Spectral view of Hartman-Grobman’s theorem for nonuniform and unbounded hyperbolic flows.
In this thesis we will consider a nonautonomous linear system which admits nonuniform contraction and a nonlinear perturbation bounded at the origin. We search to establish an equivalence between the solutions of the systems before mentioned, being our main objective to construct a topological equiv...
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| Tipo de recurso: | tesis doctoral |
| Estado: | Versión publicada |
| Fecha de publicación: | 2019 |
| País: | Chile |
| OAI Identifier: | oai:repositorio.anid.cl:10533/252989 |
| Acceso en línea: | https://hdl.handle.net/10533/252989 |
| Access Level: | acceso abierto |
| Palabra clave: | Ciencias Naturales Matemáticas Matemáticas Puras |
| Sumario: | In this thesis we will consider a nonautonomous linear system which admits nonuniform contraction and a nonlinear perturbation bounded at the origin. We search to establish an equivalence between the solutions of the systems before mentioned, being our main objective to construct a topological equivalence between them. In order to obtain such a result we review the spectral theory associated with the nonuniform hyperbolicity, specifically we consider nonuniform exponential dichotomy of the linear system. In addition we highlight some fundamental results of the spectral theory such as: i) under certain hypotheses the nonuniform spectrum of the linear system can be written as the finite union of compact intervals, ii) the linear system is equivalent by means of a nonuniform kinematic similarity to a new linear system composed of blocks, where the spectrum of each of these blocks corresponds to one of the connected components of the original linear system spectrum. On the other hand and thanks to the aforementioned spectral theory, we will show that the initial linear system is nonuniformly contracted to its spectrum when it is nonuniformly kinemically similar to a linear system composed of the sum of one diagonal matrix, where its elements are functions whose images belong to the spectrum, and a matrix whose norm can be chosen sufficiently small. We will call this property almost nonuniform reducibility. Finally, if the linear system admits nonuniform contraction, we use the previous results combined with Lyapunov's theory of functions to establish the existence of homeomorphism that relates the solutions of both systems. |
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