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- Title
- THE SITNIKOV PROBLEM, LOW ENERGY TRANSFERS, AND THE ECONOMIC FEASIBILITY OF ASTEROID MINING.
- Creator
- Blanco, Dominic, Mireles-James, Jason, Harriet L. Wilkes Honors College, Florida Atlantic University
- Abstract/Description
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Asteroid mining can be profitable; however, it is currently not economically feasible. Space companies have reduced the cost of missions by using low energy transfer. Low energy transfer uses connecting orbits requiring much less energy to move a spacecraft. To demonstrate low energy transfer, I investigate the Sitnikov Problem with eccentricity of 0.2 and 0.9. The Sitnikov Problem is a form of the gravitational three-body problem with two heavy bodies orbiting in a plane while a light third...
Show moreAsteroid mining can be profitable; however, it is currently not economically feasible. Space companies have reduced the cost of missions by using low energy transfer. Low energy transfer uses connecting orbits requiring much less energy to move a spacecraft. To demonstrate low energy transfer, I investigate the Sitnikov Problem with eccentricity of 0.2 and 0.9. The Sitnikov Problem is a form of the gravitational three-body problem with two heavy bodies orbiting in a plane while a light third body moves perpendicular to the plane. I compute the Poincaré map and find connecting orbits. I then compare past missions that used low energy transfer to similar missions which did not. In all cases, using low energy transfer lowered the cost. This shows that we should investigate the use of low energy transfer in asteroid mining missions to reduce cost.
Show less - Date Issued
- 2022
- PURL
- http://purl.flvc.org/fau/fd/FAUHT00188
- Format
- Document (PDF)
- 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
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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.
Show less - Date Issued
- 2021
- PURL
- http://purl.flvc.org/fau/fd/FA00013758
- Subject Headings
- Boundary value problems, Invariant manifolds, Applied mathematics
- Format
- Document (PDF)
- 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
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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].
Show less - 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)
- Title
- On the Study of the Aizawa System.
- Creator
- Fleurantin, Emmanuel, Mireles-James, Jason D., Florida Atlantic University, Charles E. Schmidt College of Science, Department of Mathematical Sciences
- Abstract/Description
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In this report we study the Aizawa field by first computing a Taylor series expansion for the solution of an initial value problem. We then look for singularities (equilibrium points) of the field and plot the set of solutions which lie in the linear subspace spanned by the eigenvectors. Finally, we use the Parameterization Method to compute one and two dimensional stable and unstable manifolds of equilibria for the system.
- Date Issued
- 2018
- PURL
- http://purl.flvc.org/fau/fd/FA00005994
- Subject Headings
- Series, Mathematics, Eigenvectors, Aizawa field
- Format
- Document (PDF)
- 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
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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.
Show less - 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)
- 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
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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.
Show less - Date Issued
- 2021
- PURL
- http://purl.flvc.org/fau/fd/FA00013812
- Subject Headings
- Invariants, Manifolds (Mathematics), Dynamical systems
- Format
- Document (PDF)
- Title
- SOLVING V.I. ARNOLD’S PROBLEM ABOUT ASYMPTOTIC ENUMERATION OF MORSE FUNCTIONS ON THE 2-SPHERE: A COMBINATORIAL AND ANALYTIC APPROACH WITH COMPUTER ASSISTED PROOFS.
- Creator
- Dhakal, Bishal, Mireles-James, Jason, Florida Atlantic University, Department of Mathematical Sciences, Charles E. Schmidt College of Science
- Abstract/Description
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The goal of this dissertation is to estimate the precise asymptotics for the number of geometric equivalence classes of Morse functions on the 2-sphere. Our approach involves utilizing the Lagrange inversion formula, Cauchy’s coefficient formula, and the saddle point method for the asymptotic analysis of contour integrals to analyze the generating function derived by L. Nicolaescu, expressed as the inverse of an elliptic integral. We utilize complex analysis, nonlinear functional analysis in...
Show moreThe goal of this dissertation is to estimate the precise asymptotics for the number of geometric equivalence classes of Morse functions on the 2-sphere. Our approach involves utilizing the Lagrange inversion formula, Cauchy’s coefficient formula, and the saddle point method for the asymptotic analysis of contour integrals to analyze the generating function derived by L. Nicolaescu, expressed as the inverse of an elliptic integral. We utilize complex analysis, nonlinear functional analysis in infinite sequence spaces, and interval arithmetic to write all the necessary MATLAB programs that validate our results. This work answers questions posed by Arnold and Nicolaescu, furthering our understanding of the topological properties of Morse functions on two-dimensional manifolds. It also demonstrates the effectiveness of a computer assisted approach for asymptotic analysis.
Show less - Date Issued
- 2023
- PURL
- http://purl.flvc.org/fau/fd/FA00014264
- Subject Headings
- Manifolds (Mathematics), Morse theory, Combinatorial analysis
- Format
- Document (PDF)
- Title
- IDENTIFIABILITY ANALYSIS AND OPTIMAL CONTROL OF INFECTIOUS DISEASES EPIDEMICS AND PARAMETERIZATION METHOD FOR (UN)STABLE MANIFOLDS OF IMPLICITLY DEFINED DYNAMICAL SYSTEMS.
- Creator
- Neupane Timsina, Archana, Tuncer, Necibe, Mireles James, Jason D., Florida Atlantic University, Department of Mathematical Sciences, Charles E. Schmidt College of Science
- Abstract/Description
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This dissertation is a study about applied dynamical systems on two concentrations. First, on the basis of the growing association between opioid addiction and HIV infection, a compartmental model is developed to study dynamics and optimal control of two epidemics; opioid addiction and HIV infection. We show that the disease-free-equilibrium is locally asymptotically stable when the basic reproduction number R0 = max(Ru0; Rv0) 1 and it is locally asymptotically stable when the invasion...
Show moreThis dissertation is a study about applied dynamical systems on two concentrations. First, on the basis of the growing association between opioid addiction and HIV infection, a compartmental model is developed to study dynamics and optimal control of two epidemics; opioid addiction and HIV infection. We show that the disease-free-equilibrium is locally asymptotically stable when the basic reproduction number R0 = max(Ru0; Rv0) < 1; here Rv0 is the reproduction number of the HIV infection, and Ru0 is the reproduction number of the opioid addiction. The addiction-only boundary equilibrium exists when Ru0 > 1 and it is locally asymptotically stable when the invasion number of the opioid addiction is Ruinv < 1: Similarly, HIV-only boundary equilibrium exists when Rv0 > 1 and it is locally asymptotically stable when the invasion number of the HIV infection is Rvinv < 1. We study structural identifiability of the parameters, estimate parameters employing yearly reported data from Central for Disease Control and Prevention (CDC), and study practical identifiability of estimated parameters. We observe the basic reproduction number R0 using the parameters. Next, we introduce four distinct controls in the model for the sake of control approach, including treatment for addictions, health care education about not sharing syringes, highly active anti-retroviral therapy (HAART), and rehab treatment for opiate addicts who are HIV infected. US population using CDC data, first applying a single control in the model and observing the results, we better understand the influence of individual control. After completing each of the four applications, we apply them together at the same time in the model and compare the outcomes using different control bounds and state variable weights. We conclude the results by presenting several graphs.
Show less - Date Issued
- 2022
- PURL
- http://purl.flvc.org/fau/fd/FA00013970
- Subject Headings
- Dynamical systems, Infectious diseases, Parameter estimation
- Format
- Document (PDF)
