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Título:
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Stability of R 3-dynamical systems with symmetry.
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Autores:
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Gonzalez Gascón, F. ;
Romero Ruiz del Portal, Francisco
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Tipo de documento:
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texto impreso
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Editorial:
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Società Italiana di Fisica, 1999
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Palabras clave:
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Estado = Publicado
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Materia = Ciencias: Matemáticas: Topología
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Tipo = Artículo
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Resumen:
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The study of the stability of a periodic solution p of a vector field using either the linear variational equations (associated to the vector field at p ), or the Poincaré map on a cross section, is known to present some difficulties. This work provides some techniques to ascertain the stability of the closed curve C={p 0 (t): t?R} in the case of an R 3 analytic vector field X ? possessing symmetries. It is assumed that one or more symmetry vectors S ? are known (the Lie derivative of S ? along the streamlines of X ? , L X ? (S ? ) , is zero modulus X ? ). One of the cases for which the stability of the closed curve can be determined is that of a divergence-free field X ? having a known symmetry S ? satisfying L X ? (S ? )=?(x)X ? and divS ? =?(x) . This is an interesting case because many devices used in the confinement of plasma possess symmetries of this type (X ? is the magnetic induction vector B ? ) with ?(x)=0 . This type of symmetry implies torus-like magnetic surfaces. It is noted that it constitutes an interesting (and difficult) problem to find examples of vector fields with symmetries for which ??0 . All the proofs are simple, and the technique is very nice.
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