL NONLINEAR OSCILLATIONS IN A FIELD WITH PERIODIC PERTURBATION , 1968 .

[10]  M. E. Novodvorskii,et al.  INFINITE-DIMENSIONAL ABELIAN VARIETIES AND UNITARY REPRESENTATIONS OF GROUPS , 1968 .

[11]  A study of asymptotic solutions in the vicinity of the collinear libration points of the restricted three-body problem , 1977 .

[12]  V. Alekseev.  QUASIRANDOM DYNAMICAL SYSTEMS. I. QUASIRANDOM DIFFEOMORPHISMS , 1968 .

[13]  G. F. Bryant.  On a generalization of a theorem of Lyapunov , 1975 .

[14]  Victor Szebehely,et al.  Theory of Orbits. , 1967 .

[15]  Robert L. Devaney,et al.  Singularities in Classical Mechanical Systems , 1981 .

[16]  J. Moser,et al.  On the generalization of a theorem of A. Liapounoff , 1958 .