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HOMOCLINIC DYNAMICS IN A SPATIAL RESTRICTED FOUR BODY PROBLEM
 Date Issued:
 2021
 Abstract/Description:
 The set of transverse homoclinic intersections for a saddlefocus equilibrium in the planar equilateral restricted four body problem admits certain simple homoclinic orbits which form the skeleton of the complete homoclinic intersection, or homoclinic web. In this thesis, the planar restricted four body problem is viewed as an invariant subsystem of the spatial problem, and the influence of this planar homoclinic skeleton on the spatial dynamics is studied from a numerical point of view. Starting from the vertical Lyapunov families emanating from saddle focus equilibria, we compute the stable/unstable manifolds of these spatial periodic orbits and look for intersections between these manifolds near the fundamental planar homoclinics. In this way, we are able to continue all of the basic planar homoclinic motions into the spatial problem as homoclinics for appropriate vertical Lyapunov orbits which, by the Smale Tangle theorem, suggest the existence of chaotic motions in the spatial problem. While the saddlefocus equilibrium solutions in the planar problems occur only at a discrete set of energy levels, the cycletocycle homoclinics in the spatial problem are robust with respect to small changes in energy. The method uses high order FourierTaylor and Chebyshev series approximations in conjunction with the parameterization method, a general functional analytic framework for invariant manifolds. Tools that admit a natural notion of aposteriori error analysis. Finally, we develop and implement a validation algorithm which we later use to obtain Theorems confirming the existence of homoclinic dynamics. This approach, known as the Radii polynomial, is a contraction mapping argument which can be applied to both the parameterized manifold and the Chebyshev arcs. When the Theorem applies, it guarantees the existence of a true solution near the approximation and it provides an upper bound on the C0 norm of the truncation error.
Title:  HOMOCLINIC DYNAMICS IN A SPATIAL RESTRICTED FOUR BODY PROBLEM. 
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Name(s): 
Murray, Maxime, author James, Jason Mireles, Thesis advisor Florida Atlantic University, Degree grantor Department of Mathematical Sciences Charles E. Schmidt College of Science 

Type of Resource:  text  
Genre:  Electronic Thesis Or Dissertation  
Date Created:  2021  
Date Issued:  2021  
Publisher:  Florida Atlantic University  
Place of Publication:  Boca Raton, Fla.  
Physical Form:  application/pdf  
Extent:  137 p.  
Language(s):  English  
Abstract/Description:  The set of transverse homoclinic intersections for a saddlefocus equilibrium in the planar equilateral restricted four body problem admits certain simple homoclinic orbits which form the skeleton of the complete homoclinic intersection, or homoclinic web. In this thesis, the planar restricted four body problem is viewed as an invariant subsystem of the spatial problem, and the influence of this planar homoclinic skeleton on the spatial dynamics is studied from a numerical point of view. Starting from the vertical Lyapunov families emanating from saddle focus equilibria, we compute the stable/unstable manifolds of these spatial periodic orbits and look for intersections between these manifolds near the fundamental planar homoclinics. In this way, we are able to continue all of the basic planar homoclinic motions into the spatial problem as homoclinics for appropriate vertical Lyapunov orbits which, by the Smale Tangle theorem, suggest the existence of chaotic motions in the spatial problem. While the saddlefocus equilibrium solutions in the planar problems occur only at a discrete set of energy levels, the cycletocycle homoclinics in the spatial problem are robust with respect to small changes in energy. The method uses high order FourierTaylor and Chebyshev series approximations in conjunction with the parameterization method, a general functional analytic framework for invariant manifolds. Tools that admit a natural notion of aposteriori error analysis. Finally, we develop and implement a validation algorithm which we later use to obtain Theorems confirming the existence of homoclinic dynamics. This approach, known as the Radii polynomial, is a contraction mapping argument which can be applied to both the parameterized manifold and the Chebyshev arcs. When the Theorem applies, it guarantees the existence of a true solution near the approximation and it provides an upper bound on the C0 norm of the truncation error.  
Identifier:  FA00013758 (IID)  
Degree granted:  Dissertation (Ph.D.)Florida Atlantic University, 2021.  
Collection:  FAU Electronic Theses and Dissertations Collection  
Note(s):  Includes bibliography.  
Subject(s): 
Boundary value problems Invariant manifolds Applied mathematics 

Persistent Link to This Record:  http://purl.flvc.org/fau/fd/FA00013758  
Use and Reproduction:  Copyright © is held by the author with permission granted to Florida Atlantic University to digitize, archive and distribute this item for nonprofit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.  
Use and Reproduction:  http://rightsstatements.org/vocab/InC/1.0/  
Host Institution:  FAU  
Is Part of Series:  Florida Atlantic University Digital Library Collections. 