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Title: Parameterization of invariant manifolds : the Lorenz manifold
Author: Roma Gimeno, Irene
Director/Tutor: Haro, Àlex
Keywords: Equacions diferencials ordinàries
Treballs de fi de grau
Caos (Teoria de sistemes)
Anàlisi numèrica
Varietats (Matemàtica)
Sistemes dinàmics diferenciables
Ordinary differential equations
Bachelor's thesis
Chaotic behavior in systems
Numerical analysis
Manifolds (Mathematics)
Differentiable dynamical systems
Issue Date: 23-Jun-2016
Abstract: This work is composed of three different parts. First of all, a deep study of the Lorenz equations is done, beginning with its physical deduction, continuing with its dynamical properties and ending with the discussion of three typical properties of chaotic attractors (Volume contraction, Local instability and global stability and how they are illustrated by the Lorenz system. The second part is based on Taylor’s method as a numerical integration method for the Lorenz differential equation system. The order of the expansion and the step size are the parameters to determine in order to have an error below a certain tolerance and a high computational efficiency. The last part is the one which gives the title to this project. Once we have a deep understanding of the dynamical system and a way to integrate it we can proceed to find an approximation for the invariant stable manifold using the parameterization method. A general theorem for the analytic case is first introduced and then the method is adapted to the Lorenz model, and hence obtaining a plot of this manifold.
Note: Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2016, Director: Àlex Haro
Appears in Collections:Treballs Finals de Grau (TFG) - Matemàtiques

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