Publication: Topología y dinámica de conjuntos non-saddle e índice de Conley en variedades
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2018-11-12
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Universidad Complutense de Madrid
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Abstract
Esta tesis está dedicada al estudio, utilizando técnicas topológicas, de la estructura cualitativa de un flujo cerca de un compacto invariante. Este fue uno delos primeros temas clásicos desarrollados por H. Poincaré, I. Bendixson, A. Andronovy S. Lefschetz al inicio de la teoría cualitativa de ecuaciones diferenciales,con contribuciones de autores como D.M. Grobman, P. Hartman, J.K. Hale y A.Stokes y muchos otros. Cabe destacar, por ejemplo, la descripción presentadapor T. Ura e I. Kimura en [97] o las teorías de índice de Wazewski y Conley que encapsulan, en un sentido topológico, propiedades del flujo cerca de un compacto invariante (aislado).La importancia del estudio de la estructura de un flujo cerca de un compacto invariante queda patente en las palabras de J. Auslander, N.P. Bhatia y P. Seibert[5] recogidas en el célebre artículo sobre el concepto de un atractor [60] de J.Milnor:“En el estudio de las propiedades topológicas de las ecuaciones diferenciales ordinarias, la teoría de estabilidad de compactos invariantes (que podrían ser considerados generalizaciones de puntos críticos y ciclos límite) juega un papel central”...
This dissertation is devoted to the study, using topological techniques, of thequalitative structure of a flow near a compact invariant set. This was one of thefirst classical subjects dealt with by H. Poincaré, I. Bendixson, A. Andronov andS. Lefschetz at the beginning of the qualitative theory of differential equations,with further contributions by authors such as D.M. Grobman, P. Hartman, J.K.Hale and A. Stokes and many others. We highlight, for instance, the descriptionprovided by T. Ura and I. Kimura in [97] or the index theories by Wazewski andConley which encapsulate, in a topological way, local properties of the flow near(isolated) invariant sets.The importance of studying the structure of a flow near a compact invariantset is evident in the words of J. Auslander, N.P. Bhatia and P. Seibert [5] includedin the celebrated paper on the concept of an attractor [60] by J. Milnor:“In the study of topological properties of ordinary differential equations, thestability theory of compact invariant sets (which may be regarded as generalizationsof critical points and limit cycles) plays a central role”...
This dissertation is devoted to the study, using topological techniques, of thequalitative structure of a flow near a compact invariant set. This was one of thefirst classical subjects dealt with by H. Poincaré, I. Bendixson, A. Andronov andS. Lefschetz at the beginning of the qualitative theory of differential equations,with further contributions by authors such as D.M. Grobman, P. Hartman, J.K.Hale and A. Stokes and many others. We highlight, for instance, the descriptionprovided by T. Ura and I. Kimura in [97] or the index theories by Wazewski andConley which encapsulate, in a topological way, local properties of the flow near(isolated) invariant sets.The importance of studying the structure of a flow near a compact invariantset is evident in the words of J. Auslander, N.P. Bhatia and P. Seibert [5] includedin the celebrated paper on the concept of an attractor [60] by J. Milnor:“In the study of topological properties of ordinary differential equations, thestability theory of compact invariant sets (which may be regarded as generalizationsof critical points and limit cycles) plays a central role”...
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Tesis de la Universidad Complutense de Madrid, Facultad de Ciencias Matemáticas, Departamento de Geometría y Topología, leída el 01-12-2017