A new understanding of L4 and L5 axial orbits through the torus structure

2020 ◽  
Vol 498 (4) ◽  
pp. 5343-5352
Author(s):  
Yi Qi ◽  
Anton de Ruiter

ABSTRACT In this paper, through the critical isosurface of the pseudo-Hamiltonian of co-orbital motions in the torus space, we provide a new understanding of L4 and L5 axial orbits and their invariant manifolds in the circular restricted three-body problem. The contact points on the critical isosurface of the pseudo-Hamiltonian correspond to the locations of L4 and L5 axial orbits in the torus space, and provide a set of good initial guesses of L4 and L5 axial orbits for the multiple shooting method. Furthermore, we calculate and analyse orbital behaviours of L4 and L5 axial orbit families. Based on the topological structure of the critical isosurface of the pseudo-Hamiltonian, compound dynamical motions of invariant manifolds associated with L4 and L5 axial orbits are discussed. We present an approximate estimation for libration amplitudes of different co-orbital portions of invariant manifolds. Results obtained from numerical integration demonstrate the validity of our semi-analytical approach in the torus space..

2007 ◽  
Vol 17 (04) ◽  
pp. 1151-1169 ◽  
Author(s):  
MARIAN GIDEA ◽  
JOSEP J. MASDEMONT

The stable and unstable invariant manifolds associated with Lyapunov orbits about the libration point L1between the primaries in the planar circular restricted three-body problem with equal masses are considered. The behavior of the intersections of these invariant manifolds for values of the energy between that of L1and the other collinear libration points L2, L3is studied using symbolic dynamics. Homoclinic orbits are classified according to the number of turns about the primaries.


Nonlinearity ◽  
2004 ◽  
Vol 17 (5) ◽  
pp. 1571-1606 ◽  
Author(s):  
G Gómez ◽  
W S Koon ◽  
M W Lo ◽  
J E Marsden ◽  
J Masdemont ◽  
...  

2013 ◽  
Vol 12 (4) ◽  
pp. 225-232
Author(s):  
Ryszard Hołubowski ◽  
Andrzej Merena

The application of multiple shooting method in stability analysis of non-prismatic multi-segment columns with pinned ends loaded with a concentrated force applied to the upper node has been presented. Numerical analyses were carried out for an exemplary three-segment column by solving the system of differential equations with variable coefficients and parameter. The results were compared with the solution obtained by using SOFiSTiK software based on the finite element method. The analyses show that considering the stiffness changes along the length can have a significant influence on the values of critical loads and thus change the resistance of the column. The advantage of the proposed method is its high efficiency and easy description of stiffness changes.


Author(s):  
David Demailly ◽  
Fabrice Thouverez ◽  
Louis Jézéquel ◽  
Jérôme Bonini

Abstract In this paper the Multiple Shooting Method is briefly exposed. This method is then applied to a simplified model of a rotor/stator system including a bearing with clearance under static and unbalance forces. With bearing having a small amount of clearance, it is pointed out that the orbits are no longer circular at certain frequencies as for linear systems. If the clearance is large, the frequency response curve can be divided into three zones: a non-contact zone, a zone with one or several gains and looses of contact per period and a full contact zone. For both cases we emphasize on the evolution of the orbits while passing through transition zones.


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