Limit Cycles in Two Dimensional Systems of Ordinary Differential Equations

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Rowe, A. (2018). Limit Cycles in Two Dimensional Systems of Ordinary Differential Equations. Unc Charlotte Electronic Theses And Dissertations.
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Abstract

This thesis concerns cycles, i.e., topological ovals, in the phase portraits of systems of first order ordinary differential equationsin the plane, with an emphasis on limit cycles, cycles that are isolated from all other cycles. These are of fundamental importance because when asymptotically stable they correspond to limiting periodic behavior in the underlying system of differential equations. We treat basic theorems, with their proofs, concerning existence, non-existence, and unicity of cycles, and culminate with a general theorem guaranteeing existence of an asymptotically stable limit cycle in the phase portrait of systems of first order equations that correspond to differential equations of the form $\ddot x + f(x) \dot x + g(x) = 0$. The thesis includes examples that illustrate the theorems, including the historically important Volterra-Lotka family and the van der Pol oscillator.

Details
Author

Rowe, Ashley

Title

Limit Cycles in Two Dimensional Systems of Ordinary Differential Equations

Physical Description

1 online resource (64 pages) : PDF

Date

2018

Degree Granting Institution

University of North Carolina at Charlotte

Abstract

This thesis concerns cycles, i.e., topological ovals, in the phase portraits of systems of first order ordinary differential equationsin the plane, with an emphasis on limit cycles, cycles that are isolated from all other cycles. These are of fundamental importance because when asymptotically stable they correspond to limiting periodic behavior in the underlying system of differential equations. We treat basic theorems, with their proofs, concerning existence, non-existence, and unicity of cycles, and culminate with a general theorem guaranteeing existence of an asymptotically stable limit cycle in the phase portrait of systems of first order equations that correspond to differential equations of the form $\ddot x + f(x) \dot x + g(x) = 0$. The thesis includes examples that illustrate the theorems, including the historically important Volterra-Lotka family and the van der Pol oscillator.

Genre

masters theses

Subjects--Topics

Mathematics

Degree

M.S.

Keywords

Cycles
Lienard
Limit
Limit Cycles
Ordinary Differential Equations
Van Der Pol

Subject Area

Mathematics

Advisor(s)

Shafer, Douglas

Committee Members

McGoff, Kevin
Vainberg, Boris

Degree Note

Thesis (M.S.)--University of North Carolina at Charlotte, 2018.

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Rights Holder Information

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Identifier

Rowe_uncc_0694N_11938

Permalink

http://hdl.handle.net/20.500.13093/etd:1194