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Title: The Center-Focus and Ciclicity Problems: An Implementation of the Lyapunov Method and the Interpolation Technique
Author: Sánchez Sánchez, Iván
Director: Jordi Villadelprat Yagüe
Joan Torregrosa Arús
Others: Universitat Oberta de Catalunya
Keywords: polynomial differential equations
center characterization
Lyapunov quantities
Issue Date: Sep-2017
Publisher: Universitat Oberta de Catalunya
Abstract: The 16th Hilbert problem aims to determine the maximum number of isolated periodic solutions which a system of polynomial differential equations in the plane has. A first approach to this are the center-focus and ciclicity problems, which consist in identifying whether the origin of a system is a center or a focus and determining the maximum number of limit cycles, respectively. Here, we aim to study these problems for polynomial differential equations and this means to analyze the stability in a neighbourhood of a monodromic non-degenerate point. An essential mathematical object to deal with these issues are the Lyapunov quantities, which determine whether the origin is a center or a focus and its stability. In this project, we introduce two procedures to find these quantities: the Lyapunov method and an interpolation technique. Both algorithms are computationally implemented, and the differences between both methods are discussed. A parallelization example of the interpolation technique is also introduced, which shows the necessity of considering a parallel approach for the computation of the Lyapunov quantities in terms of efficiency and computational speed. Finally, the obtained code is used to analyze the center-focus and ciclicity problems for some particular polynomial differential equations: a cubic family, a quadratic family and a polyomial family with homogeneous nonlinearities example.
Language: English
Appears in Collections:Bachelor thesis, research projects, etc.

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