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- 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
- 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
- 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)