Current Search: Fleurantin, Emmanuel (x)
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Title
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On the Study of the Aizawa System.
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Creator
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Fleurantin, Emmanuel, Mireles-James, Jason D., Florida Atlantic University, Charles E. Schmidt College of Science, Department of Mathematical Sciences
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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.
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Date Issued
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2018
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PURL
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http://purl.flvc.org/fau/fd/FA00005994
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Subject Headings
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Series, Mathematics, Eigenvectors, Aizawa field
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Format
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Document (PDF)
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Title
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FORMATION, EVOLUTION, AND BREAKDOWN OF INVARIANT TORI IN DISSIPATIVE SYSTEMS: FROM VISUALIZATION TO COMPUTER ASSISTED PROOFS.
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Creator
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Fleurantin, Emmanuel, Mireles-James, Jason, Florida Atlantic University, Department of Mathematical Sciences, Charles E. Schmidt College of Science
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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.
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Date Issued
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2021
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PURL
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http://purl.flvc.org/fau/fd/FA00013812
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Subject Headings
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Invariants, Manifolds (Mathematics), Dynamical systems
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Format
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Document (PDF)