The center and cyclicity problems in a family of three dimensional polynomial systems of ordinary differential equations.

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Hounkanli, K. (2013). The center and cyclicity problems in a family of three dimensional polynomial systems of ordinary differential equations. Unc Charlotte Electronic Theses And Dissertations.
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Abstract

This dissertation is mainly a study of the center problem in the context of a family of three dimensional systems of ordinary differential equations of the form&middot {u} = &minus v + P(u,v,w), &middot {v} = u+Q(u,v,w), &middot {w}=&minus &lambda &times w+R(u,v,w),for which the right-hand sides are polynomials and &lambda is nonzero. Such systems are called polynomial systems. There is a two dimensional local center manifold through the origin. It is invariant under the flow. The problem is to decide whether there is a focus or a center at the origin for the flow restricted to the local center manifold. We first generalize ideas and methods used to study the center and cyclicity problems in the two-dimensional setting to the three-dimensional context. This will involve generalizing to this setting the concepts of the complexification of real systems, normal forms and the center variety, described for two-dimensional systems by Valery G. Romanovski and Douglas S. Shafer in the Center and Cyclicity Problems: A Computational Algebra Approach. We then apply our results to solve the center and cyclicity problems for the Moon-Rand family of systems that arise naturally in an engineering context.

Details
Author

Hounkanli, Kokouvi

Title

The center and cyclicity problems in a family of three dimensional polynomial systems of ordinary differential equations.

Physical Description

1 online resource (83 pages) : PDF

Date

2013

Degree Granting Institution

University of North Carolina at Charlotte

Abstract

This dissertation is mainly a study of the center problem in the context of a family of three dimensional systems of ordinary differential equations of the form&middot {u} = &minus v + P(u,v,w), &middot {v} = u+Q(u,v,w), &middot {w}=&minus &lambda &times w+R(u,v,w),for which the right-hand sides are polynomials and &lambda is nonzero. Such systems are called polynomial systems. There is a two dimensional local center manifold through the origin. It is invariant under the flow. The problem is to decide whether there is a focus or a center at the origin for the flow restricted to the local center manifold. We first generalize ideas and methods used to study the center and cyclicity problems in the two-dimensional setting to the three-dimensional context. This will involve generalizing to this setting the concepts of the complexification of real systems, normal forms and the center variety, described for two-dimensional systems by Valery G. Romanovski and Douglas S. Shafer in the Center and Cyclicity Problems: A Computational Algebra Approach. We then apply our results to solve the center and cyclicity problems for the Moon-Rand family of systems that arise naturally in an engineering context.

Genre

doctoral dissertations

Subjects--Topics

Mathematics

Degree

Ph.D.

Keywords

Center and Cyclicity Problems
Dynamical Systems
Moon-Rand Systems

Subject Area

Applied Mathematics

Advisor(s)

Shafer, Douglas

Committee Members

Avrin, Joel
Lucas, Thomas
Lu, Aidong

Degree Note

Thesis (Ph.D.)--University of North Carolina at Charlotte, 2013.

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Rights Holder Information

Copyright is held by the author unless otherwise indicated.

Identifier

Hounkanli_uncc_0694D_10527

Permalink

http://hdl.handle.net/20.500.13093/etd:443