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FORMATION, EVOLUTION, AND BREAKDOWN OF INVARIANT TORI IN DISSIPATIVE SYSTEMS: FROM VISUALIZATION TO COMPUTER ASSISTED PROOFS
Title
FORMATION, EVOLUTION, AND BREAKDOWN OF INVARIANT TORI IN DISSIPATIVE SYSTEMS: FROM VISUALIZATION TO COMPUTER ASSISTED PROOFS.
Creator
Fleurantin, Emmanuel, Mireles-James, Jason, Florida Atlantic University, Department of Mathematical Sciences, Charles E. Schmidt College of Science
Abstract/Description
The goal of this work is to study smooth invariant sets using high order approximation schemes. Whenever possible, existence of invariant sets are established using computer-assisted proofs. This provides a new set of tools for mathematically rigorous analysis of the invariant objects. The dissertation focuses on application of these tools to a family of three dimensional dissipative vector fields, derived from the normal form of a cusp-Hopf bifurcation. The vector field displays a Neimark...
Show moreThe goal of this work is to study smooth invariant sets using high order approximation schemes. Whenever possible, existence of invariant sets are established using computer-assisted proofs. This provides a new set of tools for mathematically rigorous analysis of the invariant objects. The dissertation focuses on application of these tools to a family of three dimensional dissipative vector fields, derived from the normal form of a cusp-Hopf bifurcation. The vector field displays a Neimark-Sacker bifurcation giving rise to an attracting invariant torus. We examine the torus via parameter continuation from its appearance to its breakdown, scrutinizing its dynamics between these events. We also study the embeddings of the stable/unstable manifolds of the hyperbolic equilibrium solutions over this parameter range. We focus on the role of the invariant manifolds as transport barriers and their participation in global bifurcations. We then study the existence and regularity properties for attracting invariant tori in three dimensional dissipative systems of ordinary differential equations and lay out a constructive method of computer assisted proof which pertains to explicit problems in non-perturbative regimes. We get verifiable lower bounds on the regularity of the attractor in terms of the ratio of the expansion rate on the torus with the contraction rate near the torus. We look at two important cases of rotational and resonant tori. Finally, we study the related problem of approximating two dimensional subcenter manifolds of conservative systems. As an application, we compare two methods for computing the Taylor series expansion of the graph of the subcenter manifold near a saddle-center equilibrium solution of a Hamiltonian system.
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Date Issued
2021
PURL
http://purl.flvc.org/fau/fd/FA00013812
Subject Headings
Invariants, Manifolds (Mathematics), Dynamical systems
Format
Document (PDF)
DIFFEOMORPHISM INVARIANT COSMOLOGICAL SECTOR IN LOOP QUANTUM GRAVITY
Title
DIFFEOMORPHISM INVARIANT COSMOLOGICAL SECTOR IN LOOP QUANTUM GRAVITY.
Creator
Mendonca, Phillip, Engle, Jonathan, Florida Atlantic University, Charles E. Schmidt College of Science, Department of Physics
Abstract/Description
1In this dissertation we work out in detail a new proposal to define rigorously a sector of loop quantum gravity at the diffeomorphism invariant level corresponding to homogeneous and isotropic cosmologies, and propose how to compare in detail the physics of this sector with that of loop quantum cosmology. The key technical steps we have completed are (a) to formulate conditions for homogeneity and isotropy in a diffeomorphism covariant way on the classical phase space of general relativity,...
Show more1In this dissertation we work out in detail a new proposal to define rigorously a sector of loop quantum gravity at the diffeomorphism invariant level corresponding to homogeneous and isotropic cosmologies, and propose how to compare in detail the physics of this sector with that of loop quantum cosmology. The key technical steps we have completed are (a) to formulate conditions for homogeneity and isotropy in a diffeomorphism covariant way on the classical phase space of general relativity, and (b) to translate these conditions consistently using well-understood techniques to loop quantum gravity. To impose the symmetry at the quantum level, on both the connection and its conjugate momentum, the method used necessarily has similiarities to the Gupta-Bleuler method of quantizing the electromagnetic field. Lastly, a strategy for embedding states of loop quantum cosmology into this new homogeneous isotropic sector, and using this embedding to compare the physics, is presented.
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Date Issued
2019
PURL
http://purl.flvc.org/fau/fd/FA00013334
Subject Headings
Diffeomorphisms, Quantum gravity, Quantum cosmology, Invariants, Isotropy
Format
Document (PDF)
HOMOCLINIC DYNAMICS IN A SPATIAL RESTRICTED FOUR BODY PROBLEM
Title
HOMOCLINIC DYNAMICS IN A SPATIAL RESTRICTED FOUR BODY PROBLEM.
Creator
Murray, Maxime, James, Jason Mireles, Florida Atlantic University, Department of Mathematical Sciences, Charles E. Schmidt College of Science
Abstract/Description
The set of transverse homoclinic intersections for a saddle-focus 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....
Show moreThe set of transverse homoclinic intersections for a saddle-focus 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 saddle-focus equilibrium solutions in the planar problems occur only at a discrete set of energy levels, the cycle-to-cycle homoclinics in the spatial problem are robust with respect to small changes in energy. The method uses high order Fourier-Taylor 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 a-posteriori 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.
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Date Issued
2021
PURL
http://purl.flvc.org/fau/fd/FA00013758
Subject Headings
Boundary value problems, Invariant manifolds, Applied mathematics
Format
Document (PDF)
visual rotation invariant in fixated motion
Title
A visual rotation invariant in fixated motion.
Creator
Ozery, Nissim Jossef., Florida Atlantic University, Raviv, Daniel, College of Engineering and Computer Science, Department of Computer and Electrical Engineering and Computer Science
Abstract/Description
This thesis studies the 2-D-based visual invariant that exists during relative motion between a camera and a 3-D object. We show that during fixation there is a measurable nonlinear function of optical flow that produces the same value for all points of a stationary environment regardless of the 3-D shape of the environment. During fixated camera motion relative to a rigid object, e.g., a stationary environment, the projection of the fixated point remains (by definition) at the same location...
Show moreThis thesis studies the 2-D-based visual invariant that exists during relative motion between a camera and a 3-D object. We show that during fixation there is a measurable nonlinear function of optical flow that produces the same value for all points of a stationary environment regardless of the 3-D shape of the environment. During fixated camera motion relative to a rigid object, e.g., a stationary environment, the projection of the fixated point remains (by definition) at the same location in the image, and all other points located on the 3-D rigid object can only rotate relative to that 3-D fixation point. This rotation rate of the points is invariant for all points that lie on the particular environment, and it is measurable from a sequence of images. This new invariant is obtained from a set of monocular images and is expressed explicitly as a closed form solution.
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Date Issued
1994
PURL
http://purl.flvc.org/fcla/dt/15095
Subject Headings
Invariants, Visual perception, Motion perception (Vision)
Format
Document (PDF)
The Circular Restricted Four Body Problem is Non-Integrable: A Computer Assisted Proof
Title
The Circular Restricted Four Body Problem is Non-Integrable: A Computer Assisted Proof.
Creator
Kepley, Shane, Kalies, William D., Mireles-James, Jason D., Florida Atlantic University, Charles E. Schmidt College of Science, Department of Mathematical Sciences
Abstract/Description
Gravitational N-body problems are central in classical mathematical physics. Studying their long time behavior raises subtle questions about the interplay between regular and irregular motions and the boundary between integrable and chaotic dynamics. Over the last hundred years, concepts from the qualitative theory of dynamical systems such as stable/unstable manifolds, homoclinic and heteroclinic tangles, KAM theory, and whiskered invariant tori, have come to play an increasingly important...
Show moreGravitational N-body problems are central in classical mathematical physics. Studying their long time behavior raises subtle questions about the interplay between regular and irregular motions and the boundary between integrable and chaotic dynamics. Over the last hundred years, concepts from the qualitative theory of dynamical systems such as stable/unstable manifolds, homoclinic and heteroclinic tangles, KAM theory, and whiskered invariant tori, have come to play an increasingly important role in the discussion. In the last fty years the study of numerical methods for computing invariant objects has matured into a thriving sub-discipline. This growth is driven at least in part by the needs of the world's space programs. Recent work on validated numerical methods has begun to unify the computational and analytical perspectives, enriching both aspects of the subject. Many of these results use computer assisted proofs, a tool which has become increasingly popular in recent years. This thesis presents a proof that the circular restricted four body problem is non-integrable. The proof of this result is obtained as an application of more general rigorous numerical methods in nonlinear analysis.
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Date Issued
2017
PURL
http://purl.flvc.org/fau/fd/FA00004997
Subject Headings
Dissertations, Academic -- Florida Atlantic University, Mathematical physics., Invariants., Dynamical systems
Format
Document (PDF)
ACCURATE HIGH ORDER COMPUTATION OF INVARIANT MANIFOLDS FOR LONG PERIODIC ORBITS OF MAPS AND EQUILIBRIUM STATES OF PDE
Title
ACCURATE HIGH ORDER COMPUTATION OF INVARIANT MANIFOLDS FOR LONG PERIODIC ORBITS OF MAPS AND EQUILIBRIUM STATES OF PDE.
Creator
Gonzalez, Jorge L., Mireles-James, Jason, Florida Atlantic University, Department of Mathematical Sciences, Charles E. Schmidt College of Science
Abstract/Description
The study of the long time behavior of nonlinear systems is not effortless, but it is very rewarding. The computation of invariant objects, in particular manifolds provide the scientist with the ability to make predictions at the frontiers of science. However, due to the presence of strong nonlinearities in many important applications, understanding the propagation of errors becomes necessary in order to quantify the reliability of these predictions, and to build sound foundations for future...
Show moreThe study of the long time behavior of nonlinear systems is not effortless, but it is very rewarding. The computation of invariant objects, in particular manifolds provide the scientist with the ability to make predictions at the frontiers of science. However, due to the presence of strong nonlinearities in many important applications, understanding the propagation of errors becomes necessary in order to quantify the reliability of these predictions, and to build sound foundations for future discoveries. This dissertation develops methods for the accurate computation of high-order polynomial approximations of stable/unstable manifolds attached to long periodic orbits in discrete time dynamical systems. For this purpose a multiple shooting scheme is applied to invariance equations for the manifolds obtained using the Parameterization Method developed by Xavier Cabre, Ernest Fontich and Rafael De La Llave in [CFdlL03a, CFdlL03b, CFdlL05].
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Date Issued
2020
PURL
http://purl.flvc.org/fau/fd/FA00013468
Subject Headings
Invariant manifolds, Nonlinear systems, Diffeomorphisms, Parabolic partial differential equations, Differential equations, Partial
Format
Document (PDF)