Studies of Melnikov Method and Transversal Homoclinic Orbits in the Circular Planar Restricted Three-Body Problem

Zhu Ruzeng; Xiang Cheng

Article Contents

Article Navigation > Applied Mathematics and Mechanics > 1996 > 17(12): 1113-1122

Zhu Ruzeng, Xiang Cheng. Studies of Melnikov Method and Transversal Homoclinic Orbits in the Circular Planar Restricted Three-Body Problem[J]. Applied Mathematics and Mechanics, 1996, 17(12): 1113-1122.

Citation:

PDF( 615 KB)

Studies of Melnikov Method and Transversal Homoclinic Orbits in the Circular Planar Restricted Three-Body Problem

Zhu Ruzeng¹,
Xiang Cheng²

1.
LNM, Institute of Mechanics, Academia Sinica, Beijing 100080, P. R. China;
2.
Department of Mechanical Engineering, Yale University, New Haven, CT 06520, U. S. A

Received Date: 1995-07-23
Rev Recd Date: 1996-04-01
Publish Date: 1996-12-15

Abstract

Abstract

Non-Hamiltonian systems containing degenerate fixed points obtained from twodegrees of freedom near-integrable Hamiltonian systems through non-canonicaltransformations are dealt with in this paper. Two criteria.for determining theexistence of transversal homoclinic and heteroclinic orbits are presented. By exploitingthese criteria the existence of the transversal homoclinic orbits and so, of thetransversal homoclinic tangle.phenomenon in the near-integrable circular planarrestricted three-body problem with sufficiently small mass ratio of the two primaries isproven. Under some assumptions, the existence of the transversal heleroclinic orbits isproven. The global qualitative phase diagram is also illustrated.
- restricted three-body problem,
- near integrable Hamiltoniansystem,
- degenerate fixed point,
- Melnikov method,
- transversalhomoclinic(heteroclinic)orbit

FullText(HTML)

References(9)

References

[1]	A.Wintner,The Analytical Foundations of Celestial Mechanics,Princeton Univ.Press(1947).
[2]	T.M.Cherry,On Poincare's theorem of "the non-existence of uniform integrals of dynamical equation",Proc.Cambridge Phil.Soc.,22(1924).
[3]	V.Szebehely,Long Time Prediction in Dynamics,Ed.by W.Horton and L.Richl,Wiley,Austin(1981).
[4]	V.Szebehely,Is celestial mechanics deterministicϒ Applications of Modern Dynamics to Celestial Mechanics and Astrodynamics,Ed.by Szebehely,D.Reidel Pub.Company(1982),321-324.
[5]	R.Easton and R.McGehee,Homoclinic phenomena for orbits doubly asymptotic to an invariant three-sphere,Indiana U.Math.J.,28(1979),211-240.
[6]	R.Easton,Capture orbits and Melnikov integrals in the planar three-body problem,Cele.Mech.and Dyn.Astro.,50(1991),283-297.
[7]	R.McGehee,A stable manifold theorem for degenerate fixed points with application to celestial mechanics,J.DiJJ.Eq.,14(1973),70-88.
[8]	J.Guckenheimer and P.Holmes,Nonlinear Oscillations,Dynamical Systems,and Bifurcations of Vector Fields,Springer-Verlag(1983).
[9]	S.Wiggins,Global Bifurcations and Chaos,Springer-Verlag(1988).