Multiple Periodic Orbits Connecting a Collinear Configuration and a Double Isosceles Configuration in the Planar Equal-Mass Four-Body Problem

2017 ◽  
Vol 17 (4) ◽  
pp. 819-835 ◽  
Author(s):  
Bixiao Shi ◽  
Rongchang Liu ◽  
Duokui Yan ◽  
Tiancheng Ouyang
Keyword(s):  
Periodic Orbit ◽  
Variational Method ◽  
Periodic Orbits ◽  
Body Problem ◽  
Equal Mass ◽  
Local Action ◽  
Collision Singularity ◽  
Four Body Problem ◽  
Multiple Periodic Orbits ◽  
The One

AbstractBy applying our variational method, we show that there exist 24 local action minimizers connecting two prescribed configurations: a collinear configuration and a double isosceles configuration in {H^{1}([0,1],\chi)} in the planar equal-mass four-body problem. Among the 24 local action minimizers, we prove that the one with the smallest action has no collision singularity and it can be extended to a periodic or quasi-periodic orbit. Furthermore, if all the 24 local action minimizers are free of collision, we show that they can generate sixteen different periodic orbits.

GLOBAL BEHAVIOR OF THE CHARGED ISOSCELES THREE-BODY PROBLEM

1994 ◽  
Vol 04 (04) ◽  
pp. 865-884 ◽  
Author(s):  
PAU ATELA ◽  
ROBERT I. McLACHLAN
Keyword(s):  
Bifurcation Diagram ◽  
Periodic Orbits ◽  
Body Problem ◽  
Kepler Problem ◽  
Global Bifurcation ◽  
Three Body Problem ◽  
Anisotropic Kepler Problem ◽  
The One ◽  
Three Body ◽  
Limiting Value

We study the global bifurcation diagram of the two-parameter family of ODE’s that govern the charged isosceles three-body problem. (The classic isosceles three-body problem and the anisotropic Kepler problem (two bodies) are included in the same family.) There are two major sources of periodic orbits. On the one hand the “Kepler” orbit, a stable orbit exhibiting the generic bifurcations as the multiplier crosses rational values. This orbit turns out to be the continuation of the classical circular Kepler orbit. On the other extreme we have the collision-ejection orbit which exhibits an “infinite-furcation.” Up to a limiting value of the parameter we have finitely many periodic orbits (for each fixed numerator in the rotation number), passed this value there is a sudden birth of an infinite number of them. We find that these two bifurcations are remarkably connected forming the main “skeleton” of the global bifurcation diagram. We conjecture that this type of global connection must be present in related problems such as the classic isosceles three-body problem and the anisotropic Kepler problem.

Periodic orbits in the restricted four-body problem

1986 ◽  
Vol 13 (8) ◽  
pp. 473-479 ◽  
Author(s):  
K.C. Howell ◽  
D.B. Spencer
Keyword(s):  
Periodic Orbits ◽  
Body Problem ◽  
Four Body Problem

scholarly journals New periodic orbits in the planar equal-mass three-body problem

2018 ◽  
Vol 38 (4) ◽  
pp. 2187-2206
Author(s):  
Rongchang Liu ◽  
◽  
Jiangyuan Li ◽  
Duokui Yan
Keyword(s):  
Periodic Orbits ◽  
Body Problem ◽  
Equal Mass ◽  
Three Body Problem ◽  
Three Body

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

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.

The Existence of Retrograde Orbits for the Four-Body Problem with Various Choices of Masses

2013 ◽  
Vol 871 ◽  
pp. 101-106
Author(s):  
Chong Li
Keyword(s):  
Variational Methods ◽  
Periodic Orbits ◽  
Opposite Direction ◽  
Body Problem ◽  
Topological Type ◽  
The Other ◽  
Action Functional ◽  
Four Body Problem

In this paper, we study the planar Newtonian four-body problem with various choices of masses. We prove that there exist infinitely many periodic and quasi-periodic orbits with certain topological type, called retrograde orbits, that minimize the action functional on certain path spaces. On these orbits, two particles revolve around each other in one direction, while the other two particles travel on themselves orbits in opposite direction, respectively. Our proof is based on variational methods inspired by the work of Kuo-Chang Chen.

Global Regularization of a Restricted Four-Body Problem

2014 ◽  
Vol 24 (07) ◽  
pp. 1450092 ◽  
Author(s):  
Martha Alvarez-Ramírez ◽  
Joaquín Delgado ◽  
Claudio Vidal
Keyword(s):  
Numerical Integration ◽  
Body Problem ◽  
Binary Collision ◽  
Type Transformation ◽  
Collision Singularity ◽  
Double Collision ◽  
N Body Problem ◽  
Four Body Problem ◽  
Collision Orbits ◽  
Global Regularization

In the n-body problem, a collision singularity occurs when the position of two or more bodies coincide. By understanding the dynamics of collision motion in the regularized setting, a better understanding of the dynamics of near-collision motion is achieved. In this paper, we show that any double collision of the planar equilateral restricted four-body problem can be regularized by using a Birkhoff-type transformation. This transformation has the important property to provide a simultaneous regularization of three singularities due to binary collision. We present some ejection–collision orbits after the regularization of the restricted four-body problem (RFBP) with equal masses, which were obtained by numerical integration.

Periodic orbits and bifurcations in the Sitnikov four-body problem

2008 ◽  
Vol 100 (4) ◽  
pp. 251-266 ◽  
Author(s):  
P. S. Soulis ◽  
K. E. Papadakis ◽  
T. Bountis
Keyword(s):  
Periodic Orbits ◽  
Body Problem ◽  
Four Body Problem
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